Theorem divrecap 8964

Description: Relationship between division and reciprocal. (Contributed by Jim Kingdon, 24-Feb-2020.)

Assertion

Ref	Expression
divrecap	|- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )

Proof of Theorem divrecap

Step	Hyp	Ref	Expression
1		simp2 1025	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
2		simp1 1024	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
3		recclap 8955	. . . . 5 |- ( ( B e. CC /\ B # 0 ) -> ( 1 / B ) e. CC )
4	3	3adant1 1042	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( 1 / B ) e. CC )
5	1, 2, 4	mul12d 8427	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B x. ( A x. ( 1 / B ) ) ) = ( A x. ( B x. ( 1 / B ) ) ) )
6		recidap 8962	. . . . 5 |- ( ( B e. CC /\ B # 0 ) -> ( B x. ( 1 / B ) ) = 1 )
7	6	3adant1 1042	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B x. ( 1 / B ) ) = 1 )
8	7	oveq2d 6068	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( B x. ( 1 / B ) ) ) = ( A x. 1 ) )
9	2	mulridd 8293	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. 1 ) = A )
10	5, 8, 9	3eqtrd 2271	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B x. ( A x. ( 1 / B ) ) ) = A )
11	2, 4	mulcld 8296	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( 1 / B ) ) e. CC )
12		3simpc 1023	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B e. CC /\ B # 0 ) )
13		divmulap 8951	. . 3 |- ( ( A e. CC /\ ( A x. ( 1 / B ) ) e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( ( A / B ) = ( A x. ( 1 / B ) ) <-> ( B x. ( A x. ( 1 / B ) ) ) = A ) )
14	2, 11, 12, 13	syl3anc 1274	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A / B ) = ( A x. ( 1 / B ) ) <-> ( B x. ( A x. ( 1 / B ) ) ) = A ) )
15	10, 14	mpbird 167	1 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   x. cmul 8134   # cap 8857   / cdiv 8948

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949

This theorem is referenced by:  divrecap2  8965  divassap  8966  divdirap  8973  dividap  8977  divnegap  8982  rec11ap  8986  divdiv32ap  8996  redivclap  9007  divrecapzi  9026  divrecapi  9033  divrecapd  9069  expdivap  10956  efival  12422  ef01bndlem  12446  cos01bnd  12448  divcnap  15447