Theorem oen0 4197

Description: Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67.

Assertion

Ref	Expression
oen0	|- (((A e. On /\ B e. On) /\ (/) e. A) -> (/) e. (A ^o B))

Proof of Theorem oen0

Step	Hyp	Ref	Expression
1		opreq2 3954	. . . . . 6 |- (x = (/) -> (A ^o x) = (A ^o (/)))
2	1	eleq2d 1533	. . . . 5 |- (x = (/) -> ((/) e. (A ^o x) <-> (/) e. (A ^o (/))))
3		opreq2 3954	. . . . . 6 |- (x = y -> (A ^o x) = (A ^o y))
4	3	eleq2d 1533	. . . . 5 |- (x = y -> ((/) e. (A ^o x) <-> (/) e. (A ^o y)))
5		opreq2 3954	. . . . . 6 |- (x = suc y -> (A ^o x) = (A ^o suc y))
6	5	eleq2d 1533	. . . . 5 |- (x = suc y -> ((/) e. (A ^o x) <-> (/) e. (A ^o suc y)))
7		opreq2 3954	. . . . . 6 |- (x = B -> (A ^o x) = (A ^o B))
8	7	eleq2d 1533	. . . . 5 |- (x = B -> ((/) e. (A ^o x) <-> (/) e. (A ^o B)))
9		oe0 4145	. . . . . . 7 |- (A e. On -> (A ^o (/)) = 1o)
10		0lt1o 4131	. . . . . . 7 |- (/) e. 1o
11	9, 10	syl5eleqr 1547	. . . . . 6 |- (A e. On -> (/) e. (A ^o (/)))
12	11	adantr 389	. . . . 5 |- ((A e. On /\ (/) e. A) -> (/) e. (A ^o (/)))
13		omordi 4181	. . . . . . . . . . . 12 |- (((A e. On /\ (A ^o y) e. On) /\ (/) e. (A ^o y)) -> ((/) e. A -> ((A ^o y) .o (/)) e. ((A ^o y) .o A)))
14		om0 4140	. . . . . . . . . . . . . 14 |- ((A ^o y) e. On -> ((A ^o y) .o (/)) = (/))
15	14	eleq1d 1532	. . . . . . . . . . . . 13 |- ((A ^o y) e. On -> (((A ^o y) .o (/)) e. ((A ^o y) .o A) <-> (/) e. ((A ^o y) .o A)))
16	15	ad2antlr 405	. . . . . . . . . . . 12 |- (((A e. On /\ (A ^o y) e. On) /\ (/) e. (A ^o y)) -> (((A ^o y) .o (/)) e. ((A ^o y) .o A) <-> (/) e. ((A ^o y) .o A)))
17	13, 16	sylibd 202	. . . . . . . . . . 11 |- (((A e. On /\ (A ^o y) e. On) /\ (/) e. (A ^o y)) -> ((/) e. A -> (/) e. ((A ^o y) .o A)))
18		pm3.26 319	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> A e. On)
19		oecl 4156	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A ^o y) e. On)
20	18, 19	jca 288	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (A e. On /\ (A ^o y) e. On))
21	17, 20	sylan 448	. . . . . . . . . 10 |- (((A e. On /\ y e. On) /\ (/) e. (A ^o y)) -> ((/) e. A -> (/) e. ((A ^o y) .o A)))
22		oesuc 4150	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
23	22	eleq2d 1533	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> ((/) e. (A ^o suc y) <-> (/) e. ((A ^o y) .o A)))
24	23	adantr 389	. . . . . . . . . 10 |- (((A e. On /\ y e. On) /\ (/) e. (A ^o y)) -> ((/) e. (A ^o suc y) <-> (/) e. ((A ^o y) .o A)))
25	21, 24	sylibrd 204	. . . . . . . . 9 |- (((A e. On /\ y e. On) /\ (/) e. (A ^o y)) -> ((/) e. A -> (/) e. (A ^o suc y)))
26	25	exp31 376	. . . . . . . 8 |- (A e. On -> (y e. On -> ((/) e. (A ^o y) -> ((/) e. A -> (/) e. (A ^o suc y)))))
27	26	com12 11	. . . . . . 7 |- (y e. On -> (A e. On -> ((/) e. (A ^o y) -> ((/) e. A -> (/) e. (A ^o suc y)))))
28	27	com34 36	. . . . . 6 |- (y e. On -> (A e. On -> ((/) e. A -> ((/) e. (A ^o y) -> (/) e. (A ^o suc y)))))
29	28	imp3a 361	. . . . 5 |- (y e. On -> ((A e. On /\ (/) e. A) -> ((/) e. (A ^o y) -> (/) e. (A ^o suc y))))
30		0ellim 3021	. . . . . . . . . . . 12 |- (Lim x -> (/) e. x)
31		eqimss2 2100	. . . . . . . . . . . . 13 |- ((A ^o (/)) = 1o -> 1o (_ (A ^o (/)))
32	9, 31	syl 10	. . . . . . . . . . . 12 |- (A e. On -> 1o (_ (A ^o (/)))
33	30, 32	anim12i 333	. . . . . . . . . . 11 |- ((Lim x /\ A e. On) -> ((/) e. x /\ 1o (_ (A ^o (/))))
34		opreq2 3954	. . . . . . . . . . . . 13 |- (y = (/) -> (A ^o y) = (A ^o (/)))
35	34	sseq2d 2079	. . . . . . . . . . . 12 |- (y = (/) -> (1o (_ (A ^o y) <-> 1o (_ (A ^o (/))))
36	35	rcla4ev 1868	. . . . . . . . . . 11 |- (((/) e. x /\ 1o (_ (A ^o (/))) -> E.y e. x 1o (_ (A ^o y))
37		ssiun 2582	. . . . . . . . . . 11 |- (E.y e. x 1o (_ (A ^o y) -> 1o (_ U_y e. x (A ^o y))
38	33, 36, 37	3syl 20	. . . . . . . . . 10 |- ((Lim x /\ A e. On) -> 1o (_ U_y e. x (A ^o y))
39	38	adantrr 395	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> 1o (_ U_y e. x (A ^o y))
40		visset 1804	. . . . . . . . . . . 12 |- x e. V
41		oelim 4153	. . . . . . . . . . . 12 |- (((A e. On /\ (x e. V /\ Lim x)) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
42	40, 41	mpanlr1 709	. . . . . . . . . . 11 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
43	42	anasss 440	. . . . . . . . . 10 |- ((A e. On /\ (Lim x /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
44	43	an1s 485	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
45	39, 44	sseqtr4d 2088	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> 1o (_ (A ^o x))
46		oecl 4156	. . . . . . . . . . . 12 |- ((A e. On /\ x e. On) -> (A ^o x) e. On)
47	46	ancoms 436	. . . . . . . . . . 11 |- ((x e. On /\ A e. On) -> (A ^o x) e. On)
48		limelon 3022	. . . . . . . . . . . 12 |- ((x e. V /\ Lim x) -> x e. On)
49	40, 48	mpan 693	. . . . . . . . . . 11 |- (Lim x -> x e. On)
50	47, 49	sylan 448	. . . . . . . . . 10 |- ((Lim x /\ A e. On) -> (A ^o x) e. On)
51		eloni 2948	. . . . . . . . . 10 |- ((A ^o x) e. On -> Ord (A ^o x))
52		ordgt0ge1 4128	. . . . . . . . . 10 |- (Ord (A ^o x) -> ((/) e. (A ^o x) <-> 1o (_ (A ^o x)))
53	50, 51, 52	3syl 20	. . . . . . . . 9 |- ((Lim x /\ A e. On) -> ((/) e. (A ^o x) <-> 1o (_ (A ^o x)))
54	53	adantrr 395	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> ((/) e. (A ^o x) <-> 1o (_ (A ^o x)))
55	45, 54	mpbird 196	. . . . . . 7 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (/) e. (A ^o x))
56	55	ex 373	. . . . . 6 |- (Lim x -> ((A e. On /\ (/) e. A) -> (/) e. (A ^o x)))
57	56	a1dd 42	. . . . 5 |- (Lim x -> ((A e. On /\ (/) e. A) -> (A.y e. x (/) e. (A ^o y) -> (/) e. (A ^o x))))
58	2, 4, 6, 8, 12, 29, 57	tfinds3 3156	. . . 4 |- (B e. On -> ((A e. On /\ (/) e. A) -> (/) e. (A ^o B)))
59	58	exp3a 375	. . 3 |- (B e. On -> (A e. On -> ((/) e. A -> (/) e. (A ^o B))))
60	59	com12 11	. 2 |- (A e. On -> (B e. On -> ((/) e. A -> (/) e. (A ^o B))))
61	60	imp31 362	1 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (/) e. (A ^o B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638  Vcvv 1802   (_ wss 2037  (/)c0 2270  U_ciun 2556  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940  (class class class)co 3948  1oc1o 4112   .o comu 4115   ^o coe 4116

This theorem is referenced by:  oeordi 4198  oeordsuc 4205

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1o 4117  df-oadd 4119  df-omul 4120  df-oexp 4121

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