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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 2728 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 4, 5, 6 | uvcvval 21720 | . . 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 1371 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g‘𝑅))) |
9 | eqid 2728 | . . 3 ⊢ 𝐽 = 𝐽 | |
10 | iftrue 4535 | . . 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 2768 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ifcif 4529 ‘cfv 6548 (class class class)co 7420 0gc0g 17421 1rcur 20121 unitVec cuvc 21716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-uvc 21717 |
This theorem is referenced by: uvcf1 21726 uvcresum 21727 frlmssuvc2 21729 frlmup2 21733 uvcn0 41772 0prjspnrel 42051 |
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