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Theorem receuap 8574
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 8558 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> E. y e. CC ( B x. y ) = 1 )
213adant1 1010 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> E. y e. CC ( B x. y ) = 1 )
3 simprl 526 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> y e. CC )
4 simpll 524 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> A e. CC )
53, 4mulcld 7927 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( y x. A ) e. CC )
6 oveq1 5857 . . . . . . . 8 |- ( ( B x. y ) = 1 -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
76ad2antll 488 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( ( B x. y ) x. A ) = ( 1 x. A ) )
8 simplr 525 . . . . . . . 8 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> B e. CC )
98, 3, 4mulassd 7930 . . . . . . 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 7925 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( 1 x. A ) = A )
117, 9, 103eqtr3d 2211 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( B x. ( y x. A ) ) = A )
12 oveq2 5858 . . . . . . . 8 |- ( x = ( y x. A ) -> ( B x. x ) = ( B x. ( y x. A ) ) )
1312eqeq1d 2179 . . . . . . 7 |- ( x = ( y x. A ) -> ( ( B x. x ) = A <-> ( B x. ( y x. A ) ) = A ) )
1413rspcev 2834 . . . . . 6 |- ( ( ( y x. A ) e. CC /\ ( B x. ( y x. A ) ) = A ) -> E. x e. CC ( B x. x ) = A )
155, 11, 14syl2anc 409 . . . . 5 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> E. x e. CC ( B x. x ) = A )
1615rexlimdvaa 2588 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( E. y e. CC ( B x. y ) = 1 -> E. x e. CC ( B x. x ) = A ) )
17163adant3 1012 . . 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 2190 . . . . . . 7 |- ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> ( B x. x ) = ( B x. y ) )
20 mulcanap 8570 . . . . . . 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 1198 . . . . 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 1010 . . 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 2551 . 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 5858 . . . 4 |- ( x = y -> ( B x. x ) = ( B x. y ) )
2726eqeq1d 2179 . . 3 |- ( x = y -> ( ( B x. x ) = A <-> ( B x. y ) = A ) )
2827reu4 2924 . 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 415 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 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   E!wreu 2450   class class class wbr 3987  (class class class)co 5850   CCcc 7759   0cc0 7761   1c1 7762   x. cmul 7766   # cap 8487
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
This theorem is referenced by:  divvalap  8578  divmulap  8579  divclap  8582
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