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Theorem cnegexlem3 7352
Description: Existence of real number difference. Lemma for cnegex 7353. (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 7161 . . . . . 6  |-  ( ( b  e.  RR  /\  x  e.  RR )  ->  ( b  +  x
)  e.  RR )
2 ax-rnegex 7147 . . . . . 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 461 . . . 4  |-  ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  ->  E. c  e.  RR  ( ( b  +  x )  +  c )  =  0 )
54adantr 270 . . 3  |-  ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  ->  E. c  e.  RR  ( ( b  +  x )  +  c )  =  0 )
6 recn 7168 . . . . . . . 8  |-  ( b  e.  RR  ->  b  e.  CC )
7 recn 7168 . . . . . . . 8  |-  ( y  e.  RR  ->  y  e.  CC )
86, 7anim12i 331 . . . . . . 7  |-  ( ( b  e.  RR  /\  y  e.  RR )  ->  ( b  e.  CC  /\  y  e.  CC ) )
98anim1i 333 . . . . . 6  |-  ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  ->  ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR ) )
109anim1i 333 . . . . 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 7168 . . . . 5  |-  ( c  e.  RR  ->  c  e.  CC )
12 recn 7168 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
13 add32 7334 . . . . . . . . . . . 12  |-  ( ( b  e.  CC  /\  x  e.  CC  /\  c  e.  CC )  ->  (
( b  +  x
)  +  c )  =  ( ( b  +  c )  +  x ) )
14133expa 1139 . . . . . . . . . . 11  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( ( b  +  x )  +  c )  =  ( ( b  +  c )  +  x ) )
15 addcl 7160 . . . . . . . . . . . . 13  |-  ( ( b  e.  CC  /\  c  e.  CC )  ->  ( b  +  c )  e.  CC )
16 addcom 7312 . . . . . . . . . . . . 13  |-  ( ( ( b  +  c )  e.  CC  /\  x  e.  CC )  ->  ( ( b  +  c )  +  x
)  =  ( x  +  ( b  +  c ) ) )
1715, 16sylan 277 . . . . . . . . . . . 12  |-  ( ( ( b  e.  CC  /\  c  e.  CC )  /\  x  e.  CC )  ->  ( ( b  +  c )  +  x )  =  ( x  +  ( b  +  c ) ) )
1817an32s 533 . . . . . . . . . . 11  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( ( b  +  c )  +  x )  =  ( x  +  ( b  +  c ) ) )
1914, 18eqtr2d 2115 . . . . . . . . . 10  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( x  +  ( b  +  c ) )  =  ( ( b  +  x
)  +  c ) )
2012, 19sylanl2 395 . . . . . . . . 9  |-  ( ( ( b  e.  CC  /\  x  e.  RR )  /\  c  e.  CC )  ->  ( x  +  ( b  +  c ) )  =  ( ( b  +  x
)  +  c ) )
2120adantllr 465 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  (
x  +  ( b  +  c ) )  =  ( ( b  +  x )  +  c ) )
2221adantlr 461 . . . . . . 7  |-  ( ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  CC )  ->  ( x  +  ( b  +  c ) )  =  ( ( b  +  x )  +  c ) )
23 addcom 7312 . . . . . . . . . . . 12  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
2423ancoms 264 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  x  e.  CC )  ->  ( x  +  y )  =  ( y  +  x ) )
2512, 24sylan2 280 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  x  e.  RR )  ->  ( x  +  y )  =  ( y  +  x ) )
26 id 19 . . . . . . . . . 10  |-  ( ( y  +  x )  =  0  ->  (
y  +  x )  =  0 )
2725, 26sylan9eq 2134 . . . . . . . . 9  |-  ( ( ( y  e.  CC  /\  x  e.  RR )  /\  ( y  +  x )  =  0 )  ->  ( x  +  y )  =  0 )
2827adantlll 464 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  -> 
( x  +  y )  =  0 )
2928adantr 270 . . . . . . 7  |-  ( ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  CC )  ->  ( x  +  y )  =  0 )
3022, 29eqeq12d 2096 . . . . . 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 497 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  x  e.  RR )
3215adantlr 461 . . . . . . . . 9  |-  ( ( ( b  e.  CC  /\  y  e.  CC )  /\  c  e.  CC )  ->  ( b  +  c )  e.  CC )
3332adantlr 461 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  (
b  +  c )  e.  CC )
34 simpllr 501 . . . . . . . 8  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  y  e.  CC )
35 cnegexlem1 7350 . . . . . . . 8  |-  ( ( x  e.  RR  /\  ( b  +  c )  e.  CC  /\  y  e.  CC )  ->  ( ( x  +  ( b  +  c ) )  =  ( x  +  y )  <-> 
( b  +  c )  =  y ) )
3631, 33, 34, 35syl3anc 1170 . . . . . . 7  |-  ( ( ( ( b  e.  CC  /\  y  e.  CC )  /\  x  e.  RR )  /\  c  e.  CC )  ->  (
( x  +  ( b  +  c ) )  =  ( x  +  y )  <->  ( b  +  c )  =  y ) )
3736adantlr 461 . . . . . 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 188 . . . . 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 283 . . . 4  |-  ( ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  /\  c  e.  RR )  ->  ( ( ( b  +  x )  +  c )  =  0  <-> 
( b  +  c )  =  y ) )
4039rexbidva 2366 . . 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 145 . 2  |-  ( ( ( ( b  e.  RR  /\  y  e.  RR )  /\  x  e.  RR )  /\  (
y  +  x )  =  0 )  ->  E. c  e.  RR  ( b  +  c )  =  y )
42 ax-rnegex 7147 . . 3  |-  ( y  e.  RR  ->  E. x  e.  RR  ( y  +  x )  =  0 )
4342adantl 271 . 2  |-  ( ( b  e.  RR  /\  y  e.  RR )  ->  E. x  e.  RR  ( y  +  x
)  =  0 )
4441, 43r19.29a 2499 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2350  (class class class)co 5543   CCcc 7041   RRcr 7042   0cc0 7043    + caddc 7046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  cnegex  7353
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