Theorem oeworde 4278

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 4027	. . . . . 6 |- (x = (/) -> (A ^o x) = (A ^o (/)))
3	1, 2	sseq12d 2141	. . . . 5 |- (x = (/) -> (x (_ (A ^o x) <-> (/) (_ (A ^o (/))))
4		id 59	. . . . . 6 |- (x = y -> x = y)
5		opreq2 4027	. . . . . 6 |- (x = y -> (A ^o x) = (A ^o y))
6	4, 5	sseq12d 2141	. . . . 5 |- (x = y -> (x (_ (A ^o x) <-> y (_ (A ^o y)))
7		id 59	. . . . . 6 |- (x = suc y -> x = suc y)
8		opreq2 4027	. . . . . 6 |- (x = suc y -> (A ^o x) = (A ^o suc y))
9	7, 8	sseq12d 2141	. . . . 5 |- (x = suc y -> (x (_ (A ^o x) <-> suc y (_ (A ^o suc y)))
10		id 59	. . . . . 6 |- (x = B -> x = B)
11		opreq2 4027	. . . . . 6 |- (x = B -> (A ^o x) = (A ^o B))
12	10, 11	sseq12d 2141	. . . . 5 |- (x = B -> (x (_ (A ^o x) <-> B (_ (A ^o B)))
13		0ss 2353	. . . . . 6 |- (/) (_ (A ^o (/))
14	13	a1i 8	. . . . 5 |- ((A e. On /\ 1o e. A) -> (/) (_ (A ^o (/)))
15		ordsucsssuc 3131	. . . . . . . . 9 |- ((Ord y /\ Ord (A ^o y)) -> (y (_ (A ^o y) <-> suc y (_ suc (A ^o y)))
16		eloni 3015	. . . . . . . . . 10 |- (y e. On -> Ord y)
17	16	adantr 398	. . . . . . . . 9 |- ((y e. On /\ A e. On) -> Ord y)
18		oecl 4230	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (A ^o y) e. On)
19	18	ancoms 447	. . . . . . . . . 10 |- ((y e. On /\ A e. On) -> (A ^o y) e. On)
20		eloni 3015	. . . . . . . . . 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 482	. . . . . . . 8 |- ((y e. On /\ A e. On) -> (y (_ (A ^o y) <-> suc y (_ suc (A ^o y)))
23	22	adantrr 404	. . . . . . 7 |- ((y e. On /\ (A e. On /\ 1o e. A)) -> (y (_ (A ^o y) <-> suc y (_ suc (A ^o y)))
24		sstr2 2122	. . . . . . . 8 |- (suc y (_ suc (A ^o y) -> (suc (A ^o y) (_ (A ^o suc y) -> suc y (_ (A ^o suc y)))
25		visset 1860	. . . . . . . . . . . . 13 |- y e. V
26	25	sucid 3108	. . . . . . . . . . . 12 |- y e. suc y
27		oeordi 4272	. . . . . . . . . . . 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 451	. . . . . . . . . 10 |- ((suc y e. On /\ (A e. On /\ 1o e. A)) -> (A ^o y) e. (A ^o suc y))
30		oecl 4230	. . . . . . . . . . . . . 14 |- ((A e. On /\ suc y e. On) -> (A ^o suc y) e. On)
31		eloni 3015	. . . . . . . . . . . . . 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 447	. . . . . . . . . . . 12 |- ((suc y e. On /\ A e. On) -> Ord (A ^o suc y))
34		ordsucss 3126	. . . . . . . . . . . 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 404	. . . . . . . . . 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 3125	. . . . . . . . 9 |- (y e. On <-> suc y e. On)
39	37, 38	sylanb 460	. . . . . . . 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 210	. . . . . 6 |- ((y e. On /\ (A e. On /\ 1o e. A)) -> (y (_ (A ^o y) -> suc y (_ (A ^o suc y)))
42	41	ex 380	. . . . 5 |- (y e. On -> ((A e. On /\ 1o e. A) -> (y (_ (A ^o y) -> suc y (_ (A ^o suc y))))
43		limuni 3086	. . . . . . . . . . 11 |- (Lim x -> x = U.x)
44		uniiun 2655	. . . . . . . . . . 11 |- U.x = U_y e. x y
45	43, 44	syl6eq 1570	. . . . . . . . . 10 |- (Lim x -> x = U_y e. x y)
46	45	adantr 398	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> x = U_y e. x y)
47		visset 1860	. . . . . . . . . . . 12 |- x e. V
48		oelim 4227	. . . . . . . . . . . 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 723	. . . . . . . . . . 11 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
50	49	anasss 451	. . . . . . . . . 10 |- ((A e. On /\ (Lim x /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
51	50	an1s 497	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
52	46, 51	sseq12d 2141	. . . . . . . 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 2631	. . . . . . . 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 217	. . . . . . 7 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A.y e. x y (_ (A ^o y) -> x (_ (A ^o x)))
55	54	ex 380	. . . . . 6 |- (Lim x -> ((A e. On /\ (/) e. A) -> (A.y e. x y (_ (A ^o y) -> x (_ (A ^o x))))
56		on0eln0 3081	. . . . . . . 8 |- (A e. On -> ((/) e. A <-> A =/= (/)))
57		ne0i 2337	. . . . . . . 8 |- (1o e. A -> A =/= (/))
58	56, 57	syl5bir 217	. . . . . . 7 |- (A e. On -> (1o e. A -> (/) e. A))
59	58	imdistani 454	. . . . . 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 3223	. . . 4 |- (B e. On -> ((A e. On /\ 1o e. A) -> B (_ (A ^o B)))
62	61	exp3a 383	. . 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 369	1 |- (((A e. On /\ B e. On) /\ 1o e. A) -> B (_ (A ^o B))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997   e. wcel 999   =/= wne 1632  A.wral 1692  Vcvv 1858   (_ wss 2098  (/)c0 2331  U.cuni 2557  U_ciun 2620  Ord word 3004  Oncon0 3005  Lim wlim 3006  suc csuc 3007  (class class class)co 4021  1oc1o 4186   ^o coe 4190

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-oadd 4193  df-omul 4194  df-oexp 4195

Copyright terms: Public domain