Theorem vsfval 8262

Description: Value of the function for the vector subtraction operation on a normed complex vector space.

Hypotheses

Ref	Expression
vsfval.2	⊢ G = ( +v ‘U)
vsfval.3	⊢ M = ( −v ‘U)

Assertion

Ref	Expression
vsfval	⊢ M = ( /g ‘G)

Proof of Theorem vsfval

Step	Hyp	Ref	Expression
1		df-va 8222	. . . . . . . 8 ⊢ +v = (1st ∘ 1st )
2	1	dmeqi 3326	. . . . . . 7 ⊢ dom +v = dom (1st ∘ 1st )
3		fo1st 4105	. . . . . . . . . . 11 ⊢ 1st :V–onto→V
4		fof 3686	. . . . . . . . . . 11 ⊢ (1st :V–onto→V → 1st :V–→V)
5	3, 4	ax-mp 7	. . . . . . . . . 10 ⊢ 1st :V–→V
6	5	fdmi 3646	. . . . . . . . 9 ⊢ dom 1st = V
7		forn 3688	. . . . . . . . . 10 ⊢ (1st :V–onto→V → ran 1st = V)
8	3, 7	ax-mp 7	. . . . . . . . 9 ⊢ ran 1st = V
9	6, 8	eqtr4 1505	. . . . . . . 8 ⊢ dom 1st = ran 1st
10		dmcoeq 3380	. . . . . . . 8 ⊢ (dom 1st = ran 1st → dom (1st ∘ 1st ) = dom 1st )
11	9, 10	ax-mp 7	. . . . . . 7 ⊢ dom (1st ∘ 1st ) = dom 1st
12	2, 11, 6	3eqtr 1506	. . . . . 6 ⊢ dom +v = V
13	12	eleq2i 1545	. . . . 5 ⊢ (U ∈ dom +v ↔ U ∈ V)
14		visset 1820	. . . . . . . . . 10 ⊢ g ∈ V
15	14	rnex 3375	. . . . . . . . 9 ⊢ ran g ∈ V
16		eqid 1482	. . . . . . . . 9 ⊢ {⟨⟨x, y⟩, z⟩∣((x ∈ ran g ⋀ y ∈ ran g) ⋀ z = (xg((inv ‘g) ‘y)))} = {⟨⟨x, y⟩, z⟩∣((x ∈ ran g ⋀ y ∈ ran g) ⋀ z = (xg((inv ‘g) ‘y)))}
17	15, 15, 16	oprabex2 4035	. . . . . . . 8 ⊢ {⟨⟨x, y⟩, z⟩∣((x ∈ ran g ⋀ y ∈ ran g) ⋀ z = (xg((inv ‘g) ‘y)))} ∈ V
18		df-gdiv 8049	. . . . . . . 8 ⊢ /g = {⟨g, f⟩∣(g ∈ Grp ⋀ f = {⟨⟨x, y⟩, z⟩∣((x ∈ ran g ⋀ y ∈ ran g) ⋀ z = (xg((inv ‘g) ‘y)))})}
19	17, 18	fnopab2 3632	. . . . . . 7 ⊢ /g Fn Grp
20		fnfun 3599	. . . . . . 7 ⊢ ( /g Fn Grp → Fun /g )
21	19, 20	ax-mp 7	. . . . . 6 ⊢ Fun /g
22		fofun 3687	. . . . . . . . 9 ⊢ (1st :V–onto→V → Fun 1st )
23	3, 22	ax-mp 7	. . . . . . . 8 ⊢ Fun 1st
24		funco 3564	. . . . . . . 8 ⊢ ((Fun 1st ⋀ Fun 1st ) → Fun (1st ∘ 1st ))
25	23, 23, 24	mp2an 701	. . . . . . 7 ⊢ Fun (1st ∘ 1st )
26		funeq 3549	. . . . . . . 8 ⊢ ( +v = (1st ∘ 1st ) → (Fun +v ↔ Fun (1st ∘ 1st )))
27	1, 26	ax-mp 7	. . . . . . 7 ⊢ (Fun +v ↔ Fun (1st ∘ 1st ))
28	25, 27	mpbir 190	. . . . . 6 ⊢ Fun +v
29		fvco 3788	. . . . . 6 ⊢ ((Fun /g ⋀ Fun +v ⋀ U ∈ dom +v ) → (( /g ∘ +v ) ‘U) = ( /g ‘( +v ‘U)))
30	21, 28, 29	mp3an12 910	. . . . 5 ⊢ (U ∈ dom +v → (( /g ∘ +v ) ‘U) = ( /g ‘( +v ‘U)))
31	13, 30	sylbir 201	. . . 4 ⊢ (U ∈ V → (( /g ∘ +v ) ‘U) = ( /g ‘( +v ‘U)))
32		df-vs 8226	. . . . 5 ⊢ −v = ( /g ∘ +v )
33	32	fveq1i 3739	. . . 4 ⊢ ( −v ‘U) = (( /g ∘ +v ) ‘U)
34	31, 33	syl5eq 1526	. . 3 ⊢ (U ∈ V → ( −v ‘U) = ( /g ‘( +v ‘U)))
35		0ngrp 8064	. . . . . . 7 ⊢ ¬ ∅ ∈ Grp
36	17, 18	dmopab2 3633	. . . . . . . 8 ⊢ dom /g = Grp
37	36	eleq2i 1545	. . . . . . 7 ⊢ (∅ ∈ dom /g ↔ ∅ ∈ Grp)
38	35, 37	mtbir 192	. . . . . 6 ⊢ ¬ ∅ ∈ dom /g
39		ndmfv 3759	. . . . . 6 ⊢ (¬ ∅ ∈ dom /g → ( /g ‘∅) = ∅)
40	38, 39	ax-mp 7	. . . . 5 ⊢ ( /g ‘∅) = ∅
41	40	a1i 8	. . . 4 ⊢ (¬ U ∈ V → ( /g ‘∅) = ∅)
42		fvprc 3735	. . . . 5 ⊢ (¬ U ∈ V → ( +v ‘U) = ∅)
43	42	fveq2d 3742	. . . 4 ⊢ (¬ U ∈ V → ( /g ‘( +v ‘U)) = ( /g ‘∅))
44		fvprc 3735	. . . 4 ⊢ (¬ U ∈ V → ( −v ‘U) = ∅)
45	41, 43, 44	3eqtr4rd 1525	. . 3 ⊢ (¬ U ∈ V → ( −v ‘U) = ( /g ‘( +v ‘U)))
46	34, 45	pm2.61i 126	. 2 ⊢ ( −v ‘U) = ( /g ‘( +v ‘U))
47		vsfval.3	. 2 ⊢ M = ( −v ‘U)
48		vsfval.2	. . 3 ⊢ G = ( +v ‘U)
49	48	fveq2i 3741	. 2 ⊢ ( /g ‘G) = ( /g ‘( +v ‘U))
50	46, 47, 49	3eqtr4 1512	1 ⊢ M = ( /g ‘G)

Syntax hints:  ¬ wn 2   ↔ wb 146   ⋀ wa 223   = wceq 960   ∈ wcel 962  Vcvv 1818  ∅c0 2289  dom cdm 3184  ran crn 3185   ∘ ccom 3188  Fun wfun 3190   Fn wfn 3191  –→wf 3192  –onto→wfo 3194   ‘cfv 3196  (class class class)co 3977  {copab2 3978  1^st c1st 4091  Grpcgr 8042  invcgn 8044   /_g cgs 8045   +_v cpv 8212   −_v cnsb 8216

This theorem is referenced by:  nvm 8270  nvmfval 8272  nvnnncan1 8276  nvnnncan2 8277  nvaddsubass 8286  nvaddsub 8287  nvmtri2 8308  va1cnlem 8353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-fo 3210  df-fv 3212  df-opr 3979  df-oprab 3980  df-1st 4093  df-grp 8046  df-gdiv 8049  df-va 8222  df-vs 8226

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