Theorem imsval 31292

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 6885	. . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
2		fveq2 6885	. . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈))
3	1, 2	coeq12d 5841	. . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)))
4		df-ims 31208	. . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
5		fvex 6898	. . . 4 ⊢ (normCV‘𝑈) ∈ V
6		fvex 6898	. . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V
7	5, 6	coex 7942	. . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V
8	3, 4, 7	fvmpt 6993	. 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 5840	. 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))
13	8, 9, 12	3eqtr4g 2820	1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ∘ ccom 5654  ‘cfv 6538  NrmCVeccnv 31191   −_𝑣 cnsb 31196  norm_CVcnmcv 31197  IndMetcims 31198

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ims 31208

This theorem is used by:  imsdval  31293  imsdf  31296  cnims  31300  hhims  31779  hhssims  31881