Theorem oacl 4160

Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57.

Assertion

Ref	Expression
oacl	|- ((A e. On /\ B e. On) -> (A +o B) e. On)

Proof of Theorem oacl

Step	Hyp	Ref	Expression
1		opreq2 3960	. . . 4 |- (x = (/) -> (A +o x) = (A +o (/)))
2	1	eleq1d 1537	. . 3 |- (x = (/) -> ((A +o x) e. On <-> (A +o (/)) e. On))
3		opreq2 3960	. . . 4 |- (x = y -> (A +o x) = (A +o y))
4	3	eleq1d 1537	. . 3 |- (x = y -> ((A +o x) e. On <-> (A +o y) e. On))
5		opreq2 3960	. . . 4 |- (x = suc y -> (A +o x) = (A +o suc y))
6	5	eleq1d 1537	. . 3 |- (x = suc y -> ((A +o x) e. On <-> (A +o suc y) e. On))
7		opreq2 3960	. . . 4 |- (x = B -> (A +o x) = (A +o B))
8	7	eleq1d 1537	. . 3 |- (x = B -> ((A +o x) e. On <-> (A +o B) e. On))
9		oa0 4145	. . . . 5 |- (A e. On -> (A +o (/)) = A)
10	9	eleq1d 1537	. . . 4 |- (A e. On -> ((A +o (/)) e. On <-> A e. On))
11	10	ibir 592	. . 3 |- (A e. On -> (A +o (/)) e. On)
12		oasuc 4153	. . . . . 6 |- ((A e. On /\ y e. On) -> (A +o suc y) = suc (A +o y))
13	12	eleq1d 1537	. . . . 5 |- ((A e. On /\ y e. On) -> ((A +o suc y) e. On <-> suc (A +o y) e. On))
14		suceloni 3057	. . . . 5 |- ((A +o y) e. On -> suc (A +o y) e. On)
15	13, 14	syl5bir 210	. . . 4 |- ((A e. On /\ y e. On) -> ((A +o y) e. On -> (A +o suc y) e. On))
16	15	expcom 374	. . 3 |- (y e. On -> (A e. On -> ((A +o y) e. On -> (A +o suc y) e. On)))
17		visset 1809	. . . . . . 7 |- x e. V
18		oalim 4157	. . . . . . 7 |- ((A e. On /\ (x e. V /\ Lim x)) -> (A +o x) = U_y e. x (A +o y))
19	17, 18	mpanr1 708	. . . . . 6 |- ((A e. On /\ Lim x) -> (A +o x) = U_y e. x (A +o y))
20	19	eleq1d 1537	. . . . 5 |- ((A e. On /\ Lim x) -> ((A +o x) e. On <-> U_y e. x (A +o y) e. On))
21		oprex 3974	. . . . . 6 |- (A +o y) e. V
22	17, 21	iunon 3900	. . . . 5 |- (A.y e. x (A +o y) e. On -> U_y e. x (A +o y) e. On)
23	20, 22	syl5bir 210	. . . 4 |- ((A e. On /\ Lim x) -> (A.y e. x (A +o y) e. On -> (A +o x) e. On))
24	23	expcom 374	. . 3 |- (Lim x -> (A e. On -> (A.y e. x (A +o y) e. On -> (A +o x) e. On)))
25	2, 4, 6, 8, 11, 16, 24	tfinds3 3161	. 2 |- (B e. On -> (A e. On -> (A +o B) e. On))
26	25	impcom 351	1 |- ((A e. On /\ B e. On) -> (A +o B) e. On)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  Vcvv 1807  (/)c0 2276  U_ciun 2561  Oncon0 2943  Lim wlim 2944  suc csuc 2945  (class class class)co 3954   +o coa 4120

This theorem is referenced by:  omcl 4161  oaord 4171  oacan 4172  oaword 4173  oawordri 4174  oawordeulem 4178  oalimcl 4184  oaass 4185  odi 4200  oancom 4613

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-oadd 4125

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