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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 21041 | . . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) |
5 | 4 | fveq1d 6806 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾)) |
6 | 5 | adantr 482 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = ((𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))‘𝐾)) |
7 | simpr 486 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → 𝐾 ∈ 𝐼) | |
8 | 2 | fvexi 6818 | . . . 4 ⊢ 1 ∈ V |
9 | 3 | fvexi 6818 | . . . 4 ⊢ 0 ∈ V |
10 | 8, 9 | ifex 4515 | . . 3 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ V |
11 | eqeq1 2740 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 = 𝐽 ↔ 𝐾 = 𝐽)) | |
12 | 11 | ifbid 4488 | . . . 4 ⊢ (𝑘 = 𝐾 → if(𝑘 = 𝐽, 1 , 0 ) = if(𝐾 = 𝐽, 1 , 0 )) |
13 | eqid 2736 | . . . 4 ⊢ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) | |
14 | 12, 13 | fvmptg 6905 | . . 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 587 | . 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 2776 | 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 397 ∧ w3a 1087 = wceq 1539 ∈ wcel 2104 Vcvv 3437 ifcif 4465 ↦ cmpt 5164 ‘cfv 6458 (class class class)co 7307 0gc0g 17199 1rcur 19786 unitVec cuvc 21038 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3305 df-rab 3306 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-uvc 21039 |
This theorem is referenced by: uvcvvcl 21043 uvcvvcl2 21044 uvcvv1 21045 uvcvv0 21046 matunitlindflem2 35822 |
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