Theorem cnring 8147

Description: The set of complex numbers is a (unital) ring. (Contributed by Steve Rodriguez, 2-Feb-2007.)

Assertion

Ref	Expression
cnring	|- <. + , x. >. e. Ring

Proof of Theorem cnring

Step	Hyp	Ref	Expression
1		mulex 5305	. . 3 |- x. e. V
2		cnaddabl 8111	. . . . . 6 |- + e. Abel
3		ablgrp 8086	. . . . . 6 |- ( + e. Abel -> + e. Grp)
4	2, 3	ax-mp 7	. . . . 5 |- + e. Grp
5		axaddopr 5252	. . . . 5 |- + :(CC X. CC)-->CC
6	4, 5	grprnOLD 8040	. . . 4 |- CC = ran +
7	6	isring 8126	. . 3 |- ( x. e. V -> (<. + , x. >. e. Ring <-> (( + e. Abel /\ x. :(CC X. CC)-->CC) /\ (A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))) /\ E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y)))))
8	1, 7	ax-mp 7	. 2 |- (<. + , x. >. e. Ring <-> (( + e. Abel /\ x. :(CC X. CC)-->CC) /\ (A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))) /\ E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y))))
9		axmulopr 5253	. . 3 |- x. :(CC X. CC)-->CC
10	2, 9	pm3.2i 285	. 2 |- ( + e. Abel /\ x. :(CC X. CC)-->CC)
11		axmulass 5265	. . . . 5 |- ((x e. CC /\ y e. CC /\ z e. CC) -> ((x x. y) x. z) = (x x. (y x. z)))
12		axdistr 5266	. . . . 5 |- ((x e. CC /\ y e. CC /\ z e. CC) -> (x x. (y + z)) = ((x x. y) + (x x. z)))
13		adddirt 5306	. . . . 5 |- ((x e. CC /\ y e. CC /\ z e. CC) -> ((x + y) x. z) = ((x x. z) + (y x. z)))
14	11, 12, 13	3jca 818	. . . 4 |- ((x e. CC /\ y e. CC /\ z e. CC) -> (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))))
15	14	rgen3 1723	. . 3 |- A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z)))
16		ax1cn 5256	. . . 4 |- 1 e. CC
17		ax1id 5269	. . . . . 6 |- (y e. CC -> (y x. 1) = y)
18		mulid2t 5404	. . . . . 6 |- (y e. CC -> (1 x. y) = y)
19	17, 18	jca 288	. . . . 5 |- (y e. CC -> ((y x. 1) = y /\ (1 x. y) = y))
20	19	rgen 1697	. . . 4 |- A.y e. CC ((y x. 1) = y /\ (1 x. y) = y)
21		opreq2 3966	. . . . . . . 8 |- (x = 1 -> (y x. x) = (y x. 1))
22	21	eqeq1d 1482	. . . . . . 7 |- (x = 1 -> ((y x. x) = y <-> (y x. 1) = y))
23		opreq1 3965	. . . . . . . 8 |- (x = 1 -> (x x. y) = (1 x. y))
24	23	eqeq1d 1482	. . . . . . 7 |- (x = 1 -> ((x x. y) = y <-> (1 x. y) = y))
25	22, 24	anbi12d 627	. . . . . 6 |- (x = 1 -> (((y x. x) = y /\ (x x. y) = y) <-> ((y x. 1) = y /\ (1 x. y) = y)))
26	25	ralbidv 1662	. . . . 5 |- (x = 1 -> (A.y e. CC ((y x. x) = y /\ (x x. y) = y) <-> A.y e. CC ((y x. 1) = y /\ (1 x. y) = y)))
27	26	rcla4ev 1875	. . . 4 |- ((1 e. CC /\ A.y e. CC ((y x. 1) = y /\ (1 x. y) = y)) -> E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y))
28	16, 20, 27	mp2an 696	. . 3 |- E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y)
29	15, 28	pm3.2i 285	. 2 |- (A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))) /\ E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y))
30	8, 10, 29	mpbir2an 729	1 |- <. + , x. >. e. Ring

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  A.wral 1644  E.wrex 1645  Vcvv 1809  <.cop 2409   X. cxp 3165  -->wf 3175  (class class class)co 3960  CCcc 5219  1c1 5222   + caddc 5224   x. cmul 5226  Grpcgr 8016  Abelcabl 8083  Ringcring 8124

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-mp 5076  df-ltp 5077  df-plpr 5151  df-mpr 5152  df-enr 5153  df-nr 5154  df-plr 5155  df-mr 5156  df-0r 5158  df-1r 5159  df-m1r 5160  df-c 5227  df-0 5228  df-1 5229  df-i 5230  df-r 5231  df-plus 5232  df-mul 5233  df-sub 5343  df-neg 5345  df-grp 8020  df-abl 8084  df-ring 8125

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