Theorem oeworde 4210

Description: Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68.

Assertion

Ref	Expression
oeworde	|- (((A e. On /\ B e. On) /\ 1o e. A) -> B (_ (A ^o B))

Proof of Theorem oeworde

Step	Hyp	Ref	Expression
1		id 59	. . . . . 6 |- (x = (/) -> x = (/))
2		opreq2 3960	. . . . . 6 |- (x = (/) -> (A ^o x) = (A ^o (/)))
3	1, 2	sseq12d 2086	. . . . 5 |- (x = (/) -> (x (_ (A ^o x) <-> (/) (_ (A ^o (/))))
4		id 59	. . . . . 6 |- (x = y -> x = y)
5		opreq2 3960	. . . . . 6 |- (x = y -> (A ^o x) = (A ^o y))
6	4, 5	sseq12d 2086	. . . . 5 |- (x = y -> (x (_ (A ^o x) <-> y (_ (A ^o y)))
7		id 59	. . . . . 6 |- (x = suc y -> x = suc y)
8		opreq2 3960	. . . . . 6 |- (x = suc y -> (A ^o x) = (A ^o suc y))
9	7, 8	sseq12d 2086	. . . . 5 |- (x = suc y -> (x (_ (A ^o x) <-> suc y (_ (A ^o suc y)))
10		id 59	. . . . . 6 |- (x = B -> x = B)
11		opreq2 3960	. . . . . 6 |- (x = B -> (A ^o x) = (A ^o B))
12	10, 11	sseq12d 2086	. . . . 5 |- (x = B -> (x (_ (A ^o x) <-> B (_ (A ^o B)))
13		0ss 2297	. . . . . 6 |- (/) (_ (A ^o (/))
14	13	a1i 8	. . . . 5 |- ((A e. On /\ 1o e. A) -> (/) (_ (A ^o (/)))
15		ordsucsssuc 3069	. . . . . . . . 9 |- ((Ord y /\ Ord (A ^o y)) -> (y (_ (A ^o y) <-> suc y (_ suc (A ^o y)))
16		eloni 2953	. . . . . . . . . 10 |- (y e. On -> Ord y)
17	16	adantr 389	. . . . . . . . 9 |- ((y e. On /\ A e. On) -> Ord y)
18		oecl 4162	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (A ^o y) e. On)
19	18	ancoms 436	. . . . . . . . . 10 |- ((y e. On /\ A e. On) -> (A ^o y) e. On)
20		eloni 2953	. . . . . . . . . 10 |- ((A ^o y) e. On -> Ord (A ^o y))
21	19, 20	syl 10	. . . . . . . . 9 |- ((y e. On /\ A e. On) -> Ord (A ^o y))
22	15, 17, 21	sylanc 471	. . . . . . . 8 |- ((y e. On /\ A e. On) -> (y (_ (A ^o y) <-> suc y (_ suc (A ^o y)))
23	22	adantrr 395	. . . . . . 7 |- ((y e. On /\ (A e. On /\ 1o e. A)) -> (y (_ (A ^o y) <-> suc y (_ suc (A ^o y)))
24		sstr2 2067	. . . . . . . 8 |- (suc y (_ suc (A ^o y) -> (suc (A ^o y) (_ (A ^o suc y) -> suc y (_ (A ^o suc y)))
25		visset 1809	. . . . . . . . . . . . 13 |- y e. V
26	25	sucid 3046	. . . . . . . . . . . 12 |- y e. suc y
27		oeordi 4204	. . . . . . . . . . . 12 |- (((suc y e. On /\ A e. On) /\ 1o e. A) -> (y e. suc y -> (A ^o y) e. (A ^o suc y)))
28	26, 27	mpi 44	. . . . . . . . . . 11 |- (((suc y e. On /\ A e. On) /\ 1o e. A) -> (A ^o y) e. (A ^o suc y))
29	28	anasss 440	. . . . . . . . . 10 |- ((suc y e. On /\ (A e. On /\ 1o e. A)) -> (A ^o y) e. (A ^o suc y))
30		oecl 4162	. . . . . . . . . . . . . 14 |- ((A e. On /\ suc y e. On) -> (A ^o suc y) e. On)
31		eloni 2953	. . . . . . . . . . . . . 14 |- ((A ^o suc y) e. On -> Ord (A ^o suc y))
32	30, 31	syl 10	. . . . . . . . . . . . 13 |- ((A e. On /\ suc y e. On) -> Ord (A ^o suc y))
33	32	ancoms 436	. . . . . . . . . . . 12 |- ((suc y e. On /\ A e. On) -> Ord (A ^o suc y))
34		ordsucss 3064	. . . . . . . . . . . 12 |- (Ord (A ^o suc y) -> ((A ^o y) e. (A ^o suc y) -> suc (A ^o y) (_ (A ^o suc y)))
35	33, 34	syl 10	. . . . . . . . . . 11 |- ((suc y e. On /\ A e. On) -> ((A ^o y) e. (A ^o suc y) -> suc (A ^o y) (_ (A ^o suc y)))
36	35	adantrr 395	. . . . . . . . . 10 |- ((suc y e. On /\ (A e. On /\ 1o e. A)) -> ((A ^o y) e. (A ^o suc y) -> suc (A ^o y) (_ (A ^o suc y)))
37	29, 36	mpd 26	. . . . . . . . 9 |- ((suc y e. On /\ (A e. On /\ 1o e. A)) -> suc (A ^o y) (_ (A ^o suc y))
38		sucelon 3063	. . . . . . . . 9 |- (y e. On <-> suc y e. On)
39	37, 38	sylanb 449	. . . . . . . 8 |- ((y e. On /\ (A e. On /\ 1o e. A)) -> suc (A ^o y) (_ (A ^o suc y))
40	24, 39	syl5com 52	. . . . . . 7 |- ((y e. On /\ (A e. On /\ 1o e. A)) -> (suc y (_ suc (A ^o y) -> suc y (_ (A ^o suc y)))
41	23, 40	sylbid 203	. . . . . 6 |- ((y e. On /\ (A e. On /\ 1o e. A)) -> (y (_ (A ^o y) -> suc y (_ (A ^o suc y)))
42	41	ex 373	. . . . 5 |- (y e. On -> ((A e. On /\ 1o e. A) -> (y (_ (A ^o y) -> suc y (_ (A ^o suc y))))
43		limuni 3024	. . . . . . . . . . 11 |- (Lim x -> x = U.x)
44		uniiun 2596	. . . . . . . . . . 11 |- U.x = U_y e. x y
45	43, 44	syl6eq 1520	. . . . . . . . . 10 |- (Lim x -> x = U_y e. x y)
46	45	adantr 389	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> x = U_y e. x y)
47		visset 1809	. . . . . . . . . . . 12 |- x e. V
48		oelim 4159	. . . . . . . . . . . 12 |- (((A e. On /\ (x e. V /\ Lim x)) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
49	47, 48	mpanlr1 710	. . . . . . . . . . 11 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
50	49	anasss 440	. . . . . . . . . 10 |- ((A e. On /\ (Lim x /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
51	50	an1s 486	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
52	46, 51	sseq12d 2086	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (x (_ (A ^o x) <-> U_y e. x y (_ U_y e. x (A ^o y)))
53		ss2iun 2572	. . . . . . . 8 |- (A.y e. x y (_ (A ^o y) -> U_y e. x y (_ U_y e. x (A ^o y))
54	52, 53	syl5bir 210	. . . . . . 7 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A.y e. x y (_ (A ^o y) -> x (_ (A ^o x)))
55	54	ex 373	. . . . . 6 |- (Lim x -> ((A e. On /\ (/) e. A) -> (A.y e. x y (_ (A ^o y) -> x (_ (A ^o x))))
56		on0eln0 3019	. . . . . . . 8 |- (A e. On -> ((/) e. A <-> A =/= (/)))
57		ne0i 2282	. . . . . . . 8 |- (1o e. A -> A =/= (/))
58	56, 57	syl5bir 210	. . . . . . 7 |- (A e. On -> (1o e. A -> (/) e. A))
59	58	imdistani 443	. . . . . 6 |- ((A e. On /\ 1o e. A) -> (A e. On /\ (/) e. A))
60	55, 59	syl5 21	. . . . 5 |- (Lim x -> ((A e. On /\ 1o e. A) -> (A.y e. x y (_ (A ^o y) -> x (_ (A ^o x))))
61	3, 6, 9, 12, 14, 42, 60	tfinds3 3161	. . . 4 |- (B e. On -> ((A e. On /\ 1o e. A) -> B (_ (A ^o B)))
62	61	exp3a 375	. . 3 |- (B e. On -> (A e. On -> (1o e. A -> B (_ (A ^o B))))
63	62	com12 11	. 2 |- (A e. On -> (B e. On -> (1o e. A -> B (_ (A ^o B))))
64	63	imp31 362	1 |- (((A e. On /\ B e. On) /\ 1o e. A) -> B (_ (A ^o B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582  A.wral 1642  Vcvv 1807   (_ wss 2043  (/)c0 2276  U.cuni 2498  U_ciun 2561  Ord word 2942  Oncon0 2943  Lim wlim 2944  suc csuc 2945  (class class class)co 3954  1oc1o 4118   ^o coe 4122

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-1o 4123  df-oadd 4125  df-omul 4126  df-oexp 4127

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