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Theorem 00id 9826
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 9661 . 2  |-  0  e.  RR
2 ax-rnegex 9628 . 2  |-  ( 0  e.  RR  ->  E. c  e.  RR  ( 0  +  c )  =  0 )
3 oveq2 6316 . . . . . . 7  |-  ( c  =  0  ->  (
0  +  c )  =  ( 0  +  0 ) )
43eqeq1d 2473 . . . . . 6  |-  ( c  =  0  ->  (
( 0  +  c )  =  0  <->  (
0  +  0 )  =  0 ) )
54biimpd 212 . . . . 5  |-  ( c  =  0  ->  (
( 0  +  c )  =  0  -> 
( 0  +  0 )  =  0 ) )
65adantld 474 . . . 4  |-  ( c  =  0  ->  (
( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 ) )
7 ax-rrecex 9629 . . . . . . 7  |-  ( ( c  e.  RR  /\  c  =/=  0 )  ->  E. y  e.  RR  ( c  x.  y
)  =  1 )
87adantlr 729 . . . . . 6  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  E. y  e.  RR  ( c  x.  y )  =  1 )
9 simplll 776 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  c  e.  RR )
109recnd 9687 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  c  e.  CC )
11 simprl 772 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  y  e.  RR )
1211recnd 9687 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  y  e.  CC )
13 0cn 9653 . . . . . . . . . . 11  |-  0  e.  CC
14 mulass 9645 . . . . . . . . . . 11  |-  ( ( c  e.  CC  /\  y  e.  CC  /\  0  e.  CC )  ->  (
( c  x.  y
)  x.  0 )  =  ( c  x.  ( y  x.  0 ) ) )
1513, 14mp3an3 1379 . . . . . . . . . 10  |-  ( ( c  e.  CC  /\  y  e.  CC )  ->  ( ( c  x.  y )  x.  0 )  =  ( c  x.  ( y  x.  0 ) ) )
1610, 12, 15syl2anc 673 . . . . . . . . 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 6315 . . . . . . . . . . 11  |-  ( ( c  x.  y )  =  1  ->  (
( c  x.  y
)  x.  0 )  =  ( 1  x.  0 ) )
1813mulid2i 9664 . . . . . . . . . . 11  |-  ( 1  x.  0 )  =  0
1917, 18syl6eq 2521 . . . . . . . . . 10  |-  ( ( c  x.  y )  =  1  ->  (
( c  x.  y
)  x.  0 )  =  0 )
2019ad2antll 743 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  y )  x.  0 )  =  0 )
2116, 20eqtr3d 2507 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  =  0 )
2221oveq1d 6323 . . . . . . 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 777 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  +  c )  =  0 )
2423oveq1d 6323 . . . . . . . . . . 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 9642 . . . . . . . . . . . . . . 15  |-  ( ( y  e.  RR  /\  0  e.  RR )  ->  ( y  x.  0 )  e.  RR )
261, 25mpan2 685 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  (
y  x.  0 )  e.  RR )
2726ad2antrl 742 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( y  x.  0 )  e.  RR )
2827recnd 9687 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( y  x.  0 )  e.  CC )
29 adddir 9652 . . . . . . . . . . . . 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 1377 . . . . . . . . . . . 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 673 . . . . . . . . . . 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 2507 . . . . . . . . . 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 6323 . . . . . . . . 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 9642 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( y  x.  0 )  e.  RR )  ->  ( 0  x.  ( y  x.  0 ) )  e.  RR )
351, 26, 34sylancr 676 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  (
0  x.  ( y  x.  0 ) )  e.  RR )
3635ad2antrl 742 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  x.  ( y  x.  0 ) )  e.  RR )
3736recnd 9687 . . . . . . . . . 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 9642 . . . . . . . . . . . 12  |-  ( ( c  e.  RR  /\  ( y  x.  0 )  e.  RR )  ->  ( c  x.  ( y  x.  0 ) )  e.  RR )
399, 27, 38syl2anc 673 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  e.  RR )
4039recnd 9687 . . . . . . . . . 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 9644 . . . . . . . . . . 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 1379 . . . . . . . . . 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 673 . . . . . . . . 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 2506 . . . . . . . 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 482 . . . . . . . . . . 11  |-  ( ( c  e.  RR  /\  y  e.  RR )  ->  ( c  x.  (
y  x.  0 ) )  e.  RR )
46 readdcl 9640 . . . . . . . . . . 11  |-  ( ( ( c  x.  (
y  x.  0 ) )  e.  RR  /\  0  e.  RR )  ->  ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR )
4745, 1, 46sylancl 675 . . . . . . . . . 10  |-  ( ( c  e.  RR  /\  y  e.  RR )  ->  ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR )
489, 11, 47syl2anc 673 . . . . . . . . 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 9825 . . . . . . . . . 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 1378 . . . . . . . . 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 673 . . . . . . . 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 215 . . . . . . 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 2507 . . . . . 6  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  +  0 )  =  0 )
548, 53rexlimddv 2875 . . . . 5  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  ( 0  +  0 )  =  0 )
5554expcom 442 . . . 4  |-  ( c  =/=  0  ->  (
( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 ) )
566, 55pm2.61ine 2726 . . 3  |-  ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 )
5756rexlimiva 2868 . 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 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-ltxr 9698
This theorem is referenced by:  mul02lem1  9827  mul02lem2  9828  addid1  9831  addid2  9834  addgt0  10121  addgegt0  10122  addgtge0  10123  addge0  10124  add20  10147  recextlem2  10265  crne0  10624  10p10e20  11144  ser0  12303  faclbnd4lem3  12518  bcpasc  12544  relexpaddg  13193  fsumadd  13882  fsumrelem  13944  arisum  13995  fsumcube  14190  sadcaddlem  14510  sadcadd  14511  sadadd2  14513  bezout  14589  nnnn0modprm0  14836  pcaddlem  14912  4sqlem19  14992  37prm  15170  139prm  15173  163prm  15174  317prm  15175  631prm  15176  1259lem1  15180  1259lem2  15181  1259lem3  15182  1259lem4  15183  2503lem1  15186  2503lem2  15187  2503lem3  15188  4001lem1  15190  4001lem2  15191  4001lem3  15192  4001lem4  15193  sylow1lem1  17328  psrbagaddcl  18671  mplcoe3  18767  cnfld0  19069  reparphti  22106  itg1addlem4  22736  ibladdlem  22856  itgaddlem1  22859  iblabslem  22864  iblabs  22865  coeaddlem  23282  dcubic  23851  log2ublem3  23953  log2ub  23954  chtublem  24218  logfacrlim  24231  dchrisumlem1  24406  chpdifbndlem2  24471  vdgr0  25707  vdgr1a  25713  1kp2ke3k  25975  dip0r  26437  pythi  26572  normpythi  26876  ocsh  27017  0lnfn  27719  lnopeq0i  27741  nlelshi  27794  unierri  27838  probun  29325  poimirlem3  32007  poimirlem4  32008  ismblfin  32045  itg2addnc  32060  ibladdnclem  32062  itgaddnclem1  32064  itgaddnclem2  32065  iblabsnclem  32069  iblabsnc  32070  iblmulc2nc  32071  ftc1anclem8  32088  ftc1anc  32089  bezoutr1  35907  relexpaddss  36381  stoweidlem44  38017  fourierdlem42  38124  fourierdlem42OLD  38125  fourierdlem103  38185  fourierdlem104  38186  sqwvfoura  38204  sqwvfourb  38205  vtxdg0e  39699
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