Theorem 00id 10211

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.

Step	Hyp	Ref	Expression
1		0re 10040	. 2 |- 0 e. RR
2		ax-rnegex 10007	. 2 |- ( 0 e. RR -> E. c e. RR ( 0 + c ) = 0 )
3		oveq2 6658	. . . . . . 7 |- ( c = 0 -> ( 0 + c ) = ( 0 + 0 ) )
4	3	eqeq1d 2624	. . . . . 6 |- ( c = 0 -> ( ( 0 + c ) = 0 <-> ( 0 + 0 ) = 0 ) )
5	4	biimpd 219	. . . . 5 |- ( c = 0 -> ( ( 0 + c ) = 0 -> ( 0 + 0 ) = 0 ) )
6	5	adantld 483	. . . 4 |- ( c = 0 -> ( ( c e. RR /\ ( 0 + c ) = 0 ) -> ( 0 + 0 ) = 0 ) )
7		ax-rrecex 10008	. . . . . . 7 |- ( ( c e. RR /\ c =/= 0 ) -> E. y e. RR ( c x. y ) = 1 )
8	7	adantlr 751	. . . . . 6 |- ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) -> E. y e. RR ( c x. y ) = 1 )
9		simplll 798	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> c e. RR )
10	9	recnd 10068	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> c e. CC )
11		simprl 794	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> y e. RR )
12	11	recnd 10068	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> y e. CC )
13		0cn 10032	. . . . . . . . . . 11 |- 0 e. CC
14		mulass 10024	. . . . . . . . . . 11 |- ( ( c e. CC /\ y e. CC /\ 0 e. CC ) -> ( ( c x. y ) x. 0 ) = ( c x. ( y x. 0 ) ) )
15	13, 14	mp3an3 1413	. . . . . . . . . 10 |- ( ( c e. CC /\ y e. CC ) -> ( ( c x. y ) x. 0 ) = ( c x. ( y x. 0 ) ) )
16	10, 12, 15	syl2anc 693	. . . . . . . . 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 6657	. . . . . . . . . . 11 |- ( ( c x. y ) = 1 -> ( ( c x. y ) x. 0 ) = ( 1 x. 0 ) )
18	13	mulid2i 10043	. . . . . . . . . . 11 |- ( 1 x. 0 ) = 0
19	17, 18	syl6eq 2672	. . . . . . . . . 10 |- ( ( c x. y ) = 1 -> ( ( c x. y ) x. 0 ) = 0 )
20	19	ad2antll 765	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( ( c x. y ) x. 0 ) = 0 )
21	16, 20	eqtr3d 2658	. . . . . . . 8 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( c x. ( y x. 0 ) ) = 0 )
22	21	oveq1d 6665	. . . . . . 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 799	. . . . . . . . . . . 12 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( 0 + c ) = 0 )
24	23	oveq1d 6665	. . . . . . . . . . 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 10021	. . . . . . . . . . . . . . 15 |- ( ( y e. RR /\ 0 e. RR ) -> ( y x. 0 ) e. RR )
26	1, 25	mpan2 707	. . . . . . . . . . . . . 14 |- ( y e. RR -> ( y x. 0 ) e. RR )
27	26	ad2antrl 764	. . . . . . . . . . . . 13 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( y x. 0 ) e. RR )
28	27	recnd 10068	. . . . . . . . . . . 12 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( y x. 0 ) e. CC )
29		adddir 10031	. . . . . . . . . . . . 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 ) ) ) )
30	13, 29	mp3an1 1411	. . . . . . . . . . . 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 ) ) ) )
31	10, 28, 30	syl2anc 693	. . . . . . . . . . 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 ) ) ) )
32	24, 31	eqtr3d 2658	. . . . . . . . . 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 ) ) ) )
33	32	oveq1d 6665	. . . . . . . . 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 10021	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ ( y x. 0 ) e. RR ) -> ( 0 x. ( y x. 0 ) ) e. RR )
35	1, 26, 34	sylancr 695	. . . . . . . . . . . 12 |- ( y e. RR -> ( 0 x. ( y x. 0 ) ) e. RR )
36	35	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( 0 x. ( y x. 0 ) ) e. RR )
37	36	recnd 10068	. . . . . . . . . 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 10021	. . . . . . . . . . . 12 |- ( ( c e. RR /\ ( y x. 0 ) e. RR ) -> ( c x. ( y x. 0 ) ) e. RR )
39	9, 27, 38	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( c x. ( y x. 0 ) ) e. RR )
40	39	recnd 10068	. . . . . . . . . 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 10023	. . . . . . . . . . 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 ) ) )
42	13, 41	mp3an3 1413	. . . . . . . . . 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 ) ) )
43	37, 40, 42	syl2anc 693	. . . . . . . . 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 ) ) )
44	33, 43	eqtr2d 2657	. . . . . . . 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 ) )
45	26, 38	sylan2 491	. . . . . . . . . . 11 |- ( ( c e. RR /\ y e. RR ) -> ( c x. ( y x. 0 ) ) e. RR )
46		readdcl 10019	. . . . . . . . . . 11 |- ( ( ( c x. ( y x. 0 ) ) e. RR /\ 0 e. RR ) -> ( ( c x. ( y x. 0 ) ) + 0 ) e. RR )
47	45, 1, 46	sylancl 694	. . . . . . . . . 10 |- ( ( c e. RR /\ y e. RR ) -> ( ( c x. ( y x. 0 ) ) + 0 ) e. RR )
48	9, 11, 47	syl2anc 693	. . . . . . . . 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 10210	. . . . . . . . . 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 ) )
50	1, 49	mp3an2 1412	. . . . . . . . 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 ) )
51	48, 36, 50	syl2anc 693	. . . . . . . 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 ) )
52	44, 51	mpbid 222	. . . . . . 7 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( ( c x. ( y x. 0 ) ) + 0 ) = 0 )
53	22, 52	eqtr3d 2658	. . . . . 6 |- ( ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) /\ ( y e. RR /\ ( c x. y ) = 1 ) ) -> ( 0 + 0 ) = 0 )
54	8, 53	rexlimddv 3035	. . . . 5 |- ( ( ( c e. RR /\ ( 0 + c ) = 0 ) /\ c =/= 0 ) -> ( 0 + 0 ) = 0 )
55	54	expcom 451	. . . 4 |- ( c =/= 0 -> ( ( c e. RR /\ ( 0 + c ) = 0 ) -> ( 0 + 0 ) = 0 ) )
56	6, 55	pm2.61ine 2877	. . 3 |- ( ( c e. RR /\ ( 0 + c ) = 0 ) -> ( 0 + 0 ) = 0 )
57	56	rexlimiva 3028	. 2 |- ( E. c e. RR ( 0 + c ) = 0 -> ( 0 + 0 ) = 0 )
58	1, 2, 57	mp2b 10	1 |- ( 0 + 0 ) = 0

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079

This theorem is referenced by:  mul02lem1  10212  mul02lem2  10213  addid1  10216  addid2  10219  addgt0  10514  addgegt0  10515  addgtge0  10516  addge0  10517  add20  10540  recextlem2  10658  crne0  11013  decaddm10  11578  10p10e20  11628  10p10e20OLD  11629  ser0  12853  faclbnd4lem3  13082  bcpasc  13108  relexpaddg  13793  fsumadd  14470  fsumrelem  14539  arisum  14592  fsumcube  14791  sadcaddlem  15179  sadcadd  15180  sadadd2  15182  bezout  15260  bezoutr1  15282  nnnn0modprm0  15511  pcaddlem  15592  4sqlem19  15667  139prm  15831  163prm  15832  317prm  15833  631prm  15834  1259lem1  15838  1259lem2  15839  1259lem4  15841  2503lem1  15844  2503lem2  15845  2503lem3  15846  4001lem1  15848  4001lem2  15849  4001lem3  15850  4001lem4  15851  sylow1lem1  18013  psrbagaddcl  19370  mplcoe3  19466  cnfld0  19770  reparphti  22797  itg1addlem4  23466  ibladdlem  23586  itgaddlem1  23589  iblabslem  23594  iblabs  23595  coeaddlem  24005  dcubic  24573  log2ublem3  24675  log2ub  24676  chtublem  24936  logfacrlim  24949  dchrisumlem1  25178  chpdifbndlem2  25243  vtxdg0e  26370  1kp2ke3k  27303  dip0r  27572  pythi  27705  normpythi  27999  ocsh  28142  0lnfn  28844  lnopeq0i  28866  nlelshi  28919  unierri  28963  probun  30481  hgt750lem2  30730  poimirlem3  33412  poimirlem4  33413  ismblfin  33450  itg2addnc  33464  ibladdnclem  33466  itgaddnclem1  33468  itgaddnclem2  33469  iblabsnclem  33473  iblabsnc  33474  iblmulc2nc  33475  ftc1anclem8  33492  ftc1anc  33493  relexpaddss  38010  stoweidlem44  40261  fourierdlem42  40366  fourierdlem103  40426  fourierdlem104  40427  sqwvfoura  40445  sqwvfourb  40446  fmtno5lem4  41468  139prmALT  41511