Theorem addid1 9810

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 9639	. 2 |- 1 e. RR
2		ax-rnegex 9607	. 2 |- ( 1 e. RR -> E. c e. RR ( 1 + c ) = 0 )
3		ax-1ne0 9605	. . . . . 6 |- 1 =/= 0
4		oveq2 6296	. . . . . . . . . 10 |- ( c = 0 -> ( 1 + c ) = ( 1 + 0 ) )
5	4	eqeq1d 2452	. . . . . . . . 9 |- ( c = 0 -> ( ( 1 + c ) = 0 <-> ( 1 + 0 ) = 0 ) )
6	5	biimpcd 228	. . . . . . . 8 |- ( ( 1 + c ) = 0 -> ( c = 0 -> ( 1 + 0 ) = 0 ) )
7		oveq2 6296	. . . . . . . . 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 9595	. . . . . . . . . . . . . . 15 |- _i e. CC
9	8, 8	mulcli 9645	. . . . . . . . . . . . . 14 |- ( _i x. _i ) e. CC
10	9, 9	mulcli 9645	. . . . . . . . . . . . 13 |- ( ( _i x. _i ) x. ( _i x. _i ) ) e. CC
11		ax-1cn 9594	. . . . . . . . . . . . 13 |- 1 e. CC
12		0cn 9632	. . . . . . . . . . . . 13 |- 0 e. CC
13	10, 11, 12	adddii 9650	. . . . . . . . . . . 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 9642	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 1 ) = ( ( _i x. _i ) x. ( _i x. _i ) )
15		mul01 9809	. . . . . . . . . . . . . . 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 9604	. . . . . . . . . . . . . 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 6300	. . . . . . . . . . . 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 2464	. . . . . . . . . 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 9648	. . . . . . . . . . . 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 9642	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. 1 ) = ( _i x. _i )
24	23	oveq2i 6299	. . . . . . . . . . . . . 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 9650	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) )
26	17	oveq2i 6299	. . . . . . . . . . . . . . . 16 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( _i x. _i ) x. 0 )
27		mul01 9809	. . . . . . . . . . . . . . . . 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 6298	. . . . . . . . . . . 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 9805	. . . . . . . . . . . 12 |- ( 0 + 0 ) = 0
35	34	eqcomi 2459	. . . . . . . . . . 11 |- 0 = ( 0 + 0 )
36	33, 35	eqeq12i 2464	. . . . . . . . . 10 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = 0 <-> ( 0 + 1 ) = ( 0 + 0 ) )
37		0re 9640	. . . . . . . . . . 11 |- 0 e. RR
38		readdcan 9804	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 0 e. RR /\ 0 e. RR ) -> ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 ) )
39	1, 37, 37, 38	mp3an 1363	. . . . . . . . . 10 |- ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 )
40	21, 36, 39	3bitri 275	. . . . . . . . 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 200	. . . . . . . 8 |- ( ( 1 + 0 ) = 0 -> 1 = 0 )
42	6, 41	syl6 34	. . . . . . 7 |- ( ( 1 + c ) = 0 -> ( c = 0 -> 1 = 0 ) )
43	42	necon3d 2644	. . . . . 6 |- ( ( 1 + c ) = 0 -> ( 1 =/= 0 -> c =/= 0 ) )
44	3, 43	mpi 20	. . . . 5 |- ( ( 1 + c ) = 0 -> c =/= 0 )
45		ax-rrecex 9608	. . . . 5 |- ( ( c e. RR /\ c =/= 0 ) -> E. x e. RR ( c x. x ) = 1 )
46	44, 45	sylan2 477	. . . 4 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> E. x e. RR ( c x. x ) = 1 )
47		simpr 463	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> A e. CC )
48		simplrl 769	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. RR )
49	48	recnd 9666	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. CC )
50	47, 49	mulcld 9660	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( A x. x ) e. CC )
51		simplll 767	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> c e. RR )
52	51	recnd 9666	. . . . . . . . 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 9664	. . . . . . . 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 9662	. . . . . . . . . . . 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 768	. . . . . . . . . . . . 13 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( 1 + c ) = 0 )
58	57	oveq1d 6303	. . . . . . . . . . . 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 9667	. . . . . . . . . . 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 9804	. . . . . . . . . . 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 1267	. . . . . . . . . 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 214	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c + 0 ) = c )
67	66	oveq2d 6304	. . . . . . . 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 9798	. . . . . . . . . 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 1267	. . . . . . . . 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 770	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c x. x ) = 1 )
72	71	oveq1d 6303	. . . . . . . . 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 9658	. . . . . . . . 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 9809	. . . . . . . . 9 |- ( ( A x. x ) e. CC -> ( ( A x. x ) x. 0 ) = 0 )
76	50, 75	syl 17	. . . . . . . 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 6306	. . . . . . 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 615	. . . . 5 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> ( x e. RR -> ( ( c x. x ) = 1 -> ( A e. CC -> ( A + 0 ) = A ) ) ) )
80	79	rexlimdv 2876	. . . 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 2874	. 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 188   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   E.wrex 2737  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537   _ici 9538   + caddc 9539   x. cmul 9541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-ltxr 9677

This theorem is referenced by:  cnegex  9811  addid2  9813  addcan2  9815  addid1i  9817  addid1d  9830  subid  9890  subid1  9891  swrdccat3blem  12846  shftval3  13132  reim0  13174  isercolllem3  13723  fsumcvg  13771  summolem2a  13774  risefac1  14079  ovolicc1  22462  brbtwn2  24928  axsegconlem1  24940  ax5seglem4  24955  axeuclid  24986  axcontlem2  24988  axcontlem4  24990  stoweidlem26  37880  2zrngamnd  39928  aacllem  40527