![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 29856 | . . 3 ⊢ ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = (1st ‘(1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩)) |
3 | opex 5465 | . . . . 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 29864 | . . . . . . 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 7983 | . . . 4 ⊢ (1st ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) = ⟨ +ℎ , ·ℎ ⟩ |
11 | 10 | fveq2i 6895 | . . 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 7983 | . . 3 ⊢ (1st ‘⟨ +ℎ , ·ℎ ⟩) = +ℎ |
15 | 2, 11, 14 | 3eqtrri 2766 | . 2 ⊢ +ℎ = ( +𝑣 ‘⟨⟨ +ℎ , ·ℎ ⟩, normℎ⟩) |
16 | 4 | fveq2i 6895 | . 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 3475 ⟨cop 4635 ‘cfv 6544 1st c1st 7973 NrmCVeccnv 29837 +𝑣 cpv 29838 +ℎ cva 30173 ·ℎ csm 30174 normℎcno 30176 |
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 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-oprab 7413 df-1st 7975 df-vc 29812 df-nv 29845 df-va 29848 |
This theorem is referenced by: h2hvs 30230 axhfvadd-zf 30235 axhvcom-zf 30236 axhvass-zf 30237 axhvaddid-zf 30239 axhvdistr1-zf 30243 axhvdistr2-zf 30244 axhis2-zf 30248 hhva 30419 |
Copyright terms: Public domain | W3C validator |