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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 2798 | . . . 4 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
2 | 1 | smfval 28388 | . . 3 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
3 | opex 5321 | . . . . 5 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
4 | h2h.1 | . . . . . . . 8 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
5 | h2h.2 | . . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec | |
6 | 4, 5 | eqeltrri 2887 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
7 | nvex 28394 | . . . . . . 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 1138 | . . . . 5 ⊢ normℎ ∈ V |
10 | 3, 9 | op1st 7679 | . . . 4 ⊢ (1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = 〈 +ℎ , ·ℎ 〉 |
11 | 10 | fveq2i 6648 | . . 3 ⊢ (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (2nd ‘〈 +ℎ , ·ℎ 〉) |
12 | 8 | simp1i 1136 | . . . 4 ⊢ +ℎ ∈ V |
13 | 8 | simp2i 1137 | . . . 4 ⊢ ·ℎ ∈ V |
14 | 12, 13 | op2nd 7680 | . . 3 ⊢ (2nd ‘〈 +ℎ , ·ℎ 〉) = ·ℎ |
15 | 2, 11, 14 | 3eqtrri 2826 | . 2 ⊢ ·ℎ = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
16 | 4 | fveq2i 6648 | . 2 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
17 | 15, 16 | eqtr4i 2824 | 1 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 NrmCVeccnv 28367 ·𝑠OLD cns 28370 +ℎ cva 28703 ·ℎ csm 28704 normℎcno 28706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-oprab 7139 df-1st 7671 df-2nd 7672 df-vc 28342 df-nv 28375 df-sm 28380 |
This theorem is referenced by: h2hvs 28760 axhfvmul-zf 28770 axhvmulid-zf 28771 axhvmulass-zf 28772 axhvdistr1-zf 28773 axhvdistr2-zf 28774 axhvmul0-zf 28775 axhis3-zf 28779 hhsm 28952 |
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