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Mirrors > Home > HSE Home > Th. List > h2hsm | Structured version Visualization version GIF version |
Description: The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h2h.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
h2h.2 | ⊢ 𝑈 ∈ NrmCVec |
Ref | Expression |
---|---|
h2hsm | ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
2 | 1 | smfval 28908 | . . 3 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
3 | opex 5378 | . . . . 5 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
4 | h2h.1 | . . . . . . . 8 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
5 | h2h.2 | . . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec | |
6 | 4, 5 | eqeltrri 2834 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
7 | nvex 28914 | . . . . . . 7 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → ( +ℎ ∈ V ∧ ·ℎ ∈ V ∧ normℎ ∈ V)) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ ( +ℎ ∈ V ∧ ·ℎ ∈ V ∧ normℎ ∈ V) |
9 | 8 | simp3i 1139 | . . . . 5 ⊢ normℎ ∈ V |
10 | 3, 9 | op1st 7817 | . . . 4 ⊢ (1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = 〈 +ℎ , ·ℎ 〉 |
11 | 10 | fveq2i 6764 | . . 3 ⊢ (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (2nd ‘〈 +ℎ , ·ℎ 〉) |
12 | 8 | simp1i 1137 | . . . 4 ⊢ +ℎ ∈ V |
13 | 8 | simp2i 1138 | . . . 4 ⊢ ·ℎ ∈ V |
14 | 12, 13 | op2nd 7818 | . . 3 ⊢ (2nd ‘〈 +ℎ , ·ℎ 〉) = ·ℎ |
15 | 2, 11, 14 | 3eqtrri 2770 | . 2 ⊢ ·ℎ = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
16 | 4 | fveq2i 6764 | . 2 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
17 | 15, 16 | eqtr4i 2768 | 1 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1085 = wceq 1539 ∈ wcel 2107 Vcvv 3427 〈cop 4569 ‘cfv 6423 1st c1st 7807 2nd c2nd 7808 NrmCVeccnv 28887 ·𝑠OLD cns 28890 +ℎ cva 29223 ·ℎ csm 29224 normℎcno 29226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7571 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3429 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-fo 6429 df-fv 6431 df-oprab 7264 df-1st 7809 df-2nd 7810 df-vc 28862 df-nv 28895 df-sm 28900 |
This theorem is referenced by: h2hvs 29280 axhfvmul-zf 29290 axhvmulid-zf 29291 axhvmulass-zf 29292 axhvdistr1-zf 29293 axhvdistr2-zf 29294 axhvmul0-zf 29295 axhis3-zf 29299 hhsm 29472 |
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