Theorem addid2 10219

Description: 0 is a left identity for addition. This used to be one of our complex number axioms, until it was discovered that it was dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
addid2	|- ( A e. CC -> ( 0 + A ) = A )

Proof of Theorem addid2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnegex 10217	. 2 |- ( A e. CC -> E. x e. CC ( A + x ) = 0 )
2		cnegex 10217	. . . 4 |- ( x e. CC -> E. y e. CC ( x + y ) = 0 )
3	2	ad2antrl 764	. . 3 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) ) -> E. y e. CC ( x + y ) = 0 )
4		0cn 10032	. . . . . . . . . 10 |- 0 e. CC
5		addass 10023	. . . . . . . . . 10 |- ( ( 0 e. CC /\ 0 e. CC /\ y e. CC ) -> ( ( 0 + 0 ) + y ) = ( 0 + ( 0 + y ) ) )
6	4, 4, 5	mp3an12 1414	. . . . . . . . 9 |- ( y e. CC -> ( ( 0 + 0 ) + y ) = ( 0 + ( 0 + y ) ) )
7	6	adantr 481	. . . . . . . 8 |- ( ( y e. CC /\ ( x + y ) = 0 ) -> ( ( 0 + 0 ) + y ) = ( 0 + ( 0 + y ) ) )
8	7	3ad2ant3 1084	. . . . . . 7 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( ( 0 + 0 ) + y ) = ( 0 + ( 0 + y ) ) )
9		00id 10211	. . . . . . . . 9 |- ( 0 + 0 ) = 0
10	9	oveq1i 6660	. . . . . . . 8 |- ( ( 0 + 0 ) + y ) = ( 0 + y )
11		simp1 1061	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> A e. CC )
12		simp2l 1087	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> x e. CC )
13		simp3l 1089	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> y e. CC )
14	11, 12, 13	addassd 10062	. . . . . . . . . 10 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( ( A + x ) + y ) = ( A + ( x + y ) ) )
15		simp2r 1088	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( A + x ) = 0 )
16	15	oveq1d 6665	. . . . . . . . . 10 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( ( A + x ) + y ) = ( 0 + y ) )
17		simp3r 1090	. . . . . . . . . . 11 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( x + y ) = 0 )
18	17	oveq2d 6666	. . . . . . . . . 10 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( A + ( x + y ) ) = ( A + 0 ) )
19	14, 16, 18	3eqtr3rd 2665	. . . . . . . . 9 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( A + 0 ) = ( 0 + y ) )
20		addid1 10216	. . . . . . . . . 10 |- ( A e. CC -> ( A + 0 ) = A )
21	20	3ad2ant1 1082	. . . . . . . . 9 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( A + 0 ) = A )
22	19, 21	eqtr3d 2658	. . . . . . . 8 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( 0 + y ) = A )
23	10, 22	syl5eq 2668	. . . . . . 7 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( ( 0 + 0 ) + y ) = A )
24	22	oveq2d 6666	. . . . . . 7 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( 0 + ( 0 + y ) ) = ( 0 + A ) )
25	8, 23, 24	3eqtr3rd 2665	. . . . . 6 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) /\ ( y e. CC /\ ( x + y ) = 0 ) ) -> ( 0 + A ) = A )
26	25	3expia 1267	. . . . 5 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) ) -> ( ( y e. CC /\ ( x + y ) = 0 ) -> ( 0 + A ) = A ) )
27	26	expd 452	. . . 4 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) ) -> ( y e. CC -> ( ( x + y ) = 0 -> ( 0 + A ) = A ) ) )
28	27	rexlimdv 3030	. . 3 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) ) -> ( E. y e. CC ( x + y ) = 0 -> ( 0 + A ) = A ) )
29	3, 28	mpd 15	. 2 |- ( ( A e. CC /\ ( x e. CC /\ ( A + x ) = 0 ) ) -> ( 0 + A ) = A )
30	1, 29	rexlimddv 3035	1 |- ( A e. CC -> ( 0 + A ) = A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650   CCcc 9934   0cc0 9936   + caddc 9939

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:  addcan  10220  addid2i  10224  addid2d  10237  negneg  10331  fz0to4untppr  12442  fzo0addel  12521  fzoaddel2  12523  divfl0  12625  modid  12695  modsumfzodifsn  12743  swrdspsleq  13449  swrds1  13451  isercolllem3  14397  sumrblem  14442  summolem2a  14446  fsum0diag2  14515  eftlub  14839  gcdid  15248  cnaddablx  18271  cnaddabl  18272  cnaddid  18273  cncrng  19767  cnlmod  22940  ptolemy  24248  logtayl  24406  leibpilem2  24668  axcontlem2  25845  cnaddabloOLD  27436  cnidOLD  27437  dvcosax  40141  2zrngamnd  41941  aacllem  42547