Theorem imsval 30621

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 6861	. . . 4 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
2		fveq2 6861	. . . 4 ⊢ (𝑢 = 𝑈 → ( −𝑣 ‘𝑢) = ( −𝑣 ‘𝑈))
3	1, 2	coeq12d 5831	. . 3 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)))
4		df-ims 30537	. . 3 ⊢ IndMet = (𝑢 ∈ NrmCVec ↦ ((normCV‘𝑢) ∘ ( −𝑣 ‘𝑢)))
5		fvex 6874	. . . 4 ⊢ (normCV‘𝑈) ∈ V
6		fvex 6874	. . . 4 ⊢ ( −𝑣 ‘𝑈) ∈ V
7	5, 6	coex 7909	. . 3 ⊢ ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈)) ∈ V
8	3, 4, 7	fvmpt 6971	. 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 5830	. 2 ⊢ (𝑁 ∘ 𝑀) = ((normCV‘𝑈) ∘ ( −𝑣 ‘𝑈))
13	8, 9, 12	3eqtr4g 2790	1 ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∘ ccom 5645  ‘cfv 6514  NrmCVeccnv 30520   −_𝑣 cnsb 30525  norm_CVcnmcv 30526  IndMetcims 30527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fv 6522  df-ims 30537

This theorem is referenced by:  imsdval  30622  imsdf  30625  cnims  30629  hhims  31108  hhssims  31210