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Theorem receuap 8192
Description: Existential uniqueness of reciprocals. (Contributed by Jim Kingdon, 21-Feb-2020.)
Assertion
Ref Expression
receuap |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E! x e. CC ( B x. x ) = A )
Distinct variable groups:   x, A   x, B
Proof of Theorem receuap
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 recexap 8176 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> E. y e. CC ( B x. y ) = 1 )
213adant1 962 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E. y e. CC ( B x. y ) = 1 )
3 simprl 499 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> y e. CC )
4 simpll 497 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> A e. CC )
53, 4mulcld 7562 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( y x. A ) e. CC )
6 oveq1 5673 . . . . . . . 8 |- ( ( B x. y ) = 1 -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
76ad2antll 476 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
8 simplr 498 . . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> B e. CC )
98, 3, 4mulassd 7565 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( ( B x. y ) x. A ) = ( B x. ( y x. A ) ) )
104mulid2d 7560 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( 1 x. A ) = A )
117, 9, 103eqtr3d 2129 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( B x. ( y x. A ) ) = A )
12 oveq2 5674 . . . . . . . 8 |- ( x = ( y x. A ) -> ( B x. x ) = ( B x. ( y x. A ) ) )
1312eqeq1d 2097 . . . . . . 7 |- ( x = ( y x. A ) -> ( ( B x. x ) = A <-> ( B x. ( y x. A ) ) = A ) )
1413rspcev 2723 . . . . . 6 |- ( ( ( y x. A ) e. CC /\ ( B x. ( y x. A ) ) = A ) -> E. x e. CC ( B x. x ) = A )
155, 11, 14syl2anc 404 . . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> E. x e. CC ( B x. x ) = A )
1615rexlimdvaa 2491 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( E. y e. CC ( B x. y ) = 1 -> E. x e. CC ( B x. x ) = A ) )
17163adant3 964 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( E. y e. CC ( B x. y ) = 1 -> E. x e. CC ( B x. x ) = A ) )
182, 17mpd 13 . 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E. x e. CC ( B x. x ) = A )
19 eqtr3 2108 . . . . . . 7 |- ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> ( B x. x ) = ( B x. y ) )
20 mulcanap 8188 . . . . . . 7 |- ( ( x e. CC /\ y e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( ( B x. x ) = ( B x. y ) <-> x = y ) )
2119, 20syl5ib 153 . . . . . 6 |- ( ( x e. CC /\ y e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
22213expa 1144 . . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
2322expcom 115 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> ( ( x e. CC /\ y e. CC ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) ) )
24233adant1 962 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( x e. CC /\ y e. CC ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) ) )
2524ralrimivv 2455 . 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A. x e. CC A. y e. CC ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
26 oveq2 5674 . . . 4 |- ( x = y -> ( B x. x ) = ( B x. y ) )
2726eqeq1d 2097 . . 3 |- ( x = y -> ( ( B x. x ) = A <-> ( B x. y ) = A ) )
2827reu4 2810 . 2 |- ( E! x e. CC ( B x. x ) = A <-> ( E. x e. CC ( B x. x ) = A /\ A. x e. CC A. y e. CC ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) ) )
2918, 25, 28sylanbrc 409 1 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E! x e. CC ( B x. x ) = A )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 925   = wceq 1290   e. wcel 1439   A.wral 2360   E.wrex 2361   E!wreu 2362   class class class wbr 3851  (class class class)co 5666   CCcc 7402   0cc0 7404   1c1 7405   x. cmul 7409   # cap 8112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113
This theorem is referenced by:  divvalap  8195  divmulap  8196  divclap  8199
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