Theorem apcxp2 13458

Description: Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)

Assertion

Ref	Expression
apcxp2	|- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( A ^c B ) # ( A ^c C ) ) )

Proof of Theorem apcxp2

Step	Hyp	Ref	Expression
1		simprl 521	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. RR )
2		simpll 519	. . . . 5 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A e. RR+ )
3	2	relogcld 13403	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. RR )
4	1, 3	remulcld 7925	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B x. ( log ` A ) ) e. RR )
5		simprr 522	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. RR )
6	5, 3	remulcld 7925	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( C x. ( log ` A ) ) e. RR )
7		reapef 13299	. . 3 |- ( ( ( B x. ( log ` A ) ) e. RR /\ ( C x. ( log ` A ) ) e. RR ) -> ( ( B x. ( log ` A ) ) # ( C x. ( log ` A ) ) <-> ( exp ` ( B x. ( log ` A ) ) ) # ( exp ` ( C x. ( log ` A ) ) ) ) )
8	4, 6, 7	syl2anc 409	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( B x. ( log ` A ) ) # ( C x. ( log ` A ) ) <-> ( exp ` ( B x. ( log ` A ) ) ) # ( exp ` ( C x. ( log ` A ) ) ) ) )
9	1	recnd 7923	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
10	5	recnd 7923	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
11	3	recnd 7923	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. CC )
12		simplr 520	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A # 1 )
13	2, 12	logrpap0d 13399	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) # 0 )
14		apmul1 8680	. . 3 |- ( ( B e. CC /\ C e. CC /\ ( ( log ` A ) e. CC /\ ( log ` A ) # 0 ) ) -> ( B # C <-> ( B x. ( log ` A ) ) # ( C x. ( log ` A ) ) ) )
15	9, 10, 11, 13, 14	syl112anc 1232	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( B x. ( log ` A ) ) # ( C x. ( log ` A ) ) ) )
16		rpcxpef 13415	. . . 4 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
17	2, 9, 16	syl2anc 409	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
18		rpcxpef 13415	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
19	2, 10, 18	syl2anc 409	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
20	17, 19	breq12d 3994	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( A ^c B ) # ( A ^c C ) <-> ( exp ` ( B x. ( log ` A ) ) ) # ( exp ` ( C x. ( log ` A ) ) ) ) )
21	8, 15, 20	3bitr4d 219	1 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( A ^c B ) # ( A ^c C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   class class class wbr 3981   ` cfv 5187  (class class class)co 5841   CCcc 7747   RRcr 7748   0cc0 7749   1c1 7750   x. cmul 7754   # cap 8475   RR+crp 9585   expce 11579   logclog 13377   ^c ccxp 13378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867  ax-arch 7868  ax-caucvg 7869  ax-pre-suploc 7870  ax-addf 7871  ax-mulf 7872

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-disj 3959  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-isom 5196  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-of 6049  df-1st 6105  df-2nd 6106  df-recs 6269  df-irdg 6334  df-frec 6355  df-1o 6380  df-oadd 6384  df-er 6497  df-map 6612  df-pm 6613  df-en 6703  df-dom 6704  df-fin 6705  df-sup 6945  df-inf 6946  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-2 8912  df-3 8913  df-4 8914  df-n0 9111  df-z 9188  df-uz 9463  df-q 9554  df-rp 9586  df-xneg 9704  df-xadd 9705  df-ioo 9824  df-ico 9826  df-icc 9827  df-fz 9941  df-fzo 10074  df-seqfrec 10377  df-exp 10451  df-fac 10635  df-bc 10657  df-ihash 10685  df-shft 10753  df-cj 10780  df-re 10781  df-im 10782  df-rsqrt 10936  df-abs 10937  df-clim 11216  df-sumdc 11291  df-ef 11585  df-e 11586  df-rest 12553  df-topgen 12572  df-psmet 12587  df-xmet 12588  df-met 12589  df-bl 12590  df-mopn 12591  df-top 12596  df-topon 12609  df-bases 12641  df-ntr 12696  df-cn 12788  df-cnp 12789  df-tx 12853  df-cncf 13158  df-limced 13225  df-dvap 13226  df-relog 13379  df-rpcxp 13380

This theorem is referenced by:  logbgcd1irraplemap  13487