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Theorem 00id 9003
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 8854 . 2  |-  0  e.  RR
2 ax-rnegex 8824 . 2  |-  ( 0  e.  RR  ->  E. c  e.  RR  ( 0  +  c )  =  0 )
3 oveq2 5882 . . . . . . 7  |-  ( c  =  0  ->  (
0  +  c )  =  ( 0  +  0 ) )
43eqeq1d 2304 . . . . . 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 8825 . . . . . . 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 8877 . . . . . . . . . . . 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 8877 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  y  e.  CC )
13 0cn 8847 . . . . . . . . . . . . 13  |-  0  e.  CC
14 mulass 8841 . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . 13  |-  ( ( c  x.  y )  =  1  ->  (
( c  x.  y
)  x.  0 )  =  ( 1  x.  0 ) )
1813mulid2i 8856 . . . . . . . . . . . . 13  |-  ( 1  x.  0 )  =  0
1917, 18syl6eq 2344 . . . . . . . . . . . 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 2330 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( c  x.  ( y  x.  0 ) )  =  0 )
2221oveq1d 5889 . . . . . . . . 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 5889 . . . . . . . . . . . . 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 8838 . . . . . . . . . . . . . . . . 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 8877 . . . . . . . . . . . . . 14  |-  ( ( ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  /\  c  =/=  0 )  /\  (
y  e.  RR  /\  ( c  x.  y
)  =  1 ) )  ->  ( y  x.  0 )  e.  CC )
29 adddir 8846 . . . . . . . . . . . . . . 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 2330 . . . . . . . . . . . 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 5889 . . . . . . . . . . 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 8838 . . . . . . . . . . . . . . 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 8877 . . . . . . . . . . . 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 8838 . . . . . . . . . . . . . 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 8877 . . . . . . . . . . . 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 8840 . . . . . . . . . . . . 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 2329 . . . . . . . . . 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 8836 . . . . . . . . . . . . 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 9002 . . . . . . . . . . . 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 2330 . . . . . . . 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 2679 . . . . . 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 2535 . . 3  |-  ( ( c  e.  RR  /\  ( 0  +  c )  =  0 )  ->  ( 0  +  0 )  =  0 )
5958rexlimiva 2675 . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758
This theorem is referenced by:  mul02lem1  9004  mul02lem2  9005  addid1  9008  addid2  9011  negdii  9146  addgt0  9276  addgegt0  9277  addgtge0  9278  addge0  9279  add20  9302  recextlem2  9415  crne0  9755  10p10e20  10210  ser0  11114  faclbnd4lem3  11324  bcpasc  11349  fsumadd  12227  fsumrelem  12281  arisum  12334  sadcaddlem  12664  sadcadd  12665  sadadd2  12667  bezout  12737  pcaddlem  12952  4sqlem19  13026  37prm  13138  139prm  13141  163prm  13142  317prm  13143  631prm  13144  1259lem1  13145  1259lem2  13146  1259lem3  13147  1259lem4  13148  2503lem1  13151  2503lem2  13152  2503lem3  13153  4001lem1  13155  4001lem2  13156  4001lem3  13157  4001lem4  13158  sylow1lem1  14925  psrbagaddcl  16132  mplcoe3  16226  cnfld0  16414  reparphti  18511  itg1addlem4  19070  ibladdlem  19190  itgaddlem1  19193  iblabslem  19198  iblabs  19199  coeaddlem  19646  dcubic  20158  log2ublem3  20260  log2ub  20261  chtublem  20466  logfacrlim  20479  dchrisumlem1  20654  chpdifbndlem2  20719  1kp2ke3k  20849  dip0r  21309  pythi  21444  normpythi  21737  ocsh  21878  0lnfn  22581  lnopeq0i  22603  nlelshi  22656  unierri  22700  probun  23637  vdgr0  23907  vdgr1a  23912  fsumcube  24867  itg2addnc  25005  ibladdnclem  25007  itgaddnclem1  25009  itgaddnclem2  25010  iblabsnclem  25014  iblabsnc  25015  iblmulc2nc  25016  bezoutr1  27176  stoweidlem44  27896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888
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