![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hlcom | Structured version Visualization version GIF version |
Description: Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hladdf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
hladdf.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
Ref | Expression |
---|---|
hlcom | ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 28461 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
2 | hladdf.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hladdf.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
4 | 2, 3 | nvcom 28190 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
5 | 1, 4 | syl3an1 1144 | 1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ‘cfv 6185 (class class class)co 6974 NrmCVeccnv 28153 +𝑣 cpv 28154 BaseSetcba 28155 CHilOLDchlo 28455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-1st 7499 df-2nd 7500 df-ablo 28114 df-vc 28128 df-nv 28161 df-va 28164 df-ba 28165 df-sm 28166 df-0v 28167 df-nmcv 28169 df-cbn 28433 df-hlo 28456 |
This theorem is referenced by: axhvcom-zf 28554 |
Copyright terms: Public domain | W3C validator |