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Mirrors > Home > HSE Home > Th. List > h2hva | Structured version Visualization version GIF version |
Description: The group (addition) 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 |
---|---|
h2hva | ⊢ +ℎ = ( +𝑣 ‘𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . 4 ⊢ ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) | |
2 | 1 | vafval 30325 | . . 3 ⊢ ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (1st ‘(1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)) |
3 | opex 5454 | . . . . 5 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ V | |
4 | h2h.1 | . . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ | |
5 | h2h.2 | . . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec | |
6 | 4, 5 | eqeltrri 2822 | . . . . . . 7 ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec |
7 | nvex 30333 | . . . . . . 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 7976 | . . . 4 ⊢ (1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = ⟨ +ℎ , ·ℎ ⟩ |
11 | 10 | fveq2i 6884 | . . 3 ⊢ (1st ‘(1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)) = (1st ‘⟨ +ℎ , ·ℎ ⟩) |
12 | 8 | simp1i 1136 | . . . 4 ⊢ +ℎ ∈ V |
13 | 8 | simp2i 1137 | . . . 4 ⊢ ·ℎ ∈ V |
14 | 12, 13 | op1st 7976 | . . 3 ⊢ (1st ‘⟨ +ℎ , ·ℎ ⟩) = +ℎ |
15 | 2, 11, 14 | 3eqtrri 2757 | . 2 ⊢ +ℎ = ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) |
16 | 4 | fveq2i 6884 | . 2 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) |
17 | 15, 16 | eqtr4i 2755 | 1 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⟨cop 4626 ‘cfv 6533 1st c1st 7966 NrmCVeccnv 30306 +𝑣 cpv 30307 +ℎ cva 30642 ·ℎ csm 30643 normℎcno 30645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fo 6539 df-fv 6541 df-oprab 7405 df-1st 7968 df-vc 30281 df-nv 30314 df-va 30317 |
This theorem is referenced by: h2hvs 30699 axhfvadd-zf 30704 axhvcom-zf 30705 axhvass-zf 30706 axhvaddid-zf 30708 axhvdistr1-zf 30712 axhvdistr2-zf 30713 axhis2-zf 30717 hhva 30888 |
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