Theorem apcxp2 13354

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 13299	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. RR )
4	1, 3	remulcld 7911	. . 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 7911	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( C x. ( log ` A ) ) e. RR )
7		reapef 13195	. . 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 7909	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
10	5	recnd 7909	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
11	3	recnd 7909	. . 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 13295	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) # 0 )
14		apmul1 8666	. . 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 1224	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( B x. ( log ` A ) ) # ( C x. ( log ` A ) ) ) )
16		rpcxpef 13311	. . . 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 13311	. . . 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 3980	. 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 1335   e. wcel 2128   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   CCcc 7733   RRcr 7734   0cc0 7735   1c1 7736   x. cmul 7740   # cap 8461   RR+crp 9567   expce 11551   logclog 13273   ^c ccxp 13274

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855  ax-pre-suploc 7856  ax-addf 7857  ax-mulf 7858

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-disj 3945  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-isom 5182  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-of 6035  df-1st 6091  df-2nd 6092  df-recs 6255  df-irdg 6320  df-frec 6341  df-1o 6366  df-oadd 6370  df-er 6483  df-map 6598  df-pm 6599  df-en 6689  df-dom 6690  df-fin 6691  df-sup 6931  df-inf 6932  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-xneg 9686  df-xadd 9687  df-ioo 9803  df-ico 9805  df-icc 9806  df-fz 9920  df-fzo 10052  df-seqfrec 10355  df-exp 10429  df-fac 10612  df-bc 10634  df-ihash 10662  df-shft 10727  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-clim 11188  df-sumdc 11263  df-ef 11557  df-e 11558  df-rest 12449  df-topgen 12468  df-psmet 12483  df-xmet 12484  df-met 12485  df-bl 12486  df-mopn 12487  df-top 12492  df-topon 12505  df-bases 12537  df-ntr 12592  df-cn 12684  df-cnp 12685  df-tx 12749  df-cncf 13054  df-limced 13121  df-dvap 13122  df-relog 13275  df-rpcxp 13276

This theorem is referenced by:  logbgcd1irraplemap  13383