Theorem imsval 31269

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 6877	. . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
2		fveq2 6877	. . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈))
3	1, 2	coeq12d 5842	. . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)))
4		df-ims 31185	. . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
5		fvex 6890	. . . 4 ⊢ (normCV‘𝑈) ∈ V
6		fvex 6890	. . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V
7	5, 6	coex 7931	. . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V
8	3, 4, 7	fvmpt 6985	. 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 5841	. 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))
13	8, 9, 12	3eqtr4g 2821	1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∘ ccom 5655  ‘cfv 6531  NrmCVeccnv 31168   −_𝑣 cnsb 31173  norm_CVcnmcv 31174  IndMetcims 31175

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6487  df-fun 6533  df-fv 6539  df-ims 31185

This theorem is used by:  imsdval  31270  imsdf  31273  cnims  31277  hhims  31756  hhssims  31858