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Theorem cnegexlem3 7638
Description: Existence of real number difference. Lemma for cnegex 7639. (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 7447 . . . . . 6  |-  ( ( b  e.  RR  /\  x  e.  RR )  ->  ( b  +  x
)  e.  RR )
2 ax-rnegex 7433 . . . . . 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 7454 . . . . . . . 8  |-  ( b  e.  RR  ->  b  e.  CC )
7 recn 7454 . . . . . . . 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 7454 . . . . 5  |-  ( c  e.  RR  ->  c  e.  CC )
12 recn 7454 . . . . . . . . . 10  |-  ( x  e.  RR  ->  x  e.  CC )
13 add32 7620 . . . . . . . . . . . 12  |-  ( ( b  e.  CC  /\  x  e.  CC  /\  c  e.  CC )  ->  (
( b  +  x
)  +  c )  =  ( ( b  +  c )  +  x ) )
14133expa 1143 . . . . . . . . . . 11  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( ( b  +  x )  +  c )  =  ( ( b  +  c )  +  x ) )
15 addcl 7446 . . . . . . . . . . . . 13  |-  ( ( b  e.  CC  /\  c  e.  CC )  ->  ( b  +  c )  e.  CC )
16 addcom 7598 . . . . . . . . . . . . 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 535 . . . . . . . . . . 11  |-  ( ( ( b  e.  CC  /\  x  e.  CC )  /\  c  e.  CC )  ->  ( ( b  +  c )  +  x )  =  ( x  +  ( b  +  c ) ) )
1914, 18eqtr2d 2121 . . . . . . . . . 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 7598 . . . . . . . . . . . 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 2140 . . . . . . . . 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 2102 . . . . . 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 7636 . . . . . . . 8  |-  ( ( x  e.  RR  /\  ( b  +  c )  e.  CC  /\  y  e.  CC )  ->  ( ( x  +  ( b  +  c ) )  =  ( x  +  y )  <-> 
( b  +  c )  =  y ) )
3631, 33, 34, 35syl3anc 1174 . . . . . . 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 2377 . . 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 7433 . . 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 2511 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 1289    e. wcel 1438   E.wrex 2360  (class class class)co 5634   CCcc 7327   RRcr 7328   0cc0 7329    + caddc 7332
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-resscn 7416  ax-1cn 7417  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-i2m1 7429  ax-0id 7432  ax-rnegex 7433
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637
This theorem is referenced by:  cnegex  7639
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