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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 2735 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 4, 5, 6 | uvcvval 21824 | . . 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 1372 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g‘𝑅))) |
9 | eqid 2735 | . . 3 ⊢ 𝐽 = 𝐽 | |
10 | iftrue 4537 | . . 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 2775 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ifcif 4531 ‘cfv 6563 (class class class)co 7431 0gc0g 17486 1rcur 20199 unitVec cuvc 21820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-uvc 21821 |
This theorem is referenced by: uvcf1 21830 uvcresum 21831 frlmssuvc2 21833 frlmup2 21837 uvcn0 42529 0prjspnrel 42614 |
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