Theorem oen0 4213

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 3969	. . . . . 6 |- (x = (/) -> (A ^o x) = (A ^o (/)))
2	1	eleq2d 1541	. . . . 5 |- (x = (/) -> ((/) e. (A ^o x) <-> (/) e. (A ^o (/))))
3		opreq2 3969	. . . . . 6 |- (x = y -> (A ^o x) = (A ^o y))
4	3	eleq2d 1541	. . . . 5 |- (x = y -> ((/) e. (A ^o x) <-> (/) e. (A ^o y)))
5		opreq2 3969	. . . . . 6 |- (x = suc y -> (A ^o x) = (A ^o suc y))
6	5	eleq2d 1541	. . . . 5 |- (x = suc y -> ((/) e. (A ^o x) <-> (/) e. (A ^o suc y)))
7		opreq2 3969	. . . . . 6 |- (x = B -> (A ^o x) = (A ^o B))
8	7	eleq2d 1541	. . . . 5 |- (x = B -> ((/) e. (A ^o x) <-> (/) e. (A ^o B)))
9		oe0 4161	. . . . . . 7 |- (A e. On -> (A ^o (/)) = 1o)
10		0lt1o 4147	. . . . . . 7 |- (/) e. 1o
11	9, 10	syl5eleqr 1555	. . . . . 6 |- (A e. On -> (/) e. (A ^o (/)))
12	11	adantr 389	. . . . 5 |- ((A e. On /\ (/) e. A) -> (/) e. (A ^o (/)))
13		omordi 4197	. . . . . . . . . . . 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 4156	. . . . . . . . . . . . . 14 |- ((A ^o y) e. On -> ((A ^o y) .o (/)) = (/))
15	14	eleq1d 1540	. . . . . . . . . . . . 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 4172	. . . . . . . . . . . 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 4166	. . . . . . . . . . . 12 |- ((A e. On /\ y e. On) -> (A ^o suc y) = ((A ^o y) .o A))
23	22	eleq2d 1541	. . . . . . . . . . 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 3031	. . . . . . . . . . . 12 |- (Lim x -> (/) e. x)
31		eqimss2 2110	. . . . . . . . . . . . 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 3969	. . . . . . . . . . . . 13 |- (y = (/) -> (A ^o y) = (A ^o (/)))
35	34	sseq2d 2089	. . . . . . . . . . . 12 |- (y = (/) -> (1o (_ (A ^o y) <-> 1o (_ (A ^o (/))))
36	35	rcla4ev 1877	. . . . . . . . . . 11 |- (((/) e. x /\ 1o (_ (A ^o (/))) -> E.y e. x 1o (_ (A ^o y))
37		ssiun 2592	. . . . . . . . . . 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 1813	. . . . . . . . . . . 12 |- x e. V
41		oelim 4169	. . . . . . . . . . . 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 711	. . . . . . . . . . 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 486	. . . . . . . . 9 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> (A ^o x) = U_y e. x (A ^o y))
45	39, 44	sseqtr4d 2098	. . . . . . . 8 |- ((Lim x /\ (A e. On /\ (/) e. A)) -> 1o (_ (A ^o x))
46		oecl 4172	. . . . . . . . . . . 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 3032	. . . . . . . . . . . 12 |- ((x e. V /\ Lim x) -> x e. On)
49	40, 48	mpan 695	. . . . . . . . . . 11 |- (Lim x -> x e. On)
50	47, 49	sylan 448	. . . . . . . . . 10 |- ((Lim x /\ A e. On) -> (A ^o x) e. On)
51		eloni 2958	. . . . . . . . . 10 |- ((A ^o x) e. On -> Ord (A ^o x))
52		ordgt0ge1 4144	. . . . . . . . . 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 3166	. . . 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 956   e. wcel 958  A.wral 1645  E.wrex 1646  Vcvv 1811   (_ wss 2047  (/)c0 2280  U_ciun 2566  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950  (class class class)co 3963  1oc1o 4128   .o comu 4131   ^o coe 4132

This theorem is referenced by:  oeordi 4214  oeordsuc 4221

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1o 4133  df-oadd 4135  df-omul 4136  df-oexp 4137

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