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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 2733 | . . . 4 ⊢ ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) | |
2 | 1 | vafval 29594 | . . 3 ⊢ ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (1st ‘(1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)) |
3 | opex 5425 | . . . . 5 ⊢ ⟨ +ℎ , ·ℎ ⟩ ∈ V | |
4 | h2h.1 | . . . . . . . 8 ⊢ 𝑈 = ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ | |
5 | h2h.2 | . . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec | |
6 | 4, 5 | eqeltrri 2831 | . . . . . . 7 ⊢ ⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩ ∈ NrmCVec |
7 | nvex 29602 | . . . . . . 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 1142 | . . . . 5 ⊢ normℎ ∈ V |
10 | 3, 9 | op1st 7933 | . . . 4 ⊢ (1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = ⟨ +ℎ , ·ℎ ⟩ |
11 | 10 | fveq2i 6849 | . . 3 ⊢ (1st ‘(1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)) = (1st ‘⟨ +ℎ , ·ℎ ⟩) |
12 | 8 | simp1i 1140 | . . . 4 ⊢ +ℎ ∈ V |
13 | 8 | simp2i 1141 | . . . 4 ⊢ ·ℎ ∈ V |
14 | 12, 13 | op1st 7933 | . . 3 ⊢ (1st ‘⟨ +ℎ , ·ℎ ⟩) = +ℎ |
15 | 2, 11, 14 | 3eqtrri 2766 | . 2 ⊢ +ℎ = ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) |
16 | 4 | fveq2i 6849 | . 2 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) |
17 | 15, 16 | eqtr4i 2764 | 1 ⊢ +ℎ = ( +𝑣 ‘𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ⟨cop 4596 ‘cfv 6500 1st c1st 7923 NrmCVeccnv 29575 +𝑣 cpv 29576 +ℎ cva 29911 ·ℎ csm 29912 normℎcno 29914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fo 6506 df-fv 6508 df-oprab 7365 df-1st 7925 df-vc 29550 df-nv 29583 df-va 29586 |
This theorem is referenced by: h2hvs 29968 axhfvadd-zf 29973 axhvcom-zf 29974 axhvass-zf 29975 axhvaddid-zf 29977 axhvdistr1-zf 29981 axhvdistr2-zf 29982 axhis2-zf 29986 hhva 30157 |
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