Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hladdid | Structured version Visualization version GIF version |
Description: Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hladdid.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
hladdid.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
hladdid.5 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
hladdid | ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 29249 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
2 | hladdid.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hladdid.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
4 | hladdid.5 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
5 | 2, 3, 4 | nv0rid 28993 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
6 | 1, 5 | sylan 580 | 1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 NrmCVeccnv 28942 +𝑣 cpv 28943 BaseSetcba 28944 0veccn0v 28946 CHilOLDchlo 29243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-1st 7824 df-2nd 7825 df-grpo 28851 df-gid 28852 df-ablo 28903 df-vc 28917 df-nv 28950 df-va 28953 df-ba 28954 df-sm 28955 df-0v 28956 df-nmcv 28958 df-cbn 29221 df-hlo 29244 |
This theorem is referenced by: axhvaddid-zf 29344 |
Copyright terms: Public domain | W3C validator |