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Mirrors > Home > MPE Home > Th. List > uvcvv1 | Structured version Visualization version GIF version |
Description: The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
uvcvv.u | ⊢ 𝑈 = (𝑅 unitVec 𝐼) |
uvcvv.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
uvcvv.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
uvcvv.j | ⊢ (𝜑 → 𝐽 ∈ 𝐼) |
uvcvv1.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
uvcvv1 | ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvcvv.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
2 | uvcvv.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | uvcvv.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼) | |
4 | uvcvv.u | . . . 4 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
5 | uvcvv1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
6 | eqid 2733 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 4, 5, 6 | uvcvval 21325 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g‘𝑅))) |
8 | 1, 2, 3, 3, 7 | syl31anc 1374 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g‘𝑅))) |
9 | eqid 2733 | . . 3 ⊢ 𝐽 = 𝐽 | |
10 | iftrue 4533 | . . 3 ⊢ (𝐽 = 𝐽 → if(𝐽 = 𝐽, 1 , (0g‘𝑅)) = 1 ) | |
11 | 9, 10 | mp1i 13 | . 2 ⊢ (𝜑 → if(𝐽 = 𝐽, 1 , (0g‘𝑅)) = 1 ) |
12 | 8, 11 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ifcif 4527 ‘cfv 6540 (class class class)co 7404 0gc0g 17381 1rcur 19996 unitVec cuvc 21321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-uvc 21322 |
This theorem is referenced by: uvcf1 21331 uvcresum 21332 frlmssuvc2 21334 frlmup2 21338 uvcn0 41062 0prjspnrel 41313 |
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