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Mirrors > Home > HSE Home > Th. List > h2hnm | Structured version Visualization version GIF version |
Description: The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h2h.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
h2h.2 | ⊢ 𝑈 ∈ NrmCVec |
Ref | Expression |
---|---|
h2hnm | ⊢ normℎ = (normCV‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h2h.1 | . . 3 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
2 | 1 | fveq2i 6910 | . 2 ⊢ (normCV‘𝑈) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
3 | eqid 2735 | . . 3 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
4 | 3 | nmcvfval 30636 | . 2 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
5 | opex 5475 | . . 3 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
6 | h2h.2 | . . . . . 6 ⊢ 𝑈 ∈ NrmCVec | |
7 | 1, 6 | eqeltrri 2836 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
8 | nvex 30640 | . . . . 5 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → ( +ℎ ∈ V ∧ ·ℎ ∈ V ∧ normℎ ∈ V)) | |
9 | 7, 8 | ax-mp 5 | . . . 4 ⊢ ( +ℎ ∈ V ∧ ·ℎ ∈ V ∧ normℎ ∈ V) |
10 | 9 | simp3i 1140 | . . 3 ⊢ normℎ ∈ V |
11 | 5, 10 | op2nd 8022 | . 2 ⊢ (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = normℎ |
12 | 2, 4, 11 | 3eqtrri 2768 | 1 ⊢ normℎ = (normCV‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ‘cfv 6563 2nd c2nd 8012 NrmCVeccnv 30613 normCVcnmcv 30619 +ℎ cva 30949 ·ℎ csm 30950 normℎcno 30952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-oprab 7435 df-2nd 8014 df-vc 30588 df-nv 30621 df-nmcv 30629 |
This theorem is referenced by: h2hmetdval 31007 hhnm 31200 |
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