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Theorem rec11 9705
Description: Reciprocal is one-to-one. (Contributed by NM, 16-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
rec11  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( 1  /  A )  =  ( 1  /  B )  <-> 
A  =  B ) )

Proof of Theorem rec11
StepHypRef Expression
1 ax-1cn 9041 . . . 4  |-  1  e.  CC
21a1i 11 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
1  e.  CC )
3 reccl 9678 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  CC )
43adantl 453 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( 1  /  B
)  e.  CC )
5 simpl 444 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
6 divmul 9674 . . 3  |-  ( ( 1  e.  CC  /\  ( 1  /  B
)  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  ->  ( (
1  /  A )  =  ( 1  /  B )  <->  ( A  x.  ( 1  /  B
) )  =  1 ) )
72, 4, 5, 6syl3anc 1184 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( 1  /  A )  =  ( 1  /  B )  <-> 
( A  x.  (
1  /  B ) )  =  1 ) )
8 simpll 731 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  A  e.  CC )
9 simprl 733 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  B  e.  CC )
10 simprr 734 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  B  =/=  0 )
11 divrec 9687 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  =  ( A  x.  (
1  /  B ) ) )
128, 9, 10, 11syl3anc 1184 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
1312eqeq1d 2444 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  B )  =  1  <-> 
( A  x.  (
1  /  B ) )  =  1 ) )
14 diveq1 9701 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( A  /  B
)  =  1  <->  A  =  B ) )
158, 9, 10, 14syl3anc 1184 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( A  /  B )  =  1  <-> 
A  =  B ) )
167, 13, 153bitr2d 273 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 ) )  -> 
( ( 1  /  A )  =  ( 1  /  B )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599  (class class class)co 6074   CCcc 8981   0cc0 8983   1c1 8984    x. cmul 8988    / cdiv 9670
This theorem is referenced by:  rec11i  9748  rec11d  9804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-po 4496  df-so 4497  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-riota 6542  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671
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