Theorem isvci 8165

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 836	. . . . . . . . . 10 |- ((x e. X /\ y e. CC /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
7	6	3expa 832	. . . . . . . . 9 |- (((x e. X /\ y e. CC) /\ z e. X) -> (yS(xGz)) = ((ySx)G(ySz)))
8	7	r19.21aiva 1712	. . . . . . . 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 840	. . . . . . . . . 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 832	. . . . . . . . 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 1712	. . . . . . . 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 1712	. . . . . 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 1696	. . . 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 817	. . 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 8065	. . . . . 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 8018	. . . 4 |- X = ran G
24	23	isvc 8164	. . 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 1542	1 |- W e. CVec

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  A.wral 1643  <.cop 2408   X. cxp 3164  dom cdm 3166  -->wf 3174  (class class class)co 3958  CCcc 5215  1c1 5218   + caddc 5220   x. cmul 5222  Grpcgr 7995  Abelcabl 8062  CVeccvc 8128

This theorem is referenced by:  cnvc 8166  hilvc 8984  hhssnv 9089

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 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

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 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fo 3192  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-sub 5339  df-neg 5341  df-grp 7999  df-gid 8000  df-ginv 8001  df-abl 8063  df-vc 8129

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