Theorem subrecap 8810

Description: Subtraction of reciprocals. (Contributed by Scott Fenton, 9-Jul-2015.)

Assertion

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

Proof of Theorem subrecap

Step	Hyp	Ref	Expression
1		1cnd 7987	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> 1 e. CC )
2		id 19	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) )
3		divsubdivap 8699	. . 3 |- ( ( ( 1 e. CC /\ 1 e. CC ) /\ ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) ) -> ( ( 1 / A ) - ( 1 / B ) ) = ( ( ( 1 x. B ) - ( 1 x. A ) ) / ( A x. B ) ) )
4	1, 1, 2, 3	syl21anc 1247	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) - ( 1 / B ) ) = ( ( ( 1 x. B ) - ( 1 x. A ) ) / ( A x. B ) ) )
5		simprl 529	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> B e. CC )
6	5	mulid2d 7990	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( 1 x. B ) = B )
7		simpll 527	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> A e. CC )
8	7	mulid2d 7990	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( 1 x. A ) = A )
9	6, 8	oveq12d 5906	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 x. B ) - ( 1 x. A ) ) = ( B - A ) )
10	9	oveq1d 5903	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( ( 1 x. B ) - ( 1 x. A ) ) / ( A x. B ) ) = ( ( B - A ) / ( A x. B ) ) )
11	4, 10	eqtrd 2220	1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) - ( 1 / B ) ) = ( ( B - A ) / ( A x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   CCcc 7823   0cc0 7825   1c1 7826   x. cmul 7830   - cmin 8142   # cap 8552   / cdiv 8643

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644

This theorem is referenced by:  subrecapi  8811  subrecapd  8812