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Theorem vsfval 30652
Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
vsfval.2 𝐺 = ( +𝑣𝑈)
vsfval.3 𝑀 = ( −𝑣𝑈)
Assertion
Ref Expression
vsfval 𝑀 = ( /𝑔𝐺)

Proof of Theorem vsfval
Dummy variables 𝑥 𝑔 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vs 30618 . . . . 5 𝑣 = ( /𝑔 ∘ +𝑣 )
21fveq1i 6907 . . . 4 ( −𝑣𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3 fo1st 8034 . . . . . . . 8 1st :V–onto→V
4 fof 6820 . . . . . . . 8 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . . . . 7 1st :V⟶V
6 fco 6760 . . . . . . 7 ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
75, 5, 6mp2an 692 . . . . . 6 (1st ∘ 1st ):V⟶V
8 df-va 30614 . . . . . . 7 +𝑣 = (1st ∘ 1st )
98feq1i 6727 . . . . . 6 ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
107, 9mpbir 231 . . . . 5 +𝑣 :V⟶V
11 fvco3 7008 . . . . 5 (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
1210, 11mpan 690 . . . 4 (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
132, 12eqtrid 2789 . . 3 (𝑈 ∈ V → ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
14 0ngrp 30530 . . . . . 6 ¬ ∅ ∈ GrpOp
15 vex 3484 . . . . . . . . . 10 𝑔 ∈ V
1615rnex 7932 . . . . . . . . 9 ran 𝑔 ∈ V
1716, 16mpoex 8104 . . . . . . . 8 (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18 df-gdiv 30515 . . . . . . . 8 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
1917, 18dmmpti 6712 . . . . . . 7 dom /𝑔 = GrpOp
2019eleq2i 2833 . . . . . 6 (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
2114, 20mtbir 323 . . . . 5 ¬ ∅ ∈ dom /𝑔
22 ndmfv 6941 . . . . 5 (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
2321, 22mp1i 13 . . . 4 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24 fvprc 6898 . . . . 5 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
2524fveq2d 6910 . . . 4 𝑈 ∈ V → ( /𝑔 ‘( +𝑣𝑈)) = ( /𝑔 ‘∅))
26 fvprc 6898 . . . 4 𝑈 ∈ V → ( −𝑣𝑈) = ∅)
2723, 25, 263eqtr4rd 2788 . . 3 𝑈 ∈ V → ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
2813, 27pm2.61i 182 . 2 ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈))
29 vsfval.3 . 2 𝑀 = ( −𝑣𝑈)
30 vsfval.2 . . 3 𝐺 = ( +𝑣𝑈)
3130fveq2i 6909 . 2 ( /𝑔𝐺) = ( /𝑔 ‘( +𝑣𝑈))
3228, 29, 313eqtr4i 2775 1 𝑀 = ( /𝑔𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  dom cdm 5685  ran crn 5686  ccom 5689  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  GrpOpcgr 30508  invcgn 30510   /𝑔 cgs 30511   +𝑣 cpv 30604  𝑣 cnsb 30608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gdiv 30515  df-va 30614  df-vs 30618
This theorem is referenced by:  nvm  30660  nvmfval  30663  nvnnncan1  30666  nvaddsub  30674
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