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Theorem axrnegex 8784
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 8808. (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 8754 . . . . 5  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
21simplbi 446 . . . 4  |-  ( A  e.  RR  ->  ( 1st `  A )  e. 
R. )
3 m1r 8704 . . . 4  |-  -1R  e.  R.
4 mulclsr 8706 . . . 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 8752 . . 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 5873 . . 3  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  (
<. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
10 addresr 8760 . . . 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 8723 . . . . . 6  |-  ( ( 1st `  A )  e.  R.  ->  (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) )  =  0R )
1312opeq1d 3802 . . . . 5  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  <. 0R ,  0R >. )
14 df-0 8744 . . . . 5  |-  0  =  <. 0R ,  0R >.
1513, 14syl6eqr 2333 . . . 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 2319 . 2  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 )
18 oveq2 5866 . . . 4  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  ( A  +  x )  =  ( A  +  <. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
1918eqeq1d 2291 . . 3  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  (
( A  +  x
)  =  0  <->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 ) )
2019rspcev 2884 . 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 1623    e. wcel 1684   E.wrex 2544   <.cop 3643   ` cfv 5255  (class class class)co 5858   1stc1st 6120   R.cnr 8489   0Rc0r 8490   -1Rcm1r 8492    +R cplr 8493    .R cmr 8494   RRcr 8736   0cc0 8737    + caddc 8740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-1p 8606  df-plp 8607  df-mp 8608  df-ltp 8609  df-plpr 8679  df-mpr 8680  df-enr 8681  df-nr 8682  df-plr 8683  df-mr 8684  df-0r 8686  df-1r 8687  df-m1r 8688  df-c 8743  df-0 8744  df-r 8747  df-add 8748
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