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Theorem 00id 9755
Description:  0 is its own additive identity. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
00id  |-  ( 0  +  0 )  =  0

Proof of Theorem 00id
Dummy variables  y 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0re 9597 . 2  |-  0  e.  RR
2 ax-rnegex 9564 . 2  |-  ( 0  e.  RR  ->  E. c  e.  RR  ( 0  +  c )  =  0 )
3 oveq2 6293 . . . . . . 7  |-  ( c  =  0  ->  (
0  +  c )  =  ( 0  +  0 ) )
43eqeq1d 2469 . . . . . 6  |-  ( c  =  0  ->  (
( 0  +  c )  =  0  <->  (
0  +  0 )  =  0 ) )
54biimpd 207 . . . . 5  |-  ( c  =  0  ->  (
( 0  +  c )  =  0  -> 
( 0  +  0 )  =  0 ) )
65adantld 467 . . . 4  |-  ( c  =  0  ->  (
( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 ) )
7 ax-rrecex 9565 . . . . . . 7  |-  ( ( c  e.  RR  /\  c  =/=  0 )  ->  E. y  e.  RR  ( c  x.  y
)  =  1 )
87adantlr 714 . . . . . 6  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  E. y  e.  RR  ( c  x.  y )  =  1 )
9 simplll 757 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  c  e.  RR )
109recnd 9623 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  c  e.  CC )
11 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  y  e.  RR )
1211recnd 9623 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  y  e.  CC )
13 0cn 9589 . . . . . . . . . . 11  |-  0  e.  CC
14 mulass 9581 . . . . . . . . . . 11  |-  ( ( c  e.  CC  /\  y  e.  CC  /\  0  e.  CC )  ->  (
( c  x.  y
)  x.  0 )  =  ( c  x.  ( y  x.  0 ) ) )
1513, 14mp3an3 1313 . . . . . . . . . 10  |-  ( ( c  e.  CC  /\  y  e.  CC )  ->  ( ( c  x.  y )  x.  0 )  =  ( c  x.  ( y  x.  0 ) ) )
1610, 12, 15syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  y )  x.  0 )  =  ( c  x.  (
y  x.  0 ) ) )
17 oveq1 6292 . . . . . . . . . . 11  |-  ( ( c  x.  y )  =  1  ->  (
( c  x.  y
)  x.  0 )  =  ( 1  x.  0 ) )
1813mulid2i 9600 . . . . . . . . . . 11  |-  ( 1  x.  0 )  =  0
1917, 18syl6eq 2524 . . . . . . . . . 10  |-  ( ( c  x.  y )  =  1  ->  (
( c  x.  y
)  x.  0 )  =  0 )
2019ad2antll 728 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  y )  x.  0 )  =  0 )
2116, 20eqtr3d 2510 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  =  0 )
2221oveq1d 6300 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  ( y  x.  0 ) )  +  0 )  =  ( 0  +  0 ) )
23 simpllr 758 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  +  c )  =  0 )
2423oveq1d 6300 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
0  +  c )  x.  ( y  x.  0 ) )  =  ( 0  x.  (
y  x.  0 ) ) )
25 remulcl 9578 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR  /\  0  e.  RR )  ->  ( y  x.  0 )  e.  RR )
261, 25mpan2 671 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  (
y  x.  0 )  e.  RR )
2726ad2antrl 727 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( y  x.  0 )  e.  RR )
2827recnd 9623 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( y  x.  0 )  e.  CC )
29 adddir 9588 . . . . . . . . . . . . 13  |-  ( ( 0  e.  CC  /\  c  e.  CC  /\  (
y  x.  0 )  e.  CC )  -> 
( ( 0  +  c )  x.  (
y  x.  0 ) )  =  ( ( 0  x.  ( y  x.  0 ) )  +  ( c  x.  ( y  x.  0 ) ) ) )
3013, 29mp3an1 1311 . . . . . . . . . . . 12  |-  ( ( c  e.  CC  /\  ( y  x.  0 )  e.  CC )  ->  ( ( 0  +  c )  x.  ( y  x.  0 ) )  =  ( ( 0  x.  (
y  x.  0 ) )  +  ( c  x.  ( y  x.  0 ) ) ) )
3110, 28, 30syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
0  +  c )  x.  ( y  x.  0 ) )  =  ( ( 0  x.  ( y  x.  0 ) )  +  ( c  x.  ( y  x.  0 ) ) ) )
3224, 31eqtr3d 2510 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  x.  ( y  x.  0 ) )  =  ( ( 0  x.  ( y  x.  0 ) )  +  ( c  x.  ( y  x.  0 ) ) ) )
3332oveq1d 6300 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
0  x.  ( y  x.  0 ) )  +  0 )  =  ( ( ( 0  x.  ( y  x.  0 ) )  +  ( c  x.  (
y  x.  0 ) ) )  +  0 ) )
34 remulcl 9578 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( y  x.  0 )  e.  RR )  ->  ( 0  x.  ( y  x.  0 ) )  e.  RR )
351, 26, 34sylancr 663 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  (
0  x.  ( y  x.  0 ) )  e.  RR )
3635ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  x.  ( y  x.  0 ) )  e.  RR )
3736recnd 9623 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  x.  ( y  x.  0 ) )  e.  CC )
38 remulcl 9578 . . . . . . . . . . . 12  |-  ( ( c  e.  RR  /\  ( y  x.  0 )  e.  RR )  ->  ( c  x.  ( y  x.  0 ) )  e.  RR )
399, 27, 38syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  e.  RR )
4039recnd 9623 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  e.  CC )
41 addass 9580 . . . . . . . . . . 11  |-  ( ( ( 0  x.  (
y  x.  0 ) )  e.  CC  /\  ( c  x.  (
y  x.  0 ) )  e.  CC  /\  0  e.  CC )  ->  ( ( ( 0  x.  ( y  x.  0 ) )  +  ( c  x.  (
y  x.  0 ) ) )  +  0 )  =  ( ( 0  x.  ( y  x.  0 ) )  +  ( ( c  x.  ( y  x.  0 ) )  +  0 ) ) )
4213, 41mp3an3 1313 . . . . . . . . . 10  |-  ( ( ( 0  x.  (
y  x.  0 ) )  e.  CC  /\  ( c  x.  (
y  x.  0 ) )  e.  CC )  ->  ( ( ( 0  x.  ( y  x.  0 ) )  +  ( c  x.  ( y  x.  0 ) ) )  +  0 )  =  ( ( 0  x.  (
y  x.  0 ) )  +  ( ( c  x.  ( y  x.  0 ) )  +  0 ) ) )
4337, 40, 42syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
( 0  x.  (
y  x.  0 ) )  +  ( c  x.  ( y  x.  0 ) ) )  +  0 )  =  ( ( 0  x.  ( y  x.  0 ) )  +  ( ( c  x.  (
y  x.  0 ) )  +  0 ) ) )
4433, 43eqtr2d 2509 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
0  x.  ( y  x.  0 ) )  +  ( ( c  x.  ( y  x.  0 ) )  +  0 ) )  =  ( ( 0  x.  ( y  x.  0 ) )  +  0 ) )
4526, 38sylan2 474 . . . . . . . . . . 11  |-  ( ( c  e.  RR  /\  y  e.  RR )  ->  ( c  x.  (
y  x.  0 ) )  e.  RR )
46 readdcl 9576 . . . . . . . . . . 11  |-  ( ( ( c  x.  (
y  x.  0 ) )  e.  RR  /\  0  e.  RR )  ->  ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR )
4745, 1, 46sylancl 662 . . . . . . . . . 10  |-  ( ( c  e.  RR  /\  y  e.  RR )  ->  ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR )
489, 11, 47syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  ( y  x.  0 ) )  +  0 )  e.  RR )
49 readdcan 9754 . . . . . . . . . 10  |-  ( ( ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR  /\  0  e.  RR  /\  (
0  x.  ( y  x.  0 ) )  e.  RR )  -> 
( ( ( 0  x.  ( y  x.  0 ) )  +  ( ( c  x.  ( y  x.  0 ) )  +  0 ) )  =  ( ( 0  x.  (
y  x.  0 ) )  +  0 )  <-> 
( ( c  x.  ( y  x.  0 ) )  +  0 )  =  0 ) )
501, 49mp3an2 1312 . . . . . . . . 9  |-  ( ( ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR  /\  ( 0  x.  (
y  x.  0 ) )  e.  RR )  ->  ( ( ( 0  x.  ( y  x.  0 ) )  +  ( ( c  x.  ( y  x.  0 ) )  +  0 ) )  =  ( ( 0  x.  ( y  x.  0 ) )  +  0 )  <->  ( ( c  x.  ( y  x.  0 ) )  +  0 )  =  0 ) )
5148, 36, 50syl2anc 661 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
( 0  x.  (
y  x.  0 ) )  +  ( ( c  x.  ( y  x.  0 ) )  +  0 ) )  =  ( ( 0  x.  ( y  x.  0 ) )  +  0 )  <->  ( (
c  x.  ( y  x.  0 ) )  +  0 )  =  0 ) )
5244, 51mpbid 210 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  ( y  x.  0 ) )  +  0 )  =  0 )
5322, 52eqtr3d 2510 . . . . . 6  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  +  0 )  =  0 )
548, 53rexlimddv 2959 . . . . 5  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  ( 0  +  0 )  =  0 )
5554expcom 435 . . . 4  |-  ( c  =/=  0  ->  (
( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 ) )
566, 55pm2.61ine 2780 . . 3  |-  ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 )
5756rexlimiva 2951 . 2  |-  ( E. c  e.  RR  (
0  +  c )  =  0  ->  (
0  +  0 )  =  0 )
581, 2, 57mp2b 10 1  |-  ( 0  +  0 )  =  0
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815  (class class class)co 6285   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569
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-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-ltxr 9634
This theorem is referenced by:  mul02lem1  9756  mul02lem2  9757  addid1  9760  addid2  9763  negdii  9904  addgt0  10039  addgegt0  10040  addgtge0  10041  addge0  10042  add20  10065  recextlem2  10181  crne0  10530  10p10e20  11047  ser0  12128  faclbnd4lem3  12342  bcpasc  12368  fsumadd  13527  fsumrelem  13587  arisum  13637  sadcaddlem  13969  sadcadd  13970  sadadd2  13972  bezout  14042  nnnn0modprm0  14193  pcaddlem  14269  4sqlem19  14343  37prm  14467  139prm  14470  163prm  14471  317prm  14472  631prm  14473  1259lem1  14474  1259lem2  14475  1259lem3  14476  1259lem4  14477  2503lem1  14480  2503lem2  14481  2503lem3  14482  4001lem1  14484  4001lem2  14485  4001lem3  14486  4001lem4  14487  sylow1lem1  16433  psrbagaddcl  17831  psrbagaddclOLD  17832  mplcoe3  17939  mplcoe3OLD  17940  cnfld0  18253  reparphti  21324  itg1addlem4  21933  ibladdlem  22053  itgaddlem1  22056  iblabslem  22061  iblabs  22062  coeaddlem  22472  dcubic  23002  log2ublem3  23104  log2ub  23105  chtublem  23311  logfacrlim  23324  dchrisumlem1  23499  chpdifbndlem2  23564  vdgr0  24673  vdgr1a  24679  1kp2ke3k  24941  dip0r  25403  pythi  25538  normpythi  25832  ocsh  25974  0lnfn  26677  lnopeq0i  26699  nlelshi  26752  unierri  26796  probun  28109  fsumcube  29675  ismblfin  29908  itg2addnc  29922  ibladdnclem  29924  itgaddnclem1  29926  itgaddnclem2  29927  iblabsnclem  29931  iblabsnc  29932  iblmulc2nc  29933  ftc1anclem8  29950  ftc1anc  29951  bezoutr1  30755  stoweidlem44  31571  fourierdlem42  31676  fourierdlem103  31737  fourierdlem104  31738  sqwvfoura  31756  sqwvfourb  31757
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