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Theorem apexp1 10861
Description: Exponentiation and apartness. (Contributed by Jim Kingdon, 9-Jul-2024.)
Assertion
Ref Expression
apexp1 |- ( ( A e. CC /\ B e. CC /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
Proof of Theorem apexp1
Dummy variables k w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5951 . . . . . . 7 |- ( w = 1 -> ( A ^ w ) = ( A ^ 1 ) )
2 oveq2 5951 . . . . . . 7 |- ( w = 1 -> ( B ^ w ) = ( B ^ 1 ) )
31, 2breq12d 4056 . . . . . 6 |- ( w = 1 -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ 1 ) # ( B ^ 1 ) ) )
43imbi1d 231 . . . . 5 |- ( w = 1 -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) ) )
54imbi2d 230 . . . 4 |- ( w = 1 -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) ) ) )
6 oveq2 5951 . . . . . . 7 |- ( w = k -> ( A ^ w ) = ( A ^ k ) )
7 oveq2 5951 . . . . . . 7 |- ( w = k -> ( B ^ w ) = ( B ^ k ) )
86, 7breq12d 4056 . . . . . 6 |- ( w = k -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ k ) # ( B ^ k ) ) )
98imbi1d 231 . . . . 5 |- ( w = k -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) )
109imbi2d 230 . . . 4 |- ( w = k -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) ) )
11 oveq2 5951 . . . . . . 7 |- ( w = ( k + 1 ) -> ( A ^ w ) = ( A ^ ( k + 1 ) ) )
12 oveq2 5951 . . . . . . 7 |- ( w = ( k + 1 ) -> ( B ^ w ) = ( B ^ ( k + 1 ) ) )
1311, 12breq12d 4056 . . . . . 6 |- ( w = ( k + 1 ) -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) )
1413imbi1d 231 . . . . 5 |- ( w = ( k + 1 ) -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) )
1514imbi2d 230 . . . 4 |- ( w = ( k + 1 ) -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) ) )
16 oveq2 5951 . . . . . . 7 |- ( w = N -> ( A ^ w ) = ( A ^ N ) )
17 oveq2 5951 . . . . . . 7 |- ( w = N -> ( B ^ w ) = ( B ^ N ) )
1816, 17breq12d 4056 . . . . . 6 |- ( w = N -> ( ( A ^ w ) # ( B ^ w ) <-> ( A ^ N ) # ( B ^ N ) ) )
1918imbi1d 231 . . . . 5 |- ( w = N -> ( ( ( A ^ w ) # ( B ^ w ) -> A # B ) <-> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) )
2019imbi2d 230 . . . 4 |- ( w = N -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ w ) # ( B ^ w ) -> A # B ) ) <-> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) ) )
21 simpl 109 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2221exp1d 10811 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A ^ 1 ) = A )
23 simpr 110 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
2423exp1d 10811 . . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( B ^ 1 ) = B )
2522, 24breq12d 4056 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) <-> A # B ) )
2625biimpd 144 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) # ( B ^ 1 ) -> A # B ) )
27 simpr 110 . . . . . . . 8 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ ( A ^ k ) # ( B ^ k ) ) -> ( A ^ k ) # ( B ^ k ) )
28 simpllr 534 . . . . . . . 8 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ ( A ^ k ) # ( B ^ k ) ) -> ( ( A ^ k ) # ( B ^ k ) -> A # B ) )
2927, 28mpd 13 . . . . . . 7 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ ( A ^ k ) # ( B ^ k ) ) -> A # B )
30 simpr 110 . . . . . . 7 |- ( ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) /\ A # B ) -> A # B )
31 simpr 110 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) )
3221ad3antlr 493 . . . . . . . . . 10 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> A e. CC )
33 nnnn0 9301 . . . . . . . . . . 11 |- ( k e. NN -> k e. NN0 )
3433ad3antrrr 492 . . . . . . . . . 10 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> k e. NN0 )
3532, 34expp1d 10817 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
3623ad3antlr 493 . . . . . . . . . 10 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> B e. CC )
3736, 34expp1d 10817 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( B ^ ( k + 1 ) ) = ( ( B ^ k ) x. B ) )
3831, 35, 373brtr3d 4074 . . . . . . . 8 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( ( A ^ k ) x. A ) # ( ( B ^ k ) x. B ) )
3932, 34expcld 10816 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( A ^ k ) e. CC )
4036, 34expcld 10816 . . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( B ^ k ) e. CC )
41 mulext 8686 . . . . . . . . 9 |- ( ( ( ( A ^ k ) e. CC /\ A e. CC ) /\ ( ( B ^ k ) e. CC /\ B e. CC ) ) -> ( ( ( A ^ k ) x. A ) # ( ( B ^ k ) x. B ) -> ( ( A ^ k ) # ( B ^ k ) \/ A # B ) ) )
4239, 32, 40, 36, 41syl22anc 1250 . . . . . . . 8 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( ( ( A ^ k ) x. A ) # ( ( B ^ k ) x. B ) -> ( ( A ^ k ) # ( B ^ k ) \/ A # B ) ) )
4338, 42mpd 13 . . . . . . 7 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> ( ( A ^ k ) # ( B ^ k ) \/ A # B ) )
4429, 30, 43mpjaodan 799 . . . . . 6 |- ( ( ( ( k e. NN /\ ( A e. CC /\ B e. CC ) ) /\ ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) /\ ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) ) -> A # B )
4544exp41 370 . . . . 5 |- ( k e. NN -> ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ k ) # ( B ^ k ) -> A # B ) -> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) ) )
4645a2d 26 . . . 4 |- ( k e. NN -> ( ( ( A e. CC /\ B e. CC ) -> ( ( A ^ k ) # ( B ^ k ) -> A # B ) ) -> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ ( k + 1 ) ) # ( B ^ ( k + 1 ) ) -> A # B ) ) ) )
475, 10, 15, 20, 26, 46nnind 9051 . . 3 |- ( N e. NN -> ( ( A e. CC /\ B e. CC ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) ) )
4847impcom 125 . 2 |- ( ( ( A e. CC /\ B e. CC ) /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
49483impa 1196 1 |- ( ( A e. CC /\ B e. CC /\ N e. NN ) -> ( ( A ^ N ) # ( B ^ N ) -> A # B ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 980   = wceq 1372   e. wcel 2175   class class class wbr 4043  (class class class)co 5943   CCcc 7922   1c1 7925   + caddc 7927   x. cmul 7929   # cap 8653   NNcn 9035   NN0cn0 9294   ^cexp 10681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-seqfrec 10591  df-exp 10682
This theorem is referenced by:  logbgcd1irraplemap  15412
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