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Mirrors > Home > MPE Home > Th. List > uvcvval | Structured version Visualization version GIF version |
Description: Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
uvcfval.u | ⊢ 𝑈 = (𝑅 unitVec 𝐼) |
uvcfval.o | ⊢ 1 = (1r‘𝑅) |
uvcfval.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
uvcvval | ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvcfval.u | . . . . 5 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
2 | uvcfval.o | . . . . 5 ⊢ 1 = (1r‘𝑅) | |
3 | uvcfval.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | uvcval 21675 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) |
5 | 4 | fveq1d 6886 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾)) |
6 | 5 | adantr 480 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾)) |
7 | simpr 484 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → 𝐾 ∈ 𝐼) | |
8 | 2 | fvexi 6898 | . . . 4 ⊢ 1 ∈ V |
9 | 3 | fvexi 6898 | . . . 4 ⊢ 0 ∈ V |
10 | 8, 9 | ifex 4573 | . . 3 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ V |
11 | eqeq1 2730 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 = 𝐽 ↔ 𝐾 = 𝐽)) | |
12 | 11 | ifbid 4546 | . . . 4 ⊢ (𝑘 = 𝐾 → if(𝑘 = 𝐽, 1 , 0 ) = if(𝐾 = 𝐽, 1 , 0 )) |
13 | eqid 2726 | . . . 4 ⊢ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) | |
14 | 12, 13 | fvmptg 6989 | . . 3 ⊢ ((𝐾 ∈ 𝐼 ∧ if(𝐾 = 𝐽, 1 , 0 ) ∈ V) → ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
15 | 7, 10, 14 | sylancl 585 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
16 | 6, 15 | eqtrd 2766 | 1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ifcif 4523 ↦ cmpt 5224 ‘cfv 6536 (class class class)co 7404 0gc0g 17391 1rcur 20083 unitVec cuvc 21672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-uvc 21673 |
This theorem is referenced by: uvcvvcl 21677 uvcvvcl2 21678 uvcvv1 21679 uvcvv0 21680 matunitlindflem2 36997 |
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