Theorem imsval 31015

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 6883	. . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
2		fveq2 6883	. . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈))
3	1, 2	coeq12d 5852	. . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)))
4		df-ims 30931	. . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
5		fvex 6896	. . . 4 ⊢ (normCV‘𝑈) ∈ V
6		fvex 6896	. . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V
7	5, 6	coex 7928	. . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V
8	3, 4, 7	fvmpt 6991	. 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 5851	. 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))
13	8, 9, 12	3eqtr4g 2823	1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143   ∘ ccom 5667  ‘cfv 6538  NrmCVeccnv 30914   −_𝑣 cnsb 30919  norm_CVcnmcv 30920  IndMetcims 30921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-ims 30931

This theorem is referenced by:  imsdval  31016  imsdf  31019  cnims  31023  hhims  31502  hhssims  31604