Theorem oecan 4200

Description: Left cancellation law for ordinal exponentiation.

Assertion

Ref	Expression
oecan	|- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> ((A ^o B) = (A ^o C) <-> B = C))

Proof of Theorem oecan

Step	Hyp	Ref	Expression
1		oeordi 4198	. . . . . . . . 9 |- (((C e. On /\ A e. On) /\ 1o e. A) -> (B e. C -> (A ^o B) e. (A ^o C)))
2	1	ex 373	. . . . . . . 8 |- ((C e. On /\ A e. On) -> (1o e. A -> (B e. C -> (A ^o B) e. (A ^o C))))
3	2	ancoms 436	. . . . . . 7 |- ((A e. On /\ C e. On) -> (1o e. A -> (B e. C -> (A ^o B) e. (A ^o C))))
4	3	3adant2 796	. . . . . 6 |- ((A e. On /\ B e. On /\ C e. On) -> (1o e. A -> (B e. C -> (A ^o B) e. (A ^o C))))
5	4	imp 350	. . . . 5 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> (B e. C -> (A ^o B) e. (A ^o C)))
6		oeordi 4198	. . . . . . . . 9 |- (((B e. On /\ A e. On) /\ 1o e. A) -> (C e. B -> (A ^o C) e. (A ^o B)))
7	6	ex 373	. . . . . . . 8 |- ((B e. On /\ A e. On) -> (1o e. A -> (C e. B -> (A ^o C) e. (A ^o B))))
8	7	ancoms 436	. . . . . . 7 |- ((A e. On /\ B e. On) -> (1o e. A -> (C e. B -> (A ^o C) e. (A ^o B))))
9	8	3adant3 797	. . . . . 6 |- ((A e. On /\ B e. On /\ C e. On) -> (1o e. A -> (C e. B -> (A ^o C) e. (A ^o B))))
10	9	imp 350	. . . . 5 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> (C e. B -> (A ^o C) e. (A ^o B)))
11	5, 10	orim12d 563	. . . 4 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> ((B e. C \/ C e. B) -> ((A ^o B) e. (A ^o C) \/ (A ^o C) e. (A ^o B))))
12	11	con3d 95	. . 3 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> (-. ((A ^o B) e. (A ^o C) \/ (A ^o C) e. (A ^o B)) -> -. (B e. C \/ C e. B)))
13		ordtri3 2973	. . . . . 6 |- ((Ord (A ^o B) /\ Ord (A ^o C)) -> ((A ^o B) = (A ^o C) <-> -. ((A ^o B) e. (A ^o C) \/ (A ^o C) e. (A ^o B))))
14		oecl 4156	. . . . . . 7 |- ((A e. On /\ B e. On) -> (A ^o B) e. On)
15		eloni 2948	. . . . . . 7 |- ((A ^o B) e. On -> Ord (A ^o B))
16	14, 15	syl 10	. . . . . 6 |- ((A e. On /\ B e. On) -> Ord (A ^o B))
17		oecl 4156	. . . . . . 7 |- ((A e. On /\ C e. On) -> (A ^o C) e. On)
18		eloni 2948	. . . . . . 7 |- ((A ^o C) e. On -> Ord (A ^o C))
19	17, 18	syl 10	. . . . . 6 |- ((A e. On /\ C e. On) -> Ord (A ^o C))
20	13, 16, 19	syl2an 454	. . . . 5 |- (((A e. On /\ B e. On) /\ (A e. On /\ C e. On)) -> ((A ^o B) = (A ^o C) <-> -. ((A ^o B) e. (A ^o C) \/ (A ^o C) e. (A ^o B))))
21	20	3impdi 877	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> ((A ^o B) = (A ^o C) <-> -. ((A ^o B) e. (A ^o C) \/ (A ^o C) e. (A ^o B))))
22	21	adantr 389	. . 3 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> ((A ^o B) = (A ^o C) <-> -. ((A ^o B) e. (A ^o C) \/ (A ^o C) e. (A ^o B))))
23		ordtri3 2973	. . . . . 6 |- ((Ord B /\ Ord C) -> (B = C <-> -. (B e. C \/ C e. B)))
24		eloni 2948	. . . . . 6 |- (B e. On -> Ord B)
25		eloni 2948	. . . . . 6 |- (C e. On -> Ord C)
26	23, 24, 25	syl2an 454	. . . . 5 |- ((B e. On /\ C e. On) -> (B = C <-> -. (B e. C \/ C e. B)))
27	26	3adant1 795	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (B = C <-> -. (B e. C \/ C e. B)))
28	27	adantr 389	. . 3 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> (B = C <-> -. (B e. C \/ C e. B)))
29	12, 22, 28	3imtr4d 541	. 2 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> ((A ^o B) = (A ^o C) -> B = C))
30		opreq2 3954	. 2 |- (B = C -> (A ^o B) = (A ^o C))
31	29, 30	impbid1 515	1 |- (((A e. On /\ B e. On /\ C e. On) /\ 1o e. A) -> ((A ^o B) = (A ^o C) <-> B = C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  Ord word 2937  Oncon0 2938  (class class class)co 3948  1oc1o 4112   ^o coe 4116

This theorem is referenced by:  oeword 4201

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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