Theorem cnid 8127

Description: The group identity element of complex number addition is zero. (Contributed by Steve Rodriguez, 3-Dec-2006.)

Assertion

Ref	Expression
cnid	|- 0 = (Id` + )

Proof of Theorem cnid

Step	Hyp	Ref	Expression
1		addid2t 5329	. . . 4 |- (x e. CC -> (0 + x) = x)
2	1	rgen 1698	. . 3 |- A.x e. CC (0 + x) = x
3		0cn 5328	. . . 4 |- 0 e. CC
4		opreq1 3968	. . . . . . . 8 |- (y = 0 -> (y + x) = (0 + x))
5	4	eqeq1d 1483	. . . . . . 7 |- (y = 0 -> ((y + x) = x <-> (0 + x) = x))
6	5	ralbidv 1663	. . . . . 6 |- (y = 0 -> (A.x e. CC (y + x) = x <-> A.x e. CC (0 + x) = x))
7		cnaddabl 8126	. . . . . . . . . . 11 |- + e. Abel
8		ablgrp 8102	. . . . . . . . . . 11 |- ( + e. Abel -> + e. Grp)
9	7, 8	ax-mp 7	. . . . . . . . . 10 |- + e. Grp
10		axaddopr 5265	. . . . . . . . . . . . 13 |- + :(CC X. CC)-->CC
11	10	fdmi 3632	. . . . . . . . . . . 12 |- dom + = (CC X. CC)
12	9, 11	grprn 8056	. . . . . . . . . . 11 |- CC = ran +
13		eqid 1475	. . . . . . . . . . 11 |- (Id` + ) = (Id` + )
14	12, 13	grpidval 8058	. . . . . . . . . 10 |- ( + e. Grp -> (Id` + ) = U.{y e. CC | A.x e. CC (y + x) = x})
15	9, 14	ax-mp 7	. . . . . . . . 9 |- (Id` + ) = U.{y e. CC | A.x e. CC (y + x) = x}
16	15	eqcomi 1479	. . . . . . . 8 |- U.{y e. CC | A.x e. CC (y + x) = x} = (Id` + )
17	16	a1i 8	. . . . . . 7 |- (y = 0 -> U.{y e. CC | A.x e. CC (y + x) = x} = (Id` + ))
18		id 59	. . . . . . 7 |- (y = 0 -> y = 0)
19	17, 18	eqeq12d 1489	. . . . . 6 |- (y = 0 -> (U.{y e. CC | A.x e. CC (y + x) = x} = y <-> (Id` + ) = 0))
20	6, 19	bibi12d 629	. . . . 5 |- (y = 0 -> ((A.x e. CC (y + x) = x <-> U.{y e. CC | A.x e. CC (y + x) = x} = y) <-> (A.x e. CC (0 + x) = x <-> (Id` + ) = 0)))
21	12	grpideu 8053	. . . . . . 7 |- ( + e. Grp -> E!y e. CC A.x e. CC (y + x) = x)
22	9, 21	ax-mp 7	. . . . . 6 |- E!y e. CC A.x e. CC (y + x) = x
23		reuuni1 2882	. . . . . 6 |- ((y e. CC /\ E!y e. CC A.x e. CC (y + x) = x) -> (A.x e. CC (y + x) = x <-> U.{y e. CC | A.x e. CC (y + x) = x} = y))
24	22, 23	mpan2 696	. . . . 5 |- (y e. CC -> (A.x e. CC (y + x) = x <-> U.{y e. CC | A.x e. CC (y + x) = x} = y))
25	20, 24	vtoclga 1852	. . . 4 |- (0 e. CC -> (A.x e. CC (0 + x) = x <-> (Id` + ) = 0))
26	3, 25	ax-mp 7	. . 3 |- (A.x e. CC (0 + x) = x <-> (Id` + ) = 0)
27	2, 26	mpbi 189	. 2 |- (Id` + ) = 0
28	27	eqcomi 1479	1 |- 0 = (Id` + )

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  A.wral 1645  E!wreu 1647  {crab 1648  U.cuni 2503   X. cxp 3168  ` cfv 3182  (class class class)co 3963  CCcc 5232  0cc0 5234   + caddc 5237  Grpcgr 8033  Idcgi 8034  Abelcabl 8099

This theorem is referenced by:  addinv 8128  readdsubg 8129  zaddsubg 8130  cnnv 8307

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-sub 5356  df-neg 5358  df-grp 8037  df-gid 8038  df-abl 8100

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