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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 6899 | . 2 ⊢ (normCV‘𝑈) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
3 | eqid 2725 | . . 3 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
4 | 3 | nmcvfval 30489 | . 2 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
5 | opex 5466 | . . 3 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
6 | h2h.2 | . . . . . 6 ⊢ 𝑈 ∈ NrmCVec | |
7 | 1, 6 | eqeltrri 2822 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
8 | nvex 30493 | . . . . 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 1138 | . . 3 ⊢ normℎ ∈ V |
11 | 5, 10 | op2nd 8003 | . 2 ⊢ (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = normℎ |
12 | 2, 4, 11 | 3eqtrri 2758 | 1 ⊢ normℎ = (normCV‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3461 〈cop 4636 ‘cfv 6549 2nd c2nd 7993 NrmCVeccnv 30466 normCVcnmcv 30472 +ℎ cva 30802 ·ℎ csm 30803 normℎcno 30805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fv 6557 df-oprab 7423 df-2nd 7995 df-vc 30441 df-nv 30474 df-nmcv 30482 |
This theorem is referenced by: h2hmetdval 30860 hhnm 31053 |
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