Theorem apcxp2 13652

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 526	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. RR )
2		simpll 524	. . . . 5 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A e. RR+ )
3	2	relogcld 13597	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. RR )
4	1, 3	remulcld 7950	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B x. ( log ` A ) ) e. RR )
5		simprr 527	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. RR )
6	5, 3	remulcld 7950	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( C x. ( log ` A ) ) e. RR )
7		reapef 13493	. . 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 7948	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
10	5	recnd 7948	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
11	3	recnd 7948	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) e. CC )
12		simplr 525	. . . 4 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A # 1 )
13	2, 12	logrpap0d 13593	. . 3 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( log ` A ) # 0 )
14		apmul1 8705	. . 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 1237	. 2 |- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( B x. ( log ` A ) ) # ( C x. ( log ` A ) ) ) )
16		rpcxpef 13609	. . . 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 13609	. . . 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 4002	. 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 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   x. cmul 7779   # cap 8500   RR+crp 9610   expce 11605   logclog 13571   ^c ccxp 13572

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894  ax-pre-suploc 7895  ax-addf 7896  ax-mulf 7897

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-map 6628  df-pm 6629  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-ioo 9849  df-ico 9851  df-icc 9852  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-bc 10682  df-ihash 10710  df-shft 10779  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-e 11612  df-rest 12581  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-met 12783  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-ntr 12890  df-cn 12982  df-cnp 12983  df-tx 13047  df-cncf 13352  df-limced 13419  df-dvap 13420  df-relog 13573  df-rpcxp 13574

This theorem is referenced by:  logbgcd1irraplemap  13681