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Theorem axrnegex 8800
Description: Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 8824. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
axrnegex  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
Distinct variable group:    x, A

Proof of Theorem axrnegex
StepHypRef Expression
1 elreal2 8770 . . . . 5  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
21simplbi 446 . . . 4  |-  ( A  e.  RR  ->  ( 1st `  A )  e. 
R. )
3 m1r 8720 . . . 4  |-  -1R  e.  R.
4 mulclsr 8722 . . . 4  |-  ( ( ( 1st `  A
)  e.  R.  /\  -1R  e.  R. )  -> 
( ( 1st `  A
)  .R  -1R )  e.  R. )
52, 3, 4sylancl 643 . . 3  |-  ( A  e.  RR  ->  (
( 1st `  A
)  .R  -1R )  e.  R. )
6 opelreal 8768 . . 3  |-  ( <.
( ( 1st `  A
)  .R  -1R ) ,  0R >.  e.  RR  <->  ( ( 1st `  A
)  .R  -1R )  e.  R. )
75, 6sylibr 203 . 2  |-  ( A  e.  RR  ->  <. (
( 1st `  A
)  .R  -1R ) ,  0R >.  e.  RR )
81simprbi 450 . . . 4  |-  ( A  e.  RR  ->  A  =  <. ( 1st `  A
) ,  0R >. )
98oveq1d 5889 . . 3  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  (
<. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
10 addresr 8776 . . . 4  |-  ( ( ( 1st `  A
)  e.  R.  /\  ( ( 1st `  A
)  .R  -1R )  e.  R. )  ->  ( <. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. )  =  <. ( ( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >. )
112, 5, 10syl2anc 642 . . 3  |-  ( A  e.  RR  ->  ( <. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. )  =  <. ( ( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >. )
12 pn0sr 8739 . . . . . 6  |-  ( ( 1st `  A )  e.  R.  ->  (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) )  =  0R )
1312opeq1d 3818 . . . . 5  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  <. 0R ,  0R >. )
14 df-0 8760 . . . . 5  |-  0  =  <. 0R ,  0R >.
1513, 14syl6eqr 2346 . . . 4  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  0 )
162, 15syl 15 . . 3  |-  ( A  e.  RR  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  0 )
179, 11, 163eqtrd 2332 . 2  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 )
18 oveq2 5882 . . . 4  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  ( A  +  x )  =  ( A  +  <. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
1918eqeq1d 2304 . . 3  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  (
( A  +  x
)  =  0  <->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 ) )
2019rspcev 2897 . 2  |-  ( (
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >.  e.  RR  /\  ( A  +  <. ( ( 1st `  A
)  .R  -1R ) ,  0R >. )  =  0 )  ->  E. x  e.  RR  ( A  +  x )  =  0 )
217, 17, 20syl2anc 642 1  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   E.wrex 2557   <.cop 3656   ` cfv 5271  (class class class)co 5874   1stc1st 6136   R.cnr 8505   0Rc0r 8506   -1Rcm1r 8508    +R cplr 8509    .R cmr 8510   RRcr 8752   0cc0 8753    + caddc 8756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-ltpq 8550  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557  df-ltnq 8558  df-np 8621  df-1p 8622  df-plp 8623  df-mp 8624  df-ltp 8625  df-plpr 8695  df-mpr 8696  df-enr 8697  df-nr 8698  df-plr 8699  df-mr 8700  df-0r 8702  df-1r 8703  df-m1r 8704  df-c 8759  df-0 8760  df-r 8763  df-add 8764
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