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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 6909 | . 2 ⊢ (normCV‘𝑈) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
| 3 | eqid 2737 | . . 3 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
| 4 | 3 | nmcvfval 30626 | . 2 ⊢ (normCV‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
| 5 | opex 5469 | . . 3 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
| 6 | h2h.2 | . . . . . 6 ⊢ 𝑈 ∈ NrmCVec | |
| 7 | 1, 6 | eqeltrri 2838 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
| 8 | nvex 30630 | . . . . 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 1142 | . . 3 ⊢ normℎ ∈ V |
| 11 | 5, 10 | op2nd 8023 | . 2 ⊢ (2nd ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = normℎ |
| 12 | 2, 4, 11 | 3eqtrri 2770 | 1 ⊢ normℎ = (normCV‘𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 2nd c2nd 8013 NrmCVeccnv 30603 normCVcnmcv 30609 +ℎ cva 30939 ·ℎ csm 30940 normℎcno 30942 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-oprab 7435 df-2nd 8015 df-vc 30578 df-nv 30611 df-nmcv 30619 |
| This theorem is referenced by: h2hmetdval 30997 hhnm 31190 |
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