Theorem oelim 4169

Description: Ordinal exponentiation with a limit exponent and nonzero mantissa. Definition 8.30 of [TakeutiZaring] p. 67.

Assertion

Ref	Expression
oelim	|- (((A e. On /\ (B e. C /\ Lim B)) /\ (/) e. A) -> (A ^o B) = U_x e. B (A ^o x))

Distinct variable groups:   x,A   x,B

Proof of Theorem oelim

Step	Hyp	Ref	Expression
1		rdglim2a 3950	. . . 4 |- ((B e. On /\ Lim B) -> (rec({<.y, z>. | z = (y .o A)}, 1o)` B) = U_x e. B (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
2	1	ad2antlr 405	. . 3 |- (((A e. On /\ (B e. On /\ Lim B)) /\ (/) e. A) -> (rec({<.y, z>. | z = (y .o A)}, 1o)` B) = U_x e. B (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
3		oevn0 4154	. . . . 5 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = (rec({<.y, z>. | z = (y .o A)}, 1o)` B))
4		oevn0 4154	. . . . . . . . . 10 |- (((A e. On /\ x e. On) /\ (/) e. A) -> (A ^o x) = (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
5		onelon 2972	. . . . . . . . . 10 |- ((B e. On /\ x e. B) -> x e. On)
6	4, 5	sylanl2 461	. . . . . . . . 9 |- (((A e. On /\ (B e. On /\ x e. B)) /\ (/) e. A) -> (A ^o x) = (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
7	6	exp42 383	. . . . . . . 8 |- (A e. On -> (B e. On -> (x e. B -> ((/) e. A -> (A ^o x) = (rec({<.y, z>. | z = (y .o A)}, 1o)` x)))))
8	7	com34 36	. . . . . . 7 |- (A e. On -> (B e. On -> ((/) e. A -> (x e. B -> (A ^o x) = (rec({<.y, z>. | z = (y .o A)}, 1o)` x)))))
9	8	imp41 368	. . . . . 6 |- ((((A e. On /\ B e. On) /\ (/) e. A) /\ x e. B) -> (A ^o x) = (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
10	9	iuneq2dv 2582	. . . . 5 |- (((A e. On /\ B e. On) /\ (/) e. A) -> U_x e. B (A ^o x) = U_x e. B (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
11	3, 10	eqeq12d 1489	. . . 4 |- (((A e. On /\ B e. On) /\ (/) e. A) -> ((A ^o B) = U_x e. B (A ^o x) <-> (rec({<.y, z>. | z = (y .o A)}, 1o)` B) = U_x e. B (rec({<.y, z>. | z = (y .o A)}, 1o)` x)))
12	11	adantlrr 399	. . 3 |- (((A e. On /\ (B e. On /\ Lim B)) /\ (/) e. A) -> ((A ^o B) = U_x e. B (A ^o x) <-> (rec({<.y, z>. | z = (y .o A)}, 1o)` B) = U_x e. B (rec({<.y, z>. | z = (y .o A)}, 1o)` x)))
13	2, 12	mpbird 196	. 2 |- (((A e. On /\ (B e. On /\ Lim B)) /\ (/) e. A) -> (A ^o B) = U_x e. B (A ^o x))
14		limelon 3032	. . 3 |- ((B e. C /\ Lim B) -> B e. On)
15		pm3.27 323	. . 3 |- ((B e. C /\ Lim B) -> Lim B)
16	14, 15	jca 288	. 2 |- ((B e. C /\ Lim B) -> (B e. On /\ Lim B))
17	13, 16	sylanl2 461	1 |- (((A e. On /\ (B e. C /\ Lim B)) /\ (/) e. A) -> (A ^o B) = U_x e. B (A ^o x))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  (/)c0 2280  U_ciun 2566  {copab 2666  Oncon0 2948  Lim wlim 2949  ` cfv 3182  reccrdg 3931  (class class class)co 3963  1oc1o 4128   .o comu 4131   ^o coe 4132

This theorem is referenced by:  oecl 4172  oe1m 4179  oen0 4213  oeordi 4214  oewordri 4219  oeworde 4220  oelim2 4222

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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1o 4133  df-oexp 4137

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