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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 2737 | . . . 4 ⊢ ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
| 2 | 1 | vafval 30622 | . . 3 ⊢ ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = (1st ‘(1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉)) | 
| 3 | opex 5469 | . . . . 5 ⊢ 〈 +ℎ , ·ℎ 〉 ∈ V | |
| 4 | h2h.1 | . . . . . . . 8 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
| 5 | h2h.2 | . . . . . . . 8 ⊢ 𝑈 ∈ NrmCVec | |
| 6 | 4, 5 | eqeltrri 2838 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec | 
| 7 | nvex 30630 | . . . . . . 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 8022 | . . . 4 ⊢ (1st ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) = 〈 +ℎ , ·ℎ 〉 | 
| 11 | 10 | fveq2i 6909 | . . 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 8022 | . . 3 ⊢ (1st ‘〈 +ℎ , ·ℎ 〉) = +ℎ | 
| 15 | 2, 11, 14 | 3eqtrri 2770 | . 2 ⊢ +ℎ = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | 
| 16 | 4 | fveq2i 6909 | . 2 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | 
| 17 | 15, 16 | eqtr4i 2768 | 1 ⊢ +ℎ = ( +𝑣 ‘𝑈) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 1st c1st 8012 NrmCVeccnv 30603 +𝑣 cpv 30604 +ℎ cva 30939 ·ℎ csm 30940 normℎcno 30942 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-oprab 7435 df-1st 8014 df-vc 30578 df-nv 30611 df-va 30614 | 
| This theorem is referenced by: h2hvs 30996 axhfvadd-zf 31001 axhvcom-zf 31002 axhvass-zf 31003 axhvaddid-zf 31005 axhvdistr1-zf 31009 axhvdistr2-zf 31010 axhis2-zf 31014 hhva 31185 | 
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