Theorem addid1 9763

Description: 0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
addid1	|- ( A e. CC -> ( A + 0 ) = A )

Proof of Theorem addid1

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

Step	Hyp	Ref	Expression
1		1re 9598	. 2 |- 1 e. RR
2		ax-rnegex 9566	. 2 |- ( 1 e. RR -> E. c e. RR ( 1 + c ) = 0 )
3		ax-1ne0 9564	. . . . . 6 |- 1 =/= 0
4		oveq2 6289	. . . . . . . . . 10 |- ( c = 0 -> ( 1 + c ) = ( 1 + 0 ) )
5	4	eqeq1d 2445	. . . . . . . . 9 |- ( c = 0 -> ( ( 1 + c ) = 0 <-> ( 1 + 0 ) = 0 ) )
6	5	biimpcd 224	. . . . . . . 8 |- ( ( 1 + c ) = 0 -> ( c = 0 -> ( 1 + 0 ) = 0 ) )
7		oveq2 6289	. . . . . . . . 9 |- ( ( 1 + 0 ) = 0 -> ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. ( 1 + 0 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) )
8		ax-icn 9554	. . . . . . . . . . . . . . 15 |- _i e. CC
9	8, 8	mulcli 9604	. . . . . . . . . . . . . 14 |- ( _i x. _i ) e. CC
10	9, 9	mulcli 9604	. . . . . . . . . . . . 13 |- ( ( _i x. _i ) x. ( _i x. _i ) ) e. CC
11		ax-1cn 9553	. . . . . . . . . . . . 13 |- 1 e. CC
12		0cn 9591	. . . . . . . . . . . . 13 |- 0 e. CC
13	10, 11, 12	adddii 9609	. . . . . . . . . . . 12 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. ( 1 + 0 ) ) = ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 1 ) + ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) )
14	10	mulid1i 9601	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 1 ) = ( ( _i x. _i ) x. ( _i x. _i ) )
15		mul01 9762	. . . . . . . . . . . . . . 15 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) e. CC -> ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) = 0 )
16	10, 15	ax-mp 5	. . . . . . . . . . . . . 14 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) = 0
17		ax-i2m1 9563	. . . . . . . . . . . . . 14 |- ( ( _i x. _i ) + 1 ) = 0
18	16, 17	eqtr4i 2475	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) = ( ( _i x. _i ) + 1 )
19	14, 18	oveq12i 6293	. . . . . . . . . . . 12 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 1 ) + ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) )
20	13, 19	eqtri 2472	. . . . . . . . . . 11 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. ( 1 + 0 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) )
21	20, 16	eqeq12i 2463	. . . . . . . . . 10 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. ( 1 + 0 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) <-> ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = 0 )
22	10, 9, 11	addassi 9607	. . . . . . . . . . . 12 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) + 1 ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) )
23	9	mulid1i 9601	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. 1 ) = ( _i x. _i )
24	23	oveq2i 6292	. . . . . . . . . . . . . 14 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) )
25	9, 9, 11	adddii 9609	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) )
26	17	oveq2i 6292	. . . . . . . . . . . . . . . 16 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( _i x. _i ) x. 0 )
27		mul01 9762	. . . . . . . . . . . . . . . . 17 |- ( ( _i x. _i ) e. CC -> ( ( _i x. _i ) x. 0 ) = 0 )
28	9, 27	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( ( _i x. _i ) x. 0 ) = 0
29	26, 28	eqtri 2472	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = 0
30	25, 29	eqtr3i 2474	. . . . . . . . . . . . . 14 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) ) = 0
31	24, 30	eqtr3i 2474	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) = 0
32	31	oveq1i 6291	. . . . . . . . . . . 12 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) + 1 ) = ( 0 + 1 )
33	22, 32	eqtr3i 2474	. . . . . . . . . . 11 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = ( 0 + 1 )
34		00id 9758	. . . . . . . . . . . 12 |- ( 0 + 0 ) = 0
35	34	eqcomi 2456	. . . . . . . . . . 11 |- 0 = ( 0 + 0 )
36	33, 35	eqeq12i 2463	. . . . . . . . . 10 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = 0 <-> ( 0 + 1 ) = ( 0 + 0 ) )
37		0re 9599	. . . . . . . . . . 11 |- 0 e. RR
38		readdcan 9757	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 0 e. RR /\ 0 e. RR ) -> ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 ) )
39	1, 37, 37, 38	mp3an 1325	. . . . . . . . . 10 |- ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 )
40	21, 36, 39	3bitri 271	. . . . . . . . 9 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. ( 1 + 0 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) <-> 1 = 0 )
41	7, 40	sylib 196	. . . . . . . 8 |- ( ( 1 + 0 ) = 0 -> 1 = 0 )
42	6, 41	syl6 33	. . . . . . 7 |- ( ( 1 + c ) = 0 -> ( c = 0 -> 1 = 0 ) )
43	42	necon3d 2667	. . . . . 6 |- ( ( 1 + c ) = 0 -> ( 1 =/= 0 -> c =/= 0 ) )
44	3, 43	mpi 17	. . . . 5 |- ( ( 1 + c ) = 0 -> c =/= 0 )
45		ax-rrecex 9567	. . . . 5 |- ( ( c e. RR /\ c =/= 0 ) -> E. x e. RR ( c x. x ) = 1 )
46	44, 45	sylan2 474	. . . 4 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> E. x e. RR ( c x. x ) = 1 )
47		simpr 461	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> A e. CC )
48		simplrl 761	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. RR )
49	48	recnd 9625	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. CC )
50	47, 49	mulcld 9619	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( A x. x ) e. CC )
51		simplll 759	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> c e. RR )
52	51	recnd 9625	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> c e. CC )
53	12	a1i 11	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> 0 e. CC )
54	50, 52, 53	adddid 9623	. . . . . . . 8 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( A x. x ) x. ( c + 0 ) ) = ( ( ( A x. x ) x. c ) + ( ( A x. x ) x. 0 ) ) )
55	11	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> 1 e. CC )
56	55, 52, 53	addassd 9621	. . . . . . . . . . . 12 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( 1 + c ) + 0 ) = ( 1 + ( c + 0 ) ) )
57		simpllr 760	. . . . . . . . . . . . 13 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( 1 + c ) = 0 )
58	57	oveq1d 6296	. . . . . . . . . . . 12 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( 1 + c ) + 0 ) = ( 0 + 0 ) )
59	56, 58	eqtr3d 2486	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( 1 + ( c + 0 ) ) = ( 0 + 0 ) )
60	34, 59, 57	3eqtr4a 2510	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( 1 + ( c + 0 ) ) = ( 1 + c ) )
61	37	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> 0 e. RR )
62	51, 61	readdcld 9626	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c + 0 ) e. RR )
63	1	a1i 11	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> 1 e. RR )
64		readdcan 9757	. . . . . . . . . . 11 |- ( ( ( c + 0 ) e. RR /\ c e. RR /\ 1 e. RR ) -> ( ( 1 + ( c + 0 ) ) = ( 1 + c ) <-> ( c + 0 ) = c ) )
65	62, 51, 63, 64	syl3anc 1229	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( 1 + ( c + 0 ) ) = ( 1 + c ) <-> ( c + 0 ) = c ) )
66	60, 65	mpbid 210	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c + 0 ) = c )
67	66	oveq2d 6297	. . . . . . . 8 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( A x. x ) x. ( c + 0 ) ) = ( ( A x. x ) x. c ) )
68	54, 67	eqtr3d 2486	. . . . . . 7 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( ( A x. x ) x. c ) + ( ( A x. x ) x. 0 ) ) = ( ( A x. x ) x. c ) )
69		mul31 9751	. . . . . . . . . 10 |- ( ( A e. CC /\ x e. CC /\ c e. CC ) -> ( ( A x. x ) x. c ) = ( ( c x. x ) x. A ) )
70	47, 49, 52, 69	syl3anc 1229	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( A x. x ) x. c ) = ( ( c x. x ) x. A ) )
71		simplrr 762	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c x. x ) = 1 )
72	71	oveq1d 6296	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( c x. x ) x. A ) = ( 1 x. A ) )
73	47	mulid2d 9617	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( 1 x. A ) = A )
74	70, 72, 73	3eqtrd 2488	. . . . . . . 8 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( A x. x ) x. c ) = A )
75		mul01 9762	. . . . . . . . 9 |- ( ( A x. x ) e. CC -> ( ( A x. x ) x. 0 ) = 0 )
76	50, 75	syl 16	. . . . . . . 8 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( A x. x ) x. 0 ) = 0 )
77	74, 76	oveq12d 6299	. . . . . . 7 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( ( ( A x. x ) x. c ) + ( ( A x. x ) x. 0 ) ) = ( A + 0 ) )
78	68, 77, 74	3eqtr3d 2492	. . . . . 6 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( A + 0 ) = A )
79	78	exp42 611	. . . . 5 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> ( x e. RR -> ( ( c x. x ) = 1 -> ( A e. CC -> ( A + 0 ) = A ) ) ) )
80	79	rexlimdv 2933	. . . 4 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> ( E. x e. RR ( c x. x ) = 1 -> ( A e. CC -> ( A + 0 ) = A ) ) )
81	46, 80	mpd 15	. . 3 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> ( A e. CC -> ( A + 0 ) = A ) )
82	81	rexlimiva 2931	. 2 |- ( E. c e. RR ( 1 + c ) = 0 -> ( A e. CC -> ( A + 0 ) = A ) )
83	1, 2, 82	mp2b 10	1 |- ( A e. CC -> ( A + 0 ) = A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   E.wrex 2794  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496   _ici 9497   + caddc 9498   x. cmul 9500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-ltxr 9636

This theorem is referenced by:  cnegex  9764  addid2  9766  addcan2  9768  addid1i  9770  addid1d  9783  subid  9843  subid1  9844  swrdccat3blem  12701  shftval3  12890  reim0  12932  isercolllem3  13470  fsumcvg  13515  summolem2a  13518  ovolicc1  21904  brbtwn2  24184  axsegconlem1  24196  ax5seglem4  24211  axeuclid  24242  axcontlem2  24244  axcontlem4  24246  relexpadd  29038  risefac1  29130  stoweidlem26  31697  2zrngamnd  32457