Theorem isvci 8201

Description: Properties that determine a complex vector space.

Hypotheses

Ref	Expression
isvci.1	|- G e. Abel
isvci.2	|- dom G = (X X. X)
isvci.3	|- S:(CC X. X)-->X
isvci.4	|- (x e. X -> (1Sx) = x)
isvci.5	|- ((y e. CC /\ x e. X /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
isvci.6	|- ((y e. CC /\ z e. CC /\ x e. X) -> ((y + z)Sx) = ((ySx)G(zSx)))
isvci.7	|- ((y e. CC /\ z e. CC /\ x e. X) -> ((y x. z)Sx) = (yS(zSx)))
isvci.8	|- W = <.G, S>.

Assertion

Ref	Expression
isvci	|- W e. CVec

Distinct variable groups:   x,y,z,G   x,S,y,z   x,X,y,z

Proof of Theorem isvci

Step	Hyp	Ref	Expression
1		isvci.8	. 2 |- W = <.G, S>.
2		isvci.1	. . . 4 |- G e. Abel
3		isvci.3	. . . 4 |- S:(CC X. X)-->X
4		isvci.4	. . . . . 6 |- (x e. X -> (1Sx) = x)
5		isvci.5	. . . . . . . . . . 11 |- ((y e. CC /\ x e. X /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
6	5	3com12 837	. . . . . . . . . 10 |- ((x e. X /\ y e. CC /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
7	6	3expa 833	. . . . . . . . 9 |- (((x e. X /\ y e. CC) /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
8	7	r19.21aiva 1714	. . . . . . . 8 |- ((x e. X /\ y e. CC) -> A.z e. X (yS(xGz)) = ((ySx)G(ySz)))
9		isvci.6	. . . . . . . . . . . 12 |- ((y e. CC /\ z e. CC /\ x e. X) -> ((y + z)Sx) = ((ySx)G(zSx)))
10		isvci.7	. . . . . . . . . . . 12 |- ((y e. CC /\ z e. CC /\ x e. X) -> ((y x. z)Sx) = (yS(zSx)))
11	9, 10	jca 288	. . . . . . . . . . 11 |- ((y e. CC /\ z e. CC /\ x e. X) -> (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
12	11	3comr 841	. . . . . . . . . 10 |- ((x e. X /\ y e. CC /\ z e. CC) -> (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
13	12	3expa 833	. . . . . . . . 9 |- (((x e. X /\ y e. CC) /\ z e. CC) -> (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
14	13	r19.21aiva 1714	. . . . . . . 8 |- ((x e. X /\ y e. CC) -> A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))
15	8, 14	jca 288	. . . . . . 7 |- ((x e. X /\ y e. CC) -> (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))
16	15	r19.21aiva 1714	. . . . . 6 |- (x e. X -> A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))
17	4, 16	jca 288	. . . . 5 |- (x e. X -> ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))))
18	17	rgen 1698	. . . 4 |- A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))
19	2, 3, 18	3pm3.2i 818	. . 3 |- (G e. Abel /\ S:(CC X. X)-->X /\ A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx))))))
20		ablgrp 8102	. . . . . 6 |- (G e. Abel -> G e. Grp)
21	2, 20	ax-mp 7	. . . . 5 |- G e. Grp
22		isvci.2	. . . . 5 |- dom G = (X X. X)
23	21, 22	grprn 8056	. . . 4 |- X = ran G
24	23	isvc 8200	. . 3 |- (<.G, S>. e. CVec <-> (G e. Abel /\ S:(CC X. X)-->X /\ A.x e. X ((1Sx) = x /\ A.y e. CC (A.z e. X (yS(xGz)) = ((ySx)G(ySz)) /\ A.z e. CC (((y + z)Sx) = ((ySx)G(zSx)) /\ ((y x. z)Sx) = (yS(zSx)))))))
25	19, 24	mpbir 190	. 2 |- <.G, S>. e. CVec
26	1, 25	eqeltr 1544	1 |- W e. CVec

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  <.cop 2411   X. cxp 3168  dom cdm 3170  -->wf 3178  (class class class)co 3963  CCcc 5232  1c1 5235   + caddc 5237   x. cmul 5239  Grpcgr 8033  Abelcabl 8099  CVeccvc 8164

This theorem is referenced by:  cnvc 8202  hilvc 9029  hhssnv 9134

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-ltr 5170  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-ginv 8039  df-abl 8100  df-vc 8165

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