Theorem addid1 9671

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 9506	. 2 |- 1 e. RR
2		ax-rnegex 9474	. 2 |- ( 1 e. RR -> E. c e. RR ( 1 + c ) = 0 )
3		ax-1ne0 9472	. . . . . 6 |- 1 =/= 0
4		oveq2 6204	. . . . . . . . . 10 |- ( c = 0 -> ( 1 + c ) = ( 1 + 0 ) )
5	4	eqeq1d 2384	. . . . . . . . 9 |- ( c = 0 -> ( ( 1 + c ) = 0 <-> ( 1 + 0 ) = 0 ) )
6	5	biimpcd 224	. . . . . . . 8 |- ( ( 1 + c ) = 0 -> ( c = 0 -> ( 1 + 0 ) = 0 ) )
7		oveq2 6204	. . . . . . . . 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 9462	. . . . . . . . . . . . . . 15 |- _i e. CC
9	8, 8	mulcli 9512	. . . . . . . . . . . . . 14 |- ( _i x. _i ) e. CC
10	9, 9	mulcli 9512	. . . . . . . . . . . . 13 |- ( ( _i x. _i ) x. ( _i x. _i ) ) e. CC
11		ax-1cn 9461	. . . . . . . . . . . . 13 |- 1 e. CC
12		0cn 9499	. . . . . . . . . . . . 13 |- 0 e. CC
13	10, 11, 12	adddii 9517	. . . . . . . . . . . 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 9509	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 1 ) = ( ( _i x. _i ) x. ( _i x. _i ) )
15		mul01 9670	. . . . . . . . . . . . . . 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 9471	. . . . . . . . . . . . . 14 |- ( ( _i x. _i ) + 1 ) = 0
18	16, 17	eqtr4i 2414	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) = ( ( _i x. _i ) + 1 )
19	14, 18	oveq12i 6208	. . . . . . . . . . . 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 2411	. . . . . . . . . . 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 2402	. . . . . . . . . 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 9515	. . . . . . . . . . . 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 9509	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. 1 ) = ( _i x. _i )
24	23	oveq2i 6207	. . . . . . . . . . . . . 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 9517	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) )
26	17	oveq2i 6207	. . . . . . . . . . . . . . . 16 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( _i x. _i ) x. 0 )
27		mul01 9670	. . . . . . . . . . . . . . . . 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 2411	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = 0
30	25, 29	eqtr3i 2413	. . . . . . . . . . . . . 14 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) ) = 0
31	24, 30	eqtr3i 2413	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) = 0
32	31	oveq1i 6206	. . . . . . . . . . . 12 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) + 1 ) = ( 0 + 1 )
33	22, 32	eqtr3i 2413	. . . . . . . . . . 11 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = ( 0 + 1 )
34		00id 9666	. . . . . . . . . . . 12 |- ( 0 + 0 ) = 0
35	34	eqcomi 2395	. . . . . . . . . . 11 |- 0 = ( 0 + 0 )
36	33, 35	eqeq12i 2402	. . . . . . . . . 10 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = 0 <-> ( 0 + 1 ) = ( 0 + 0 ) )
37		0re 9507	. . . . . . . . . . 11 |- 0 e. RR
38		readdcan 9665	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 0 e. RR /\ 0 e. RR ) -> ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 ) )
39	1, 37, 37, 38	mp3an 1322	. . . . . . . . . 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 2606	. . . . . 6 |- ( ( 1 + c ) = 0 -> ( 1 =/= 0 -> c =/= 0 ) )
44	3, 43	mpi 17	. . . . 5 |- ( ( 1 + c ) = 0 -> c =/= 0 )
45		ax-rrecex 9475	. . . . 5 |- ( ( c e. RR /\ c =/= 0 ) -> E. x e. RR ( c x. x ) = 1 )
46	44, 45	sylan2 472	. . . 4 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> E. x e. RR ( c x. x ) = 1 )
47		simpr 459	. . . . . . . . . 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 9533	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. CC )
50	47, 49	mulcld 9527	. . . . . . . . 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 9533	. . . . . . . . 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 9531	. . . . . . . 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 9529	. . . . . . . . . . . 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 6211	. . . . . . . . . . . 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 2425	. . . . . . . . . . 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 2449	. . . . . . . . . 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 9534	. . . . . . . . . . 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 9665	. . . . . . . . . . 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 1226	. . . . . . . . . 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 6212	. . . . . . . 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 2425	. . . . . . 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 9659	. . . . . . . . . 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 1226	. . . . . . . . 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 6211	. . . . . . . . 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 9525	. . . . . . . . 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 2427	. . . . . . . 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 9670	. . . . . . . . 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 6214	. . . . . . 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 2431	. . . . . 6 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( A + 0 ) = A )
79	78	exp42 609	. . . . 5 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> ( x e. RR -> ( ( c x. x ) = 1 -> ( A e. CC -> ( A + 0 ) = A ) ) ) )
80	79	rexlimdv 2872	. . . 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 2870	. 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 367   = wceq 1399   e. wcel 1826   =/= wne 2577   E.wrex 2733  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404   _ici 9405   + caddc 9406   x. cmul 9408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-po 4714  df-so 4715  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-ltxr 9544

This theorem is referenced by:  cnegex  9672  addid2  9674  addcan2  9676  addid1i  9678  addid1d  9691  subid  9751  subid1  9752  swrdccat3blem  12631  shftval3  12911  reim0  12953  isercolllem3  13491  fsumcvg  13536  summolem2a  13539  ovolicc1  22012  brbtwn2  24329  axsegconlem1  24341  ax5seglem4  24356  axeuclid  24387  axcontlem2  24389  axcontlem4  24391  risefac1  29321  stoweidlem26  31974  2zrngamnd  32947  aacllem  33550