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Theorem imsval 31074

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 = (norm_CV‘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 6888	. . . 4 ⊢ (𝑢 = 𝑈 → (norm_CV‘𝑢) = (norm_CV‘𝑈))
2		fveq2 6888	. . . 4 ⊢ (𝑢 = 𝑈 → ( −_𝑣 ‘𝑢) = ( −_𝑣 ‘𝑈))
3	1, 2	coeq12d 5855	. . 3 ⊢ (𝑢 = 𝑈 → ((norm_CV‘𝑢) ∘ ( −_𝑣 ‘𝑢)) = ((norm_CV‘𝑈) ∘ ( −_𝑣 ‘𝑈)))
4		df-ims 30990	. . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((norm_CV‘𝑢) ∘ ( −_𝑣 ‘𝑢)))
5		fvex 6901	. . . 4 ⊢ (norm_CV‘𝑈) ∈ V
6		fvex 6901	. . . 4 ⊢ ( −_𝑣 ‘𝑈) ∈ V
7	5, 6	coex 7936	. . 3 ⊢ ((norm_CV‘𝑈) ∘ ( −_𝑣 ‘𝑈)) ∈ V
8	3, 4, 7	fvmpt 6996	. 2 ⊢ (𝑈 ∈ NrmCVec → (IndMet‘𝑈) = ((norm_CV‘𝑈) ∘ ( −_𝑣 ‘𝑈)))
9		imsval.8	. 2 ⊢ 𝐷 = (IndMet‘𝑈)
10		imsval.6	. . 3 ⊢ 𝑁 = (norm_CV‘𝑈)
11		imsval.3	. . 3 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
12	10, 11	coeq12i 5854	. 2 ⊢ (𝑁 ∘ 𝑀) = ((norm_CV‘𝑈) ∘ ( −_𝑣 ‘𝑈))
13	8, 9, 12	3eqtr4g 2826	1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ∘ ccom 5670 ‘cfv 6543 NrmCVeccnv 30973 −_𝑣 cnsb 30978 norm_CVcnmcv 30979 IndMetcims 30980

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6499 df-fun 6545 df-fv 6551 df-ims 30990

This theorem is used by: imsdval 31075 imsdf 31078 cnims 31082 hhims 31561 hhssims 31663

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