Theorem normvalt 5075

Description: The value of the norm of a vector in Hilbert space. Definition of norm in [Beran] p. 96. In the literature, the norm of A is usually written as "|| A ||", but we use function value notation to take advantage of our existing theorems about functions.

Assertion

Ref	Expression
normvalt	⊢ (A ∈ ℋ → (norm ‘A) = (√ ‘(A ·i A)))

Proof of Theorem normvalt

Step	Hyp	Ref	Expression
1		opreq12 3008	. . . 4 ⊢ ((x = A ∧ x = A) → (x ·i x) = (A ·i A))
2	1	anidms 332	. . 3 ⊢ (x = A → (x ·i x) = (A ·i A))
3	2	fveq2d 2836	. 2 ⊢ (x = A → (√ ‘(x ·i x)) = (√ ‘(A ·i A)))
4		df-hnorm 5074	. 2 ⊢ norm = {⟨x, y⟩∣(x ∈ ℋ ∧ y = (√ ‘(x ·i x)))}
5		fvex 2838	. 2 ⊢ (√ ‘(A ·i A)) ∈ V
6	3, 4, 5	fvopab4 2871	1 ⊢ (A ∈ ℋ → (norm ‘A) = (√ ‘(A ·i A)))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092   ‘cfv 2422  (class class class)co 3001  √csqr 4727   ℋ chil 4958   ·_i csp 4963  normcno 4964

This theorem is referenced by:  normclt 5076  normge0t 5077  normgt0t 5078  norm0 5079  normsq 5082  norm-ii 5086  norm-iii 5087  bcs 5101

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-hnorm 5074

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