Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2622 | . . . 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 2622 | . . . 4 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
6 | 1, 2, 3, 4, 5 | obsip 20065 | . . 3 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
7 | 6 | 3anidm23 1385 | . 2 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = if(𝐴 = 𝐴, 1 , (0g‘𝐹))) |
8 | eqid 2622 | . . 3 ⊢ 𝐴 = 𝐴 | |
9 | 8 | iftruei 4093 | . 2 ⊢ if(𝐴 = 𝐴, 1 , (0g‘𝐹)) = 1 |
10 | 7, 9 | syl6eq 2672 | 1 ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ifcif 4086 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑖cip 15946 0gc0g 16100 1rcur 18501 OBasiscobs 20046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-obs 20049 |
This theorem is referenced by: obsne0 20069 |
Copyright terms: Public domain | W3C validator |