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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 2823 | . . . 4 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
2 | 1 | smfval 28384 | . . 3 ⊢ ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) |
3 | opex 5358 | . . . . 5 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
4 | h2h.1 | . . . . . . . 8 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
5 | h2h.2 | . . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec | |
6 | 4, 5 | eqeltrri 2912 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
7 | nvex 28390 | . . . . . . 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 1137 | . . . . 5 ⊢ normℎ ∈ V |
10 | 3, 9 | op1st 7699 | . . . 4 ⊢ (1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = 〈 +ℎ , ·ℎ 〉 |
11 | 10 | fveq2i 6675 | . . 3 ⊢ (2nd ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) = (2nd ‘〈 +ℎ , ·ℎ 〉) |
12 | 8 | simp1i 1135 | . . . 4 ⊢ +ℎ ∈ V |
13 | 8 | simp2i 1136 | . . . 4 ⊢ ·ℎ ∈ V |
14 | 12, 13 | op2nd 7700 | . . 3 ⊢ (2nd ‘〈 +ℎ , ·ℎ 〉) = ·ℎ |
15 | 2, 11, 14 | 3eqtrri 2851 | . 2 ⊢ ·ℎ = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
16 | 4 | fveq2i 6675 | . 2 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
17 | 15, 16 | eqtr4i 2849 | 1 ⊢ ·ℎ = ( ·𝑠OLD ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ‘cfv 6357 1st c1st 7689 2nd c2nd 7690 NrmCVeccnv 28363 ·𝑠OLD cns 28366 +ℎ cva 28699 ·ℎ csm 28700 normℎcno 28702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-oprab 7162 df-1st 7691 df-2nd 7692 df-vc 28338 df-nv 28371 df-sm 28376 |
This theorem is referenced by: h2hvs 28756 axhfvmul-zf 28766 axhvmulid-zf 28767 axhvmulass-zf 28768 axhvdistr1-zf 28769 axhvdistr2-zf 28770 axhvmul0-zf 28771 axhis3-zf 28775 hhsm 28948 |
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