Theorem oelim 4175

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 3956	. . . 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 407	. . 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 4160	. . . . 5 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = (rec({<.y, z>. | z = (y .o A)}, 1o)` B))
4		oevn0 4160	. . . . . . . . . 10 |- (((A e. On /\ x e. On) /\ (/) e. A) -> (A ^o x) = (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
5		onelon 2978	. . . . . . . . . 10 |- ((B e. On /\ x e. B) -> x e. On)
6	4, 5	sylanl2 463	. . . . . . . . 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 385	. . . . . . . 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 2586	. . . . 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 1492	. . . 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 401	. . 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 3038	. . 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 463	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 958   e. wcel 960  (/)c0 2283  U_ciun 2570  {copab 2671  Oncon0 2954  Lim wlim 2955  ` cfv 3188  reccrdg 3937  (class class class)co 3969  1oc1o 4134   .o comu 4137   ^o coe 4138

This theorem is referenced by:  oecl 4178  oe1m 4185  oen0 4219  oeordi 4220  oewordri 4225  oeworde 4226  oelim2 4228

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1o 4139  df-oexp 4143

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