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Theorem subrecap 8743
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 7923 . . 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 8632 . . 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 1232 . 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 526 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  B  e.  CC )
65mulid2d 7925 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( 1  x.  B )  =  B )
7 simpll 524 . . . . 5  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  A  e.  CC )
87mulid2d 7925 . . . 4  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( 1  x.  A )  =  A )
96, 8oveq12d 5868 . . 3  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( ( 1  x.  B )  -  ( 1  x.  A
) )  =  ( B  -  A ) )
109oveq1d 5865 . 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 2203 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 1348    e. wcel 2141   class class class wbr 3987  (class class class)co 5850   CCcc 7759   0cc0 7761   1c1 7762    x. cmul 7766    - cmin 8077   # cap 8487    / cdiv 8576
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-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879
This theorem depends on definitions:  df-bi 116  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-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577
This theorem is referenced by:  subrecapi  8744  subrecapd  8745
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