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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 6106 | . 2 ⊢ (normCV‘𝑈) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
3 | eqid 2610 | . . 3 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
4 | 3 | nmcvfval 26846 | . 2 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
5 | opex 4859 | . . 3 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
6 | h2h.2 | . . . . . 6 ⊢ 𝑈 ∈ NrmCVec | |
7 | 1, 6 | eqeltrri 2685 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
8 | nvex 26850 | . . . . 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 1065 | . . 3 ⊢ normℎ ∈ V |
11 | 5, 10 | op2nd 7068 | . 2 ⊢ (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = normℎ |
12 | 2, 4, 11 | 3eqtrri 2637 | 1 ⊢ normℎ = (normCV‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ‘cfv 5804 2nd c2nd 7058 NrmCVeccnv 26823 normCVcnmcv 26829 +ℎ cva 27161 ·ℎ csm 27162 normℎcno 27164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-oprab 6553 df-2nd 7060 df-vc 26798 df-nv 26831 df-nmcv 26839 |
This theorem is referenced by: h2hmetdval 27219 hhnm 27412 |
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