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Theorem axrnegex 9551
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 9575. (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 9521 . . . . 5  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
21simplbi 460 . . . 4  |-  ( A  e.  RR  ->  ( 1st `  A )  e. 
R. )
3 m1r 9471 . . . 4  |-  -1R  e.  R.
4 mulclsr 9473 . . . 4  |-  ( ( ( 1st `  A
)  e.  R.  /\  -1R  e.  R. )  -> 
( ( 1st `  A
)  .R  -1R )  e.  R. )
52, 3, 4sylancl 662 . . 3  |-  ( A  e.  RR  ->  (
( 1st `  A
)  .R  -1R )  e.  R. )
6 opelreal 9519 . . 3  |-  ( <.
( ( 1st `  A
)  .R  -1R ) ,  0R >.  e.  RR  <->  ( ( 1st `  A
)  .R  -1R )  e.  R. )
75, 6sylibr 212 . 2  |-  ( A  e.  RR  ->  <. (
( 1st `  A
)  .R  -1R ) ,  0R >.  e.  RR )
81simprbi 464 . . . 4  |-  ( A  e.  RR  ->  A  =  <. ( 1st `  A
) ,  0R >. )
98oveq1d 6310 . . 3  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  (
<. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
10 addresr 9527 . . . 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 661 . . 3  |-  ( A  e.  RR  ->  ( <. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. )  =  <. ( ( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >. )
12 pn0sr 9490 . . . . . 6  |-  ( ( 1st `  A )  e.  R.  ->  (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) )  =  0R )
1312opeq1d 4225 . . . . 5  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  <. 0R ,  0R >. )
14 df-0 9511 . . . . 5  |-  0  =  <. 0R ,  0R >.
1513, 14syl6eqr 2526 . . . 4  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  0 )
162, 15syl 16 . . 3  |-  ( A  e.  RR  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  0 )
179, 11, 163eqtrd 2512 . 2  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 )
18 oveq2 6303 . . . 4  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  ( A  +  x )  =  ( A  +  <. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
1918eqeq1d 2469 . . 3  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  (
( A  +  x
)  =  0  <->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 ) )
2019rspcev 3219 . 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 661 1  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   E.wrex 2818   <.cop 4039   ` cfv 5594  (class class class)co 6295   1stc1st 6793   R.cnr 9255   0Rc0r 9256   -1Rcm1r 9258    +R cplr 9259    .R cmr 9260   RRcr 9503   0cc0 9504    + caddc 9507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-omul 7147  df-er 7323  df-ec 7325  df-qs 7329  df-ni 9262  df-pli 9263  df-mi 9264  df-lti 9265  df-plpq 9298  df-mpq 9299  df-ltpq 9300  df-enq 9301  df-nq 9302  df-erq 9303  df-plq 9304  df-mq 9305  df-1nq 9306  df-rq 9307  df-ltnq 9308  df-np 9371  df-1p 9372  df-plp 9373  df-mp 9374  df-ltp 9375  df-enr 9445  df-nr 9446  df-plr 9447  df-mr 9448  df-0r 9450  df-1r 9451  df-m1r 9452  df-c 9510  df-0 9511  df-r 9514  df-add 9515
This theorem is referenced by: (None)
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