Theorem oelim 4227

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 4008	. . . 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 414	. . 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 4212	. . . . 5 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = (rec({<.y, z>. | z = (y .o A)}, 1o)` B))
4		oevn0 4212	. . . . . . . . . 10 |- (((A e. On /\ x e. On) /\ (/) e. A) -> (A ^o x) = (rec({<.y, z>. | z = (y .o A)}, 1o)` x))
5		onelon 3029	. . . . . . . . . 10 |- ((B e. On /\ x e. B) -> x e. On)
6	4, 5	sylanl2 472	. . . . . . . . 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 392	. . . . . . . 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 375	. . . . . 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 2636	. . . . 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 1536	. . . 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 408	. . 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 203	. 2 |- (((A e. On /\ (B e. On /\ Lim B)) /\ (/) e. A) -> (A ^o B) = U_x e. B (A ^o x))
14		limelon 3089	. . 3 |- ((B e. C /\ Lim B) -> B e. On)
15		pm3.27 330	. . 3 |- ((B e. C /\ Lim B) -> Lim B)
16	14, 15	jca 295	. 2 |- ((B e. C /\ Lim B) -> (B e. On /\ Lim B))
17	13, 16	sylanl2 472	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 153   /\ wa 230   = wceq 997   e. wcel 999  (/)c0 2331  U_ciun 2620  {copab 2721  Oncon0 3005  Lim wlim 3006  ` cfv 3239  reccrdg 3989  (class class class)co 4021  1oc1o 4186   .o comu 4189   ^o coe 4190

This theorem is referenced by:  oecl 4230  oe1m 4237  oen0 4271  oeordi 4272  oewordri 4277  oeworde 4278  oelim2 4280

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-rep 2748  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3or 788  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-rab 1699  df-v 1859  df-sbc 1989  df-csb 2052  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-if 2414  df-pw 2454  df-sn 2464  df-pr 2465  df-tp 2467  df-op 2468  df-uni 2558  df-iun 2622  df-br 2675  df-opab 2722  df-tr 2736  df-eprel 2888  df-id 2891  df-po 2896  df-so 2906  df-fr 2974  df-we 2991  df-ord 3008  df-on 3009  df-lim 3010  df-suc 3011  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fn 3250  df-fv 3255  df-rdg 3990  df-opr 4023  df-oprab 4024  df-1o 4191  df-oexp 4195

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