Theorem addid1 9831

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 9660	. 2 |- 1 e. RR
2		ax-rnegex 9628	. 2 |- ( 1 e. RR -> E. c e. RR ( 1 + c ) = 0 )
3		ax-1ne0 9626	. . . . . 6 |- 1 =/= 0
4		oveq2 6316	. . . . . . . . . 10 |- ( c = 0 -> ( 1 + c ) = ( 1 + 0 ) )
5	4	eqeq1d 2473	. . . . . . . . 9 |- ( c = 0 -> ( ( 1 + c ) = 0 <-> ( 1 + 0 ) = 0 ) )
6	5	biimpcd 232	. . . . . . . 8 |- ( ( 1 + c ) = 0 -> ( c = 0 -> ( 1 + 0 ) = 0 ) )
7		oveq2 6316	. . . . . . . . 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 9616	. . . . . . . . . . . . . . 15 |- _i e. CC
9	8, 8	mulcli 9666	. . . . . . . . . . . . . 14 |- ( _i x. _i ) e. CC
10	9, 9	mulcli 9666	. . . . . . . . . . . . 13 |- ( ( _i x. _i ) x. ( _i x. _i ) ) e. CC
11		ax-1cn 9615	. . . . . . . . . . . . 13 |- 1 e. CC
12		0cn 9653	. . . . . . . . . . . . 13 |- 0 e. CC
13	10, 11, 12	adddii 9671	. . . . . . . . . . . 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 9663	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 1 ) = ( ( _i x. _i ) x. ( _i x. _i ) )
15		mul01 9830	. . . . . . . . . . . . . . 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 9625	. . . . . . . . . . . . . 14 |- ( ( _i x. _i ) + 1 ) = 0
18	16, 17	eqtr4i 2496	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) x. 0 ) = ( ( _i x. _i ) + 1 )
19	14, 18	oveq12i 6320	. . . . . . . . . . . 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 2493	. . . . . . . . . . 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 2485	. . . . . . . . . 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 9669	. . . . . . . . . . . 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 9663	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. 1 ) = ( _i x. _i )
24	23	oveq2i 6319	. . . . . . . . . . . . . 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 9671	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) )
26	17	oveq2i 6319	. . . . . . . . . . . . . . . 16 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = ( ( _i x. _i ) x. 0 )
27		mul01 9830	. . . . . . . . . . . . . . . . 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 2493	. . . . . . . . . . . . . . 15 |- ( ( _i x. _i ) x. ( ( _i x. _i ) + 1 ) ) = 0
30	25, 29	eqtr3i 2495	. . . . . . . . . . . . . 14 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) x. 1 ) ) = 0
31	24, 30	eqtr3i 2495	. . . . . . . . . . . . 13 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) = 0
32	31	oveq1i 6318	. . . . . . . . . . . 12 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( _i x. _i ) ) + 1 ) = ( 0 + 1 )
33	22, 32	eqtr3i 2495	. . . . . . . . . . 11 |- ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = ( 0 + 1 )
34		00id 9826	. . . . . . . . . . . 12 |- ( 0 + 0 ) = 0
35	34	eqcomi 2480	. . . . . . . . . . 11 |- 0 = ( 0 + 0 )
36	33, 35	eqeq12i 2485	. . . . . . . . . 10 |- ( ( ( ( _i x. _i ) x. ( _i x. _i ) ) + ( ( _i x. _i ) + 1 ) ) = 0 <-> ( 0 + 1 ) = ( 0 + 0 ) )
37		0re 9661	. . . . . . . . . . 11 |- 0 e. RR
38		readdcan 9825	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 0 e. RR /\ 0 e. RR ) -> ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 ) )
39	1, 37, 37, 38	mp3an 1390	. . . . . . . . . 10 |- ( ( 0 + 1 ) = ( 0 + 0 ) <-> 1 = 0 )
40	21, 36, 39	3bitri 279	. . . . . . . . 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 201	. . . . . . . 8 |- ( ( 1 + 0 ) = 0 -> 1 = 0 )
42	6, 41	syl6 33	. . . . . . 7 |- ( ( 1 + c ) = 0 -> ( c = 0 -> 1 = 0 ) )
43	42	necon3d 2664	. . . . . 6 |- ( ( 1 + c ) = 0 -> ( 1 =/= 0 -> c =/= 0 ) )
44	3, 43	mpi 20	. . . . 5 |- ( ( 1 + c ) = 0 -> c =/= 0 )
45		ax-rrecex 9629	. . . . 5 |- ( ( c e. RR /\ c =/= 0 ) -> E. x e. RR ( c x. x ) = 1 )
46	44, 45	sylan2 482	. . . 4 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> E. x e. RR ( c x. x ) = 1 )
47		simpr 468	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> A e. CC )
48		simplrl 778	. . . . . . . . . . 11 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. RR )
49	48	recnd 9687	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> x e. CC )
50	47, 49	mulcld 9681	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( A x. x ) e. CC )
51		simplll 776	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> c e. RR )
52	51	recnd 9687	. . . . . . . . 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 9685	. . . . . . . 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 9683	. . . . . . . . . . . 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 777	. . . . . . . . . . . . 13 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( 1 + c ) = 0 )
58	57	oveq1d 6323	. . . . . . . . . . . 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 2507	. . . . . . . . . . 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 2531	. . . . . . . . . 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 9688	. . . . . . . . . . 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 9825	. . . . . . . . . . 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 1292	. . . . . . . . . 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 215	. . . . . . . . 9 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c + 0 ) = c )
67	66	oveq2d 6324	. . . . . . . 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 2507	. . . . . . 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 9819	. . . . . . . . . 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 1292	. . . . . . . . 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 779	. . . . . . . . . 10 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( c x. x ) = 1 )
72	71	oveq1d 6323	. . . . . . . . 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 9679	. . . . . . . . 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 2509	. . . . . . . 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 9830	. . . . . . . . 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 6326	. . . . . . 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 2513	. . . . . 6 |- ( ( ( ( c e. RR /\ ( 1 + c ) = 0 ) /\ ( x e. RR /\ ( c x. x ) = 1 ) ) /\ A e. CC ) -> ( A + 0 ) = A )
79	78	exp42 622	. . . . 5 |- ( ( c e. RR /\ ( 1 + c ) = 0 ) -> ( x e. RR -> ( ( c x. x ) = 1 -> ( A e. CC -> ( A + 0 ) = A ) ) ) )
80	79	rexlimdv 2870	. . . 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 2868	. 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 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   E.wrex 2757  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   _ici 9559   + caddc 9560   x. cmul 9562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-ltxr 9698

This theorem is referenced by:  cnegex  9832  addid2  9834  addcan2  9836  addid1i  9838  addid1d  9851  subid  9913  subid1  9914  addid0  10060  swrdccat3blem  12905  shftval3  13216  reim0  13258  isercolllem3  13807  fsumcvg  13855  summolem2a  13858  risefac1  14163  ovolicc1  22547  brbtwn2  25014  axsegconlem1  25026  ax5seglem4  25041  axeuclid  25072  axcontlem2  25074  axcontlem4  25076  stoweidlem26  37998  2zrngamnd  40449  aacllem  41046