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Theorem 00id 8987
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 8838 . 2  |-  0  e.  RR
2 ax-rnegex 8808 . 2  |-  ( 0  e.  RR  ->  E. c  e.  RR  ( 0  +  c )  =  0 )
3 oveq2 5866 . . . . . . 7  |-  ( c  =  0  ->  (
0  +  c )  =  ( 0  +  0 ) )
43eqeq1d 2291 . . . . . 6  |-  ( c  =  0  ->  (
( 0  +  c )  =  0  <->  (
0  +  0 )  =  0 ) )
54biimpd 198 . . . . 5  |-  ( c  =  0  ->  (
( 0  +  c )  =  0  -> 
( 0  +  0 )  =  0 ) )
65adantld 453 . . . 4  |-  ( c  =  0  ->  (
( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 ) )
7 ax-rrecex 8809 . . . . . . 7  |-  ( ( c  e.  RR  /\  c  =/=  0 )  ->  E. y  e.  RR  ( c  x.  y
)  =  1 )
87adantlr 695 . . . . . 6  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  E. y  e.  RR  ( c  x.  y )  =  1 )
9 simplll 734 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  c  e.  RR )
109recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  c  e.  CC )
11 simprl 732 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  y  e.  RR )
1211recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  y  e.  CC )
13 0cn 8831 . . . . . . . . . . . . 13  |-  0  e.  CC
14 mulass 8825 . . . . . . . . . . . . 13  |-  ( ( c  e.  CC  /\  y  e.  CC  /\  0  e.  CC )  ->  (
( c  x.  y
)  x.  0 )  =  ( c  x.  ( y  x.  0 ) ) )
1513, 14mp3an3 1266 . . . . . . . . . . . 12  |-  ( ( c  e.  CC  /\  y  e.  CC )  ->  ( ( c  x.  y )  x.  0 )  =  ( c  x.  ( y  x.  0 ) ) )
1610, 12, 15syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( 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 5865 . . . . . . . . . . . . 13  |-  ( ( c  x.  y )  =  1  ->  (
( c  x.  y
)  x.  0 )  =  ( 1  x.  0 ) )
1813mulid2i 8840 . . . . . . . . . . . . 13  |-  ( 1  x.  0 )  =  0
1917, 18syl6eq 2331 . . . . . . . . . . . 12  |-  ( ( c  x.  y )  =  1  ->  (
( c  x.  y
)  x.  0 )  =  0 )
2019ad2antll 709 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  y )  x.  0 )  =  0 )
2116, 20eqtr3d 2317 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  =  0 )
2221oveq1d 5873 . . . . . . . . 9  |-  ( ( ( ( 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 735 . . . . . . . . . . . . . 14  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  +  c )  =  0 )
2423oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 8822 . . . . . . . . . . . . . . . . 17  |-  ( ( y  e.  RR  /\  0  e.  RR )  ->  ( y  x.  0 )  e.  RR )
261, 25mpan2 652 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR  ->  (
y  x.  0 )  e.  RR )
2726ad2antrl 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( y  x.  0 )  e.  RR )
2827recnd 8861 . . . . . . . . . . . . . 14  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( y  x.  0 )  e.  CC )
29 adddir 8830 . . . . . . . . . . . . . . 15  |-  ( ( 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 1264 . . . . . . . . . . . . . 14  |-  ( ( 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 642 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 2317 . . . . . . . . . . . 12  |-  ( ( ( ( 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 5873 . . . . . . . . . . 11  |-  ( ( ( ( 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 8822 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  RR  /\  ( y  x.  0 )  e.  RR )  ->  ( 0  x.  ( y  x.  0 ) )  e.  RR )
351, 26, 34sylancr 644 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  (
0  x.  ( y  x.  0 ) )  e.  RR )
3635ad2antrl 708 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  x.  ( y  x.  0 ) )  e.  RR )
3736recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  x.  ( y  x.  0 ) )  e.  CC )
38 remulcl 8822 . . . . . . . . . . . . . 14  |-  ( ( c  e.  RR  /\  ( y  x.  0 )  e.  RR )  ->  ( c  x.  ( y  x.  0 ) )  e.  RR )
399, 27, 38syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  e.  RR )
4039recnd 8861 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  e.  CC )
41 addass 8824 . . . . . . . . . . . . 13  |-  ( ( ( 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 1266 . . . . . . . . . . . 12  |-  ( ( ( 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 642 . . . . . . . . . . 11  |-  ( ( ( ( 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 2316 . . . . . . . . . 10  |-  ( ( ( ( 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 460 . . . . . . . . . . . . 13  |-  ( ( c  e.  RR  /\  y  e.  RR )  ->  ( c  x.  (
y  x.  0 ) )  e.  RR )
46 readdcl 8820 . . . . . . . . . . . . 13  |-  ( ( ( c  x.  (
y  x.  0 ) )  e.  RR  /\  0  e.  RR )  ->  ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR )
4745, 1, 46sylancl 643 . . . . . . . . . . . 12  |-  ( ( c  e.  RR  /\  y  e.  RR )  ->  ( ( c  x.  ( y  x.  0 ) )  +  0 )  e.  RR )
489, 11, 47syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ( 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 8986 . . . . . . . . . . . 12  |-  ( ( ( ( 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 1265 . . . . . . . . . . 11  |-  ( ( ( ( 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 642 . . . . . . . . . 10  |-  ( ( ( ( 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 201 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( (
c  x.  ( y  x.  0 ) )  +  0 )  =  0 )
5322, 52eqtr3d 2317 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( 0  +  0 )  =  0 )
5453exp32 588 . . . . . . 7  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  ( y  e.  RR  ->  ( (
c  x.  y )  =  1  ->  (
0  +  0 )  =  0 ) ) )
5554rexlimdv 2666 . . . . . 6  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  ( E. y  e.  RR  (
c  x.  y )  =  1  ->  (
0  +  0 )  =  0 ) )
568, 55mpd 14 . . . . 5  |-  ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0
)  ->  ( 0  +  0 )  =  0 )
5756expcom 424 . . . 4  |-  ( c  =/=  0  ->  (
( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 ) )
586, 57pm2.61ine 2522 . . 3  |-  ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 )
5958rexlimiva 2662 . 2  |-  ( E. c  e.  RR  (
0  +  c )  =  0  ->  (
0  +  0 )  =  0 )
601, 2, 59mp2b 9 1  |-  ( 0  +  0 )  =  0
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742
This theorem is referenced by:  mul02lem1  8988  mul02lem2  8989  addid1  8992  addid2  8995  negdii  9130  addgt0  9260  addgegt0  9261  addgtge0  9262  addge0  9263  add20  9286  recextlem2  9399  crne0  9739  10p10e20  10194  ser0  11098  faclbnd4lem3  11308  bcpasc  11333  fsumadd  12211  fsumrelem  12265  arisum  12318  sadcaddlem  12648  sadcadd  12649  sadadd2  12651  bezout  12721  pcaddlem  12936  4sqlem19  13010  37prm  13122  139prm  13125  163prm  13126  317prm  13127  631prm  13128  1259lem1  13129  1259lem2  13130  1259lem3  13131  1259lem4  13132  2503lem1  13135  2503lem2  13136  2503lem3  13137  4001lem1  13139  4001lem2  13140  4001lem3  13141  4001lem4  13142  sylow1lem1  14909  psrbagaddcl  16116  mplcoe3  16210  cnfld0  16398  reparphti  18495  itg1addlem4  19054  ibladdlem  19174  itgaddlem1  19177  iblabslem  19182  iblabs  19183  coeaddlem  19630  dcubic  20142  log2ublem3  20244  log2ub  20245  chtublem  20450  logfacrlim  20463  dchrisumlem1  20638  chpdifbndlem2  20703  1kp2ke3k  20833  dip0r  21293  pythi  21428  normpythi  21721  ocsh  21862  0lnfn  22565  lnopeq0i  22587  nlelshi  22640  unierri  22684  probun  23622  vdgr0  23892  vdgr1a  23897  fsumcube  24795  bezoutr1  27073  stoweidlem44  27793
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-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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-nel 2449  df-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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