MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vsfval Structured version   Visualization version   GIF version

Theorem vsfval 30662
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 30628 . . . . 5 𝑣 = ( /𝑔 ∘ +𝑣 )
21fveq1i 6908 . . . 4 ( −𝑣𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3 fo1st 8033 . . . . . . . 8 1st :V–onto→V
4 fof 6821 . . . . . . . 8 (1st :V–onto→V → 1st :V⟶V)
53, 4ax-mp 5 . . . . . . 7 1st :V⟶V
6 fco 6761 . . . . . . 7 ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
75, 5, 6mp2an 692 . . . . . 6 (1st ∘ 1st ):V⟶V
8 df-va 30624 . . . . . . 7 +𝑣 = (1st ∘ 1st )
98feq1i 6728 . . . . . 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 2787 . . 3 (𝑈 ∈ V → ( −𝑣𝑈) = ( /𝑔 ‘( +𝑣𝑈)))
14 0ngrp 30540 . . . . . 6 ¬ ∅ ∈ GrpOp
15 vex 3482 . . . . . . . . . 10 𝑔 ∈ V
1615rnex 7933 . . . . . . . . 9 ran 𝑔 ∈ V
1716, 16mpoex 8103 . . . . . . . 8 (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18 df-gdiv 30525 . . . . . . . 8 /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
1917, 18dmmpti 6713 . . . . . . 7 dom /𝑔 = GrpOp
2019eleq2i 2831 . . . . . 6 (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
2114, 20mtbir 323 . . . . 5 ¬ ∅ ∈ dom /𝑔
22 ndmfv 6942 . . . . 5 (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
2321, 22mp1i 13 . . . 4 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24 fvprc 6899 . . . . 5 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
2524fveq2d 6911 . . . 4 𝑈 ∈ V → ( /𝑔 ‘( +𝑣𝑈)) = ( /𝑔 ‘∅))
26 fvprc 6899 . . . 4 𝑈 ∈ V → ( −𝑣𝑈) = ∅)
2723, 25, 263eqtr4rd 2786 . . 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 6910 . 2 ( /𝑔𝐺) = ( /𝑔 ‘( +𝑣𝑈))
3228, 29, 313eqtr4i 2773 1 𝑀 = ( /𝑔𝐺)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  dom cdm 5689  ran crn 5690  ccom 5693  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  GrpOpcgr 30518  invcgn 30520   /𝑔 cgs 30521   +𝑣 cpv 30614  𝑣 cnsb 30618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-grpo 30522  df-gdiv 30525  df-va 30624  df-vs 30628
This theorem is referenced by:  nvm  30670  nvmfval  30673  nvnnncan1  30676  nvaddsub  30684
  Copyright terms: Public domain W3C validator