MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axrnegex Unicode version

Theorem axrnegex 8752
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 8776. (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 8722 . . . . 5  |-  ( A  e.  RR  <->  ( ( 1st `  A )  e. 
R.  /\  A  =  <. ( 1st `  A
) ,  0R >. ) )
21simplbi 448 . . . 4  |-  ( A  e.  RR  ->  ( 1st `  A )  e. 
R. )
3 m1r 8672 . . . 4  |-  -1R  e.  R.
4 mulclsr 8674 . . . 4  |-  ( ( ( 1st `  A
)  e.  R.  /\  -1R  e.  R. )  -> 
( ( 1st `  A
)  .R  -1R )  e.  R. )
52, 3, 4sylancl 646 . . 3  |-  ( A  e.  RR  ->  (
( 1st `  A
)  .R  -1R )  e.  R. )
6 opelreal 8720 . . 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 5807 . . 3  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  (
<. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
10 addresr 8728 . . . 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 645 . . 3  |-  ( A  e.  RR  ->  ( <. ( 1st `  A
) ,  0R >.  + 
<. ( ( 1st `  A
)  .R  -1R ) ,  0R >. )  =  <. ( ( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >. )
12 pn0sr 8691 . . . . . 6  |-  ( ( 1st `  A )  e.  R.  ->  (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) )  =  0R )
1312opeq1d 3776 . . . . 5  |-  ( ( 1st `  A )  e.  R.  ->  <. (
( 1st `  A
)  +R  ( ( 1st `  A )  .R  -1R ) ) ,  0R >.  =  <. 0R ,  0R >. )
14 df-0 8712 . . . . 5  |-  0  =  <. 0R ,  0R >.
1513, 14syl6eqr 2308 . . . 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 2294 . 2  |-  ( A  e.  RR  ->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 )
18 oveq2 5800 . . . 4  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  ( A  +  x )  =  ( A  +  <. ( ( 1st `  A
)  .R  -1R ) ,  0R >. ) )
1918eqeq1d 2266 . . 3  |-  ( x  =  <. ( ( 1st `  A )  .R  -1R ) ,  0R >.  ->  (
( A  +  x
)  =  0  <->  ( A  +  <. ( ( 1st `  A )  .R  -1R ) ,  0R >. )  =  0 ) )
2019rcla4ev 2859 . 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 645 1  |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   E.wrex 2519   <.cop 3617   ` cfv 4673  (class class class)co 5792   1stc1st 6054   R.cnr 8457   0Rc0r 8458   -1Rcm1r 8460    +R cplr 8461    .R cmr 8462   RRcr 8704   0cc0 8705    + caddc 8708
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ec 6630  df-qs 6634  df-ni 8464  df-pli 8465  df-mi 8466  df-lti 8467  df-plpq 8500  df-mpq 8501  df-ltpq 8502  df-enq 8503  df-nq 8504  df-erq 8505  df-plq 8506  df-mq 8507  df-1nq 8508  df-rq 8509  df-ltnq 8510  df-np 8573  df-1p 8574  df-plp 8575  df-mp 8576  df-ltp 8577  df-plpr 8647  df-mpr 8648  df-enr 8649  df-nr 8650  df-plr 8651  df-mr 8652  df-0r 8654  df-1r 8655  df-m1r 8656  df-c 8711  df-0 8712  df-r 8715  df-plus 8716
  Copyright terms: Public domain W3C validator