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Theorem subrecap 8610
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
StepHypRef Expression
1 1cnd 7794 . . 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 8500 . . 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 ) ) )
41, 1, 2, 3syl21anc 1215 . 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 520 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  B  e.  CC )
65mulid2d 7796 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( 1  x.  B )  =  B )
7 simpll 518 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  A  e.  CC )
87mulid2d 7796 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( 1  x.  A )  =  A )
96, 8oveq12d 5792 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( 1  x.  B )  -  ( 1  x.  A
) )  =  ( B  -  A ) )
109oveq1d 5789 . 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 ) ) )
114, 10eqtrd 2172 1  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( 1  /  A )  -  ( 1  /  B
) )  =  ( ( B  -  A
)  /  ( A  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7630   0cc0 7632   1c1 7633    x. cmul 7637    - cmin 7945   # cap 8355    / cdiv 8444
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445
This theorem is referenced by:  subrecapi  8611  subrecapd  8612
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