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Mirrors > Home > MPE Home > Th. List > obsipid | Structured version Visualization version GIF version |
Description: A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
obsipid.h | ⊢ , = (·𝑖‘𝑊) |
obsipid.f | ⊢ 𝐹 = (Scalar‘𝑊) |
obsipid.u | ⊢ 1 = (1r‘𝐹) |
Ref | Expression |
---|---|
obsipid | ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2820 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | obsipid.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
3 | obsipid.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | obsipid.u | . . . 4 ⊢ 1 = (1r‘𝐹) | |
5 | eqid 2820 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 1, 2, 3, 4, 5 | obsip 20860 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
7 | 6 | 3anidm23 1416 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
8 | eqid 2820 | . . 3 ⊢ 𝐴 = 𝐴 | |
9 | 8 | iftruei 4467 | . 2 ⊢ if(𝐴 = 𝐴, 1 , (0g‘𝐹)) = 1 |
10 | 7, 9 | syl6eq 2871 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ifcif 4460 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 Scalarcsca 16563 ·𝑖cip 16565 0gc0g 16708 1rcur 19246 OBasiscobs 20841 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-obs 20844 |
This theorem is referenced by: obsne0 20864 |
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