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Theorem axrnegex 8779
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 8803. (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 8749 . . . . 5  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
21simplbi 448 . . . 4  |-  ( A  e.  RR  ->  ( 1st `  A )  e. 
R. )
3 m1r 8699 . . . 4  |-  -1R  e.  R.
4 mulclsr 8701 . . . 4  |-  ( ( ( 1st `  A
)  e.  R.  /\  -1R  e.  R. )  -> 
( ( 1st `  A
)  .R  -1R )  e.  R. )
52, 3, 4sylancl 645 . . 3  |-  ( A  e.  RR  ->  (
( 1st `  A
)  .R  -1R )  e.  R. )
6 opelreal 8747 . . 3  |-  ( <.
( ( 1st `  A
)  .R  -1R ) ,  0R >.  e.  RR  <->  ( ( 1st `  A
)  .R  -1R )  e.  R. )
75, 6sylibr 205 . 2  |-  ( A  e.  RR  ->  <. (
( 1st `  A
)  .R  -1R ) ,  0R >.  e.  RR )
81simprbi 452 . . . 4  |-  ( A  e.  RR  ->  A  =  <. ( 1st `  A
) ,  0R >. )
98oveq1d 5834 . . 3  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  (
<. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
10 addresr 8755 . . . 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 644 . . 3  |-  ( A  e.  RR  ->  ( <. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. )  =  <. ( ( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >. )
12 pn0sr 8718 . . . . . 6  |-  ( ( 1st `  A )  e.  R.  ->  (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) )  =  0R )
1312opeq1d 3803 . . . . 5  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  <. 0R ,  0R >. )
14 df-0 8739 . . . . 5  |-  0  =  <. 0R ,  0R >.
1513, 14syl6eqr 2334 . . . 4  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  0 )
162, 15syl 17 . . 3  |-  ( A  e.  RR  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  0 )
179, 11, 163eqtrd 2320 . 2  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 )
18 oveq2 5827 . . . 4  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  ( A  +  x )  =  ( A  +  <. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
1918eqeq1d 2292 . . 3  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  (
( A  +  x
)  =  0  <->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 ) )
2019rspcev 2885 . 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 644 1  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   E.wrex 2545   <.cop 3644   ` cfv 5221  (class class class)co 5819   1stc1st 6081   R.cnr 8484   0Rc0r 8485   -1Rcm1r 8487    +R cplr 8488    .R cmr 8489   RRcr 8731   0cc0 8732    + caddc 8735
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ec 6657  df-qs 6661  df-ni 8491  df-pli 8492  df-mi 8493  df-lti 8494  df-plpq 8527  df-mpq 8528  df-ltpq 8529  df-enq 8530  df-nq 8531  df-erq 8532  df-plq 8533  df-mq 8534  df-1nq 8535  df-rq 8536  df-ltnq 8537  df-np 8600  df-1p 8601  df-plp 8602  df-mp 8603  df-ltp 8604  df-plpr 8674  df-mpr 8675  df-enr 8676  df-nr 8677  df-plr 8678  df-mr 8679  df-0r 8681  df-1r 8682  df-m1r 8683  df-c 8738  df-0 8739  df-r 8742  df-add 8743
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