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Theorem receuap 8816
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 8800 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> E. y e. CC ( B x. y ) = 1 )
213adant1 1039 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E. y e. CC ( B x. y ) = 1 )
3 simprl 529 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> y e. CC )
4 simpll 527 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> A e. CC )
53, 4mulcld 8167 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( y x. A ) e. CC )
6 oveq1 6008 . . . . . . . 8 |- ( ( B x. y ) = 1 -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
76ad2antll 491 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
8 simplr 528 . . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> B e. CC )
98, 3, 4mulassd 8170 . . . . . . 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 8165 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( 1 x. A ) = A )
117, 9, 103eqtr3d 2270 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( B x. ( y x. A ) ) = A )
12 oveq2 6009 . . . . . . . 8 |- ( x = ( y x. A ) -> ( B x. x ) = ( B x. ( y x. A ) ) )
1312eqeq1d 2238 . . . . . . 7 |- ( x = ( y x. A ) -> ( ( B x. x ) = A <-> ( B x. ( y x. A ) ) = A ) )
1413rspcev 2907 . . . . . 6 |- ( ( ( y x. A ) e. CC /\ ( B x. ( y x. A ) ) = A ) -> E. x e. CC ( B x. x ) = A )
155, 11, 14syl2anc 411 . . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> E. x e. CC ( B x. x ) = A )
1615rexlimdvaa 2649 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( E. y e. CC ( B x. y ) = 1 -> E. x e. CC ( B x. x ) = A ) )
17163adant3 1041 . . 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 2249 . . . . . . 7 |- ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> ( B x. x ) = ( B x. y ) )
20 mulcanap 8812 . . . . . . 7 |- ( ( x e. CC /\ y e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( ( B x. x ) = ( B x. y ) <-> x = y ) )
2119, 20imbitrid 154 . . . . . 6 |- ( ( x e. CC /\ y e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
22213expa 1227 . . . . 5 |- ( ( ( x e. CC /\ y e. CC ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) )
2322expcom 116 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> ( ( x e. CC /\ y e. CC ) -> ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> x = y ) ) )
24233adant1 1039 . . 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 2611 . 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 6009 . . . 4 |- ( x = y -> ( B x. x ) = ( B x. y ) )
2726eqeq1d 2238 . . 3 |- ( x = y -> ( ( B x. x ) = A <-> ( B x. y ) = A ) )
2827reu4 2997 . 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 417 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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   E!wreu 2510   class class class wbr 4083  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   x. cmul 8004   # cap 8728
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729
This theorem is referenced by:  divvalap  8821  divmulap  8822  divclap  8825
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