Theorem addid1 9748

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 9584	. 2 |- 1 e. RR
2		ax-rnegex 9552	. 2 |- ( 1 e. RR -> E. c e. RR ( 1 + c ) = 0 )
3		ax-1ne0 9550	. . . . . 6 |- 1 =/= 0
4		oveq2 6283	. . . . . . . . . 10 |- ( c = 0 -> ( 1 + c ) = ( 1 + 0 ) )
5	4	eqeq1d 2462	. . . . . . . . 9 |- ( c = 0 -> ( ( 1 + c ) = 0 <-> ( 1 + 0 ) = 0 ) )
6	5	biimpcd 224	. . . . . . . 8 |- ( ( 1 + c ) = 0 -> ( c = 0 -> ( 1 + 0 ) = 0 ) )
7		oveq2 6283	. . . . . . . . 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 9540	. . . . . . . . . . . . . . 15 |- _i e. CC
9	8, 8	mulcli 9590	. . . . . . . . . . . . . 14 |- ( _i x. _i ) e. CC
10	9, 9	mulcli 9590	. . . . . . . . . . . . 13 |- ( ( _i x. _i ) x. ( _i x. _i ) ) e. CC
11		ax-1cn 9539	. . . . . . . . . . . . 13 |- 1 e. CC
12		0cn 9577	. . . . . . . . . . . . 13 |- 0 e. CC
13	10, 11, 12	adddii 9595	. . . . . . . . . . . 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 9587	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 1 ) = ( ( _i x. _i ) x. ( _i x. _i ) )
15		mul01 9747	. . . . . . . . . . . . . . 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 9549	. . . . . . . . . . . . . 14 |- ( ( _i x. _i ) + 1 ) = 0
18	16, 17	eqtr4i 2492	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) = ( ( _i x. _i ) + 1 )
19	14, 18	oveq12i 6287	. . . . . . . . . . . 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 2489	. . . . . . . . . . 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 2480	. . . . . . . . . 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 9593	. . . . . . . . . . . 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 9587	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. 1 ) = ( _i x. _i )
24	23	oveq2i 6286	. . . . . . . . . . . . . 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 9595	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) )
26	17	oveq2i 6286	. . . . . . . . . . . . . . . 16 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( _i x. _i ) x. 0 )
27		mul01 9747	. . . . . . . . . . . . . . . . 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 2489	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = 0
30	25, 29	eqtr3i 2491	. . . . . . . . . . . . . 14 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) ) = 0
31	24, 30	eqtr3i 2491	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) = 0
32	31	oveq1i 6285	. . . . . . . . . . . 12 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) + 1 ) = ( 0 + 1 )
33	22, 32	eqtr3i 2491	. . . . . . . . . . 11 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = ( 0 + 1 )
34		00id 9743	. . . . . . . . . . . 12 |- ( 0 + 0 ) = 0
35	34	eqcomi 2473	. . . . . . . . . . 11 |- 0 = ( 0 + 0 )
36	33, 35	eqeq12i 2480	. . . . . . . . . 10 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = 0 <-> ( 0 + 1 ) = ( 0 + 0 ) )
37		0re 9585	. . . . . . . . . . 11 |- 0 e. RR
38		readdcan 9742	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 0 e. RR /\ 0 e. RR ) -> ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 ) )
39	1, 37, 37, 38	mp3an 1319	. . . . . . . . . 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 2684	. . . . . 6 |- ( ( 1 + c ) = 0 -> ( 1 =/= 0 -> c =/= 0 ) )
44	3, 43	mpi 17	. . . . 5 |- ( ( 1 + c ) = 0 -> c =/= 0 )
45		ax-rrecex 9553	. . . . 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 759	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. RR )
49	48	recnd 9611	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. CC )
50	47, 49	mulcld 9605	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( A x. x ) e. CC )
51		simplll 757	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> c e. RR )
52	51	recnd 9611	. . . . . . . . 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 9609	. . . . . . . 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 9607	. . . . . . . . . . . 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 758	. . . . . . . . . . . . 13 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( 1 + c ) = 0 )
58	57	oveq1d 6290	. . . . . . . . . . . 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 2503	. . . . . . . . . . 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 2527	. . . . . . . . . 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 9612	. . . . . . . . . . 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 9742	. . . . . . . . . . 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 1223	. . . . . . . . . 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 6291	. . . . . . . 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 2503	. . . . . . 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 9736	. . . . . . . . . 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 1223	. . . . . . . . 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 760	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c x. x ) = 1 )
72	71	oveq1d 6290	. . . . . . . . 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 9603	. . . . . . . . 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 2505	. . . . . . . 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 9747	. . . . . . . . 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 6293	. . . . . . 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 2509	. . . . . 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 2946	. . . 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 2944	. 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 1374   e. wcel 1762   =/= wne 2655   E.wrex 2808  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   _ici 9483   + caddc 9484   x. cmul 9486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622

This theorem is referenced by:  cnegex  9749  addid2  9751  addcan2  9753  addid1i  9755  addid1d  9768  subid  9827  subid1  9828  swrdccat3blem  12670  shftval3  12859  reim0  12901  isercolllem3  13438  fsumcvg  13483  summolem2a  13486  ovolicc1  21655  brbtwn2  23877  axsegconlem1  23889  ax5seglem4  23904  axeuclid  23935  axcontlem2  23937  axcontlem4  23939  relexpadd  28522  risefac1  28718  stoweidlem26  31281  fourierdlem103  31465