Theorem imsval 30781

Description: Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
imsval.3	⊢ 𝑀 = ( −𝑣 ‘𝑈)
imsval.6	⊢ 𝑁 = (normCV‘𝑈)
imsval.8	⊢ 𝐷 = (IndMet‘𝑈)

Assertion

Ref	Expression
imsval	⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))

Proof of Theorem imsval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
2		fveq2 6834	. . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈))
3	1, 2	coeq12d 5813	. . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)))
4		df-ims 30697	. . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
5		fvex 6847	. . . 4 ⊢ (normCV‘𝑈) ∈ V
6		fvex 6847	. . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V
7	5, 6	coex 7877	. . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V
8	3, 4, 7	fvmpt 6942	. 2 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)))
9		imsval.8	. 2 ⊢ 𝐷 = (IndMet‘𝑈)
10		imsval.6	. . 3 ⊢ 𝑁 = (normCV‘𝑈)
11		imsval.3	. . 3 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
12	10, 11	coeq12i 5812	. 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))
13	8, 9, 12	3eqtr4g 2800	1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ∘ ccom 5629  ‘cfv 6492  NrmCVeccnv 30680   −_𝑣 cnsb 30685  norm_CVcnmcv 30686  IndMetcims 30687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-ims 30697

This theorem is referenced by:  imsdval  30782  imsdf  30785  cnims  30789  hhims  31268  hhssims  31370