ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  receuap Unicode version
Theorem receuap 8779
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 8763 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> E. y e. CC ( B x. y ) = 1 )
213adant1 1018 . . 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 8130 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( y x. A ) e. CC )
6 oveq1 5976 . . . . . . . 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 8133 . . . . . . 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 8128 . . . . . . 7 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( 1 x. A ) = A )
117, 9, 103eqtr3d 2248 . . . . . 6 |- ( ( ( A e. CC /\ B e. CC ) /\ ( y e. CC /\ ( B x. y ) = 1 ) ) -> ( B x. ( y x. A ) ) = A )
12 oveq2 5977 . . . . . . . 8 |- ( x = ( y x. A ) -> ( B x. x ) = ( B x. ( y x. A ) ) )
1312eqeq1d 2216 . . . . . . 7 |- ( x = ( y x. A ) -> ( ( B x. x ) = A <-> ( B x. ( y x. A ) ) = A ) )
1413rspcev 2885 . . . . . 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 2627 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( E. y e. CC ( B x. y ) = 1 -> E. x e. CC ( B x. x ) = A ) )
17163adant3 1020 . . 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 2227 . . . . . . 7 |- ( ( ( B x. x ) = A /\ ( B x. y ) = A ) -> ( B x. x ) = ( B x. y ) )
20 mulcanap 8775 . . . . . . 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 1206 . . . . 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 1018 . . 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 2589 . 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 5977 . . . 4 |- ( x = y -> ( B x. x ) = ( B x. y ) )
2726eqeq1d 2216 . . 3 |- ( x = y -> ( ( B x. x ) = A <-> ( B x. y ) = A ) )
2827reu4 2975 . 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 981   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   E!wreu 2488   class class class wbr 4060  (class class class)co 5969   CCcc 7960   0cc0 7962   1c1 7963   x. cmul 7967   # cap 8691
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-mulrcl 8061  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-0lt1 8068  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-precex 8072  ax-cnre 8073  ax-pre-ltirr 8074  ax-pre-ltwlin 8075  ax-pre-lttrn 8076  ax-pre-apti 8077  ax-pre-ltadd 8078  ax-pre-mulgt0 8079  ax-pre-mulext 8080
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2779  df-sbc 3007  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-br 4061  df-opab 4123  df-id 4359  df-po 4362  df-iso 4363  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-iota 5252  df-fun 5293  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-pnf 8146  df-mnf 8147  df-xr 8148  df-ltxr 8149  df-le 8150  df-sub 8282  df-neg 8283  df-reap 8685  df-ap 8692
This theorem is referenced by:  divvalap  8784  divmulap  8785  divclap  8788
  Copyright terms: Public domain W3C validator