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Theorem cnegexlem3 7903
Description: Existence of real number difference. Lemma for cnegex 7904. (Contributed by Eric Schmidt, 22-May-2007.)
Assertion
Ref Expression
cnegexlem3  |-  ( ( b  e.  RR  /\  y  e.  RR )  ->  E. c  e.  RR  ( b  +  c )  =  y )
Distinct variable group:    b, c, y

Proof of Theorem cnegexlem3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 readdcl 7710 . . . . . 6  |-  ( ( b  e.  RR  /\  x  e.  RR )  ->  ( b  +  x
)  e.  RR )
2 ax-rnegex 7693 . . . . . 6  |-  ( ( b  +  x )  e.  RR  ->  E. c  e.  RR  ( ( b  +  x )  +  c )  =  0 )
31, 2syl 14 . . . . 5  |-  ( ( b  e.  RR  /\  x  e.  RR )  ->  E. c  e.  RR  ( ( b  +  x )  +  c )  =  0 )
43adantlr 466 . . . 4  |-  ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  ->  E. c  e.  RR  ( ( b  +  x )  +  c )  =  0 )
54adantr 272 . . 3  |-  ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  ->  E. c  e.  RR  ( ( b  +  x )  +  c )  =  0 )
6 recn 7717 . . . . . . . 8  |-  ( b  e.  RR  ->  b  e.  CC )
7 recn 7717 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
86, 7anim12i 334 . . . . . . 7  |-  ( ( b  e.  RR  /\  y  e.  RR )  ->  ( b  e.  CC  /\  y  e.  CC ) )
98anim1i 336 . . . . . 6  |-  ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  ->  ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR ) )
109anim1i 336 . . . . 5  |-  ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  -> 
( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 ) )
11 recn 7717 . . . . 5  |-  ( c  e.  RR  ->  c  e.  CC )
12 recn 7717 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
13 add32 7885 . . . . . . . . . . . 12  |-  ( ( b  e.  CC  /\  x  e.  CC  /\  c  e.  CC )  ->  (
( b  +  x
)  +  c )  =  ( ( b  +  c )  +  x ) )
14133expa 1164 . . . . . . . . . . 11  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( ( b  +  x )  +  c )  =  ( ( b  +  c )  +  x ) )
15 addcl 7709 . . . . . . . . . . . . 13  |-  ( ( b  e.  CC  /\  c  e.  CC )  ->  ( b  +  c )  e.  CC )
16 addcom 7863 . . . . . . . . . . . . 13  |-  ( ( ( b  +  c )  e.  CC  /\  x  e.  CC )  ->  ( ( b  +  c )  +  x
)  =  ( x  +  ( b  +  c ) ) )
1715, 16sylan 279 . . . . . . . . . . . 12  |-  ( ( ( b  e.  CC  /\  c  e.  CC )  /\  x  e.  CC )  ->  ( ( b  +  c )  +  x )  =  ( x  +  ( b  +  c ) ) )
1817an32s 540 . . . . . . . . . . 11  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( ( b  +  c )  +  x )  =  ( x  +  ( b  +  c ) ) )
1914, 18eqtr2d 2149 . . . . . . . . . 10  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( x  +  ( b  +  c ) )  =  ( ( b  +  x
)  +  c ) )
2012, 19sylanl2 398 . . . . . . . . 9  |-  ( ( ( b  e.  CC  /\  x  e.  RR )  /\  c  e.  CC )  ->  ( x  +  ( b  +  c ) )  =  ( ( b  +  x
)  +  c ) )
2120adantllr 470 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  (
x  +  ( b  +  c ) )  =  ( ( b  +  x )  +  c ) )
2221adantlr 466 . . . . . . 7  |-  ( ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  CC )  ->  ( x  +  ( b  +  c ) )  =  ( ( b  +  x )  +  c ) )
23 addcom 7863 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
2423ancoms 266 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  x  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
2512, 24sylan2 282 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  x  e.  RR )  ->  ( x  +  y )  =  ( y  +  x ) )
26 id 19 . . . . . . . . . 10  |-  ( ( y  +  x )  =  0  ->  (
y  +  x )  =  0 )
2725, 26sylan9eq 2168 . . . . . . . . 9  |-  ( ( ( y  e.  CC  /\  x  e.  RR )  /\  ( y  +  x )  =  0 )  ->  ( x  +  y )  =  0 )
2827adantlll 469 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  -> 
( x  +  y )  =  0 )
2928adantr 272 . . . . . . 7  |-  ( ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  CC )  ->  ( x  +  y )  =  0 )
3022, 29eqeq12d 2130 . . . . . 6  |-  ( ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  CC )  ->  ( ( x  +  ( b  +  c ) )  =  ( x  +  y )  <-> 
( ( b  +  x )  +  c )  =  0 ) )
31 simplr 502 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  x  e.  RR )
3215adantlr 466 . . . . . . . . 9  |-  ( ( ( b  e.  CC  /\  y  e.  CC )  /\  c  e.  CC )  ->  ( b  +  c )  e.  CC )
3332adantlr 466 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  (
b  +  c )  e.  CC )
34 simpllr 506 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  y  e.  CC )
35 cnegexlem1 7901 . . . . . . . 8  |-  ( ( x  e.  RR  /\  ( b  +  c )  e.  CC  /\  y  e.  CC )  ->  ( ( x  +  ( b  +  c ) )  =  ( x  +  y )  <-> 
( b  +  c )  =  y ) )
3631, 33, 34, 35syl3anc 1199 . . . . . . 7  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  (
( x  +  ( b  +  c ) )  =  ( x  +  y )  <->  ( b  +  c )  =  y ) )
3736adantlr 466 . . . . . 6  |-  ( ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  CC )  ->  ( ( x  +  ( b  +  c ) )  =  ( x  +  y )  <-> 
( b  +  c )  =  y ) )
3830, 37bitr3d 189 . . . . 5  |-  ( ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  CC )  ->  ( ( ( b  +  x )  +  c )  =  0  <-> 
( b  +  c )  =  y ) )
3910, 11, 38syl2an 285 . . . 4  |-  ( ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  RR )  ->  ( ( ( b  +  x )  +  c )  =  0  <-> 
( b  +  c )  =  y ) )
4039rexbidva 2409 . . 3  |-  ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  -> 
( E. c  e.  RR  ( ( b  +  x )  +  c )  =  0  <->  E. c  e.  RR  ( b  +  c )  =  y ) )
415, 40mpbid 146 . 2  |-  ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  ->  E. c  e.  RR  ( b  +  c )  =  y )
42 ax-rnegex 7693 . . 3  |-  ( y  e.  RR  ->  E. x  e.  RR  ( y  +  x )  =  0 )
4342adantl 273 . 2  |-  ( ( b  e.  RR  /\  y  e.  RR )  ->  E. x  e.  RR  ( y  +  x
)  =  0 )
4441, 43r19.29a 2550 1  |-  ( ( b  e.  RR  /\  y  e.  RR )  ->  E. c  e.  RR  ( b  +  c )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392  (class class class)co 5740   CCcc 7582   RRcr 7583   0cc0 7584    + caddc 7587
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-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-resscn 7676  ax-1cn 7677  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743
This theorem is referenced by:  cnegex  7904
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