Theorem pslem 8647

Description: Lemma for psref 8649 and others.

Assertion

Ref	Expression
pslem	|- (R e. Poset -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))

Proof of Theorem pslem

Step	Hyp	Ref	Expression
1		isps 8645	. . 3 |- (R e. Poset -> (R e. Poset <-> (Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R))))
2	1	ibi 592	. 2 |- (R e. Poset -> (Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R)))
3		breq12 2624	. . . . . . . . . 10 |- ((x = A /\ y = B) -> (xRy <-> ARB))
4	3	3adant3 799	. . . . . . . . 9 |- ((x = A /\ y = B /\ z = C) -> (xRy <-> ARB))
5		breq12 2624	. . . . . . . . . 10 |- ((y = B /\ z = C) -> (yRz <-> BRC))
6	5	3adant1 797	. . . . . . . . 9 |- ((x = A /\ y = B /\ z = C) -> (yRz <-> BRC))
7	4, 6	anbi12d 628	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> ((xRy /\ yRz) <-> (ARB /\ BRC)))
8		breq12 2624	. . . . . . . . 9 |- ((x = A /\ z = C) -> (xRz <-> ARC))
9	8	3adant2 798	. . . . . . . 8 |- ((x = A /\ y = B /\ z = C) -> (xRz <-> ARC))
10	7, 9	imbi12d 626	. . . . . . 7 |- ((x = A /\ y = B /\ z = C) -> (((xRy /\ yRz) -> xRz) <-> ((ARB /\ BRC) -> ARC)))
11	10	cla43gv 1867	. . . . . 6 |- ((A e. S /\ B e. T /\ C e. U) -> (A.xA.yA.z((xRy /\ yRz) -> xRz) -> ((ARB /\ BRC) -> ARC)))
12		breq12 2624	. . . . . . . . . . 11 |- ((x = A /\ x = A) -> (xRx <-> ARA))
13	12	anidms 434	. . . . . . . . . 10 |- (x = A -> (xRx <-> ARA))
14	13	rcla4cv 1874	. . . . . . . . 9 |- (A.x e. U.U.RxRx -> (A e. U.U.R -> ARA))
15	14	a1i 8	. . . . . . . 8 |- ((A e. S /\ B e. T) -> (A.x e. U.U.RxRx -> (A e. U.U.R -> ARA)))
16		breq12 2624	. . . . . . . . . . . 12 |- ((y = B /\ x = A) -> (yRx <-> BRA))
17	16	ancoms 436	. . . . . . . . . . 11 |- ((x = A /\ y = B) -> (yRx <-> BRA))
18	3, 17	anbi12d 628	. . . . . . . . . 10 |- ((x = A /\ y = B) -> ((xRy /\ yRx) <-> (ARB /\ BRA)))
19		eqeq12 1487	. . . . . . . . . 10 |- ((x = A /\ y = B) -> (x = y <-> A = B))
20	18, 19	imbi12d 626	. . . . . . . . 9 |- ((x = A /\ y = B) -> (((xRy /\ yRx) -> x = y) <-> ((ARB /\ BRA) -> A = B)))
21	20	cla42gv 1865	. . . . . . . 8 |- ((A e. S /\ B e. T) -> (A.xA.y((xRy /\ yRx) -> x = y) -> ((ARB /\ BRA) -> A = B)))
22	15, 21	anim12d 558	. . . . . . 7 |- ((A e. S /\ B e. T) -> ((A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)) -> ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B))))
23	22	3adant3 799	. . . . . 6 |- ((A e. S /\ B e. T /\ C e. U) -> ((A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)) -> ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B))))
24	11, 23	anim12d 558	. . . . 5 |- ((A e. S /\ B e. T /\ C e. U) -> ((A.xA.yA.z((xRy /\ yRz) -> xRz) /\ (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y))) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
25	24	com12 11	. . . 4 |- ((A.xA.yA.z((xRy /\ yRz) -> xRz) /\ (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y))) -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
26		cotr 3436	. . . 4 |- ((R o. R) (_ R <-> A.xA.yA.z((xRy /\ yRz) -> xRz))
27		asymref2 3440	. . . 4 |- ((R i^i `'R) = (I |` U.U.R) <-> (A.x e. U.U.RxRx /\ A.xA.y((xRy /\ yRx) -> x = y)))
28	25, 26, 27	syl2anb 455	. . 3 |- (((R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R)) -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
29	28	3adant1 797	. 2 |- ((Rel R /\ (R o. R) (_ R /\ (R i^i `'R) = (I |` U.U.R)) -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))
30	2, 29	syl 10	1 |- (R e. Poset -> ((A e. S /\ B e. T /\ C e. U) -> (((ARB /\ BRC) -> ARC) /\ ((A e. U.U.R -> ARA) /\ ((ARB /\ BRA) -> A = B)))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775  A.wal 954   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047  U.cuni 2503   class class class wbr 2619  Icid 2831  `'ccnv 3169   |` cres 3172   o. ccom 3174  Rel wrel 3175  Posetcps 8633

This theorem is referenced by:  psdmrn 8648  psref 8649  psasym 8651  pstr 8652

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ps 8639

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