Theorem oen0 4271

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 4027	. . . . . 6 |- (x = (/) -> (A ^o x) = (A ^o (/)))
2	1	eleq2d 1588	. . . . 5 |- (x = (/) -> ((/) e. (A ^o x) <-> (/) e. (A ^o (/))))
3		opreq2 4027	. . . . . 6 |- (x = y -> (A ^o x) = (A ^o y))
4	3	eleq2d 1588	. . . . 5 |- (x = y -> ((/) e. (A ^o x) <-> (/) e. (A ^o y)))
5		opreq2 4027	. . . . . 6 |- (x = suc y -> (A ^o x) = (A ^o suc y))
6	5	eleq2d 1588	. . . . 5 |- (x = suc y -> ((/) e. (A ^o x) <-> (/) e. (A ^o suc y)))
7		opreq2 4027	. . . . . 6 |- (x = B -> (A ^o x) = (A ^o B))
8	7	eleq2d 1588	. . . . 5 |- (x = B -> ((/) e. (A ^o x) <-> (/) e. (A ^o B)))
9		oe0 4219	. . . . . . 7 |- (A e. On -> (A ^o (/)) = 1o)
10		0lt1o 4205	. . . . . . 7 |- (/) e. 1o
11	9, 10	syl5eleqr 1602	. . . . . 6 |- (A e. On -> (/) e. (A ^o (/)))
12	11	adantr 398	. . . . 5 |- ((A e. On /\ (/) e. A) -> (/) e. (A ^o (/)))
13		omordi 4255	. . . . . . . . . . . 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 4214	. . . . . . . . . . . . . 14 |- ((A ^o y) e. On -> ((A ^o y) .o (/)) = (/))
15	14	eleq1d 1587	. . . . . . . . . . . . 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 414	. . . . . . . . . . . 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 209	. . . . . . . . . . 11 |- (((A e. On /\ (A ^o y) e. On) /\ (/) e. (A ^o y)) -> ((/) e. A -> (/) e. ((A ^o y) .o A)))
18		pm3.26 326	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> A e. On)
19		oecl 4230	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A ^o y) e. On)
20	18, 19	jca 295	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> (A e. On /\ (A ^o y) e. On))
21	17, 20	sylan 459	. . . . . . . . . 10 |- (((A e. On /\ y e. On) /\ (/) e. (A ^o y)) -> ((/) e. A -> (/) e. ((A ^o y) .o A)))
22		oesuc 4224	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
23	22	eleq2d 1588	. . . . . . . . . . 11 |- ((A e. On /\ y e. On) -> ((/) e. (A ^o suc y) <-> (/) e. ((A ^o y) .o A)))
24	23	adantr 398	. . . . . . . . . 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 211	. . . . . . . . 9 |- (((A e. On /\ y e. On) /\ (/) e. (A ^o y)) -> ((/) e. A -> (/) e. (A ^o suc y)))
26	25	exp31 385	. . . . . . . 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 368	. . . . 5 |- (y e. On -> ((A e. On /\ (/) e. A) -> ((/) e. (A ^o y) -> (/) e. (A ^o suc y))))
30		0ellim 3088	. . . . . . . . . . . 12 |- (Lim x -> (/) e. x)
31		eqimss2 2161	. . . . . . . . . . . . 13 |- ((A ^o (/)) = 1o -> 1o (_ (A ^o (/)))
32	9, 31	syl 10	. . . . . . . . . . . 12 |- (A e. On -> 1o (_ (A ^o (/)))
33	30, 32	anim12i 340	. . . . . . . . . . 11 |- ((Lim x /\ A e. On) -> ((/) e. x /\ 1o (_ (A ^o (/))))
34		opreq2 4027	. . . . . . . . . . . . 13 |- (y = (/) -> (A ^o y) = (A ^o (/)))
35	34	sseq2d 2140	. . . . . . . . . . . 12 |- (y = (/) -> (1o (_ (A ^o y) <-> 1o (_ (A ^o (/))))
36	35	rcla4ev 1924	. . . . . . . . . . 11 |- (((/) e. x /\ 1o (_ (A ^o (/))) -> E.y e. x 1o (_ (A ^o y))
37		ssiun 2646	. . . . . . . . . . 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 404	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> 1o (_ U_y e. x (A ^o y))
40		visset 1860	. . . . . . . . . . . 12 |- x e. V
41		oelim 4227	. . . . . . . . . . . 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 723	. . . . . . . . . . 11 |- (((A e. On /\ Lim x) /\ (/) e. A) -> (A ^o x) = U_y e. x (A ^o y))
43	42	anasss 451	. . . . . . . . . 10 |- ((A e. On /\ (Lim x /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
44	43	an1s 497	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
45	39, 44	sseqtr4d 2149	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> 1o (_ (A ^o x))
46		oecl 4230	. . . . . . . . . . . 12 |- ((A e. On /\ x e. On) -> (A ^o x) e. On)
47	46	ancoms 447	. . . . . . . . . . 11 |- ((x e. On /\ A e. On) -> (A ^o x) e. On)
48		limelon 3089	. . . . . . . . . . . 12 |- ((x e. V /\ Lim x) -> x e. On)
49	40, 48	mpan 707	. . . . . . . . . . 11 |- (Lim x -> x e. On)
50	47, 49	sylan 459	. . . . . . . . . 10 |- ((Lim x /\ A e. On) -> (A ^o x) e. On)
51		eloni 3015	. . . . . . . . . 10 |- ((A ^o x) e. On -> Ord (A ^o x))
52		ordgt0ge1 4202	. . . . . . . . . 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 404	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> ((/) e. (A ^o x) <-> 1o (_ (A ^o x)))
55	45, 54	mpbird 203	. . . . . . 7 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (/) e. (A ^o x))
56	55	ex 380	. . . . . 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 3223	. . . 4 |- (B e. On -> ((A e. On /\ (/) e. A) -> (/) e. (A ^o B)))
59	58	exp3a 383	. . 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 369	1 |- (((A e. On /\ B e. On) /\ (/) e. A) -> (/) e. (A ^o B))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997   e. wcel 999  A.wral 1692  E.wrex 1693  Vcvv 1858   (_ wss 2098  (/)c0 2331  U_ciun 2620  Ord word 3004  Oncon0 3005  Lim wlim 3006  suc csuc 3007  (class class class)co 4021  1oc1o 4186   .o comu 4189   ^o coe 4190

This theorem is referenced by:  oeordi 4272  oeordsuc 4279

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

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