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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 2736 | . . . 4 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
| 2 | 1 | smfval 30625 | . . 3 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | 
| 3 | opex 5468 | . . . . 5 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
| 4 | h2h.1 | . . . . . . . 8 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
| 5 | h2h.2 | . . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec | |
| 6 | 4, 5 | eqeltrri 2837 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec | 
| 7 | nvex 30631 | . . . . . . 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 1141 | . . . . 5 ⊢ normℎ ∈ V | 
| 10 | 3, 9 | op1st 8023 | . . . 4 ⊢ (1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = 〈 +ℎ , ·ℎ 〉 | 
| 11 | 10 | fveq2i 6908 | . . 3 ⊢ (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (2nd ‘〈 +ℎ , ·ℎ 〉) | 
| 12 | 8 | simp1i 1139 | . . . 4 ⊢ +ℎ ∈ V | 
| 13 | 8 | simp2i 1140 | . . . 4 ⊢ ·ℎ ∈ V | 
| 14 | 12, 13 | op2nd 8024 | . . 3 ⊢ (2nd ‘〈 +ℎ , ·ℎ 〉) = ·ℎ | 
| 15 | 2, 11, 14 | 3eqtrri 2769 | . 2 ⊢ ·ℎ = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | 
| 16 | 4 | fveq2i 6908 | . 2 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | 
| 17 | 15, 16 | eqtr4i 2767 | 1 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 Vcvv 3479 〈cop 4631 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 NrmCVeccnv 30604 ·𝑠OLD cns 30607 +ℎ cva 30940 ·ℎ csm 30941 normℎcno 30943 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-oprab 7436 df-1st 8015 df-2nd 8016 df-vc 30579 df-nv 30612 df-sm 30617 | 
| This theorem is referenced by: h2hvs 30997 axhfvmul-zf 31007 axhvmulid-zf 31008 axhvmulass-zf 31009 axhvdistr1-zf 31010 axhvdistr2-zf 31011 axhvmul0-zf 31012 axhis3-zf 31016 hhsm 31189 | 
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