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Theorem vsfval 26872
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 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-vs 26838 . . . . 5 𝑣 = ( /𝑔 ∘ +𝑣 )
21fveq1i 6104 . . . 4 ( −𝑣𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3 fo1st 7079 . . . . . . . 8 1st :V–onto→V
4 fof 6028 . . . . . . . 8 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . . . . 7 1st :V⟶V
6 fco 5971 . . . . . . 7 ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
75, 5, 6mp2an 704 . . . . . 6 (1st ∘ 1st ):V⟶V
8 df-va 26834 . . . . . . 7 +𝑣 = (1st ∘ 1st )
98feq1i 5949 . . . . . 6 ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
107, 9mpbir 220 . . . . 5 +𝑣 :V⟶V
11 fvco3 6185 . . . . 5 (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
1210, 11mpan 702 . . . 4 (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
132, 12syl5eq 2656 . . 3 (𝑈 ∈ V → ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
14 0ngrp 26749 . . . . . 6 ¬ ∅ ∈ GrpOp
15 vex 3176 . . . . . . . . . 10 𝑔 ∈ V
1615rnex 6992 . . . . . . . . 9 ran 𝑔 ∈ V
1716, 16mpt2ex 7136 . . . . . . . 8 (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18 df-gdiv 26734 . . . . . . . 8 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
1917, 18dmmpti 5936 . . . . . . 7 dom /𝑔 = GrpOp
2019eleq2i 2680 . . . . . 6 (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
2114, 20mtbir 312 . . . . 5 ¬ ∅ ∈ dom /𝑔
22 ndmfv 6128 . . . . 5 (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
2321, 22mp1i 13 . . . 4 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24 fvprc 6097 . . . . 5 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
2524fveq2d 6107 . . . 4 𝑈 ∈ V → ( /𝑔 ‘( +𝑣𝑈)) = ( /𝑔 ‘∅))
26 fvprc 6097 . . . 4 𝑈 ∈ V → ( −𝑣𝑈) = ∅)
2723, 25, 263eqtr4rd 2655 . . 3 𝑈 ∈ V → ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
2813, 27pm2.61i 175 . 2 ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈))
29 vsfval.3 . 2 𝑀 = ( −𝑣𝑈)
30 vsfval.2 . . 3 𝐺 = ( +𝑣𝑈)
3130fveq2i 6106 . 2 ( /𝑔𝐺) = ( /𝑔 ‘( +𝑣𝑈))
3228, 29, 313eqtr4i 2642 1 𝑀 = ( /𝑔𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  dom cdm 5038  ran crn 5039  ccom 5042  wf 5800  ontowfo 5802  cfv 5804  (class class class)co 6549  cmpt2 6551  1st c1st 7057  GrpOpcgr 26727  invcgn 26729   /𝑔 cgs 26730   +𝑣 cpv 26824  𝑣 cnsb 26828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gdiv 26734  df-va 26834  df-vs 26838
This theorem is referenced by:  nvm  26880  nvmfval  26883  nvnnncan1  26886  nvaddsub  26894
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