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Mirrors > Home > MPE Home > Th. List > uvcvv0 | Structured version Visualization version GIF version |
Description: The unit vector is zero 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) |
uvcvv0.k | ⊢ (𝜑 → 𝐾 ∈ 𝐼) |
uvcvv0.jk | ⊢ (𝜑 → 𝐽 ≠ 𝐾) |
uvcvv0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
uvcvv0 | ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) |
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 | uvcvv0.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝐼) | |
5 | uvcvv.u | . . . 4 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
6 | eqid 2733 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | uvcvv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
8 | 5, 6, 7 | uvcvval 21314 | . . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1r‘𝑅), 0 )) |
9 | 1, 2, 3, 4, 8 | syl31anc 1374 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1r‘𝑅), 0 )) |
10 | uvcvv0.jk | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
11 | nesym 2998 | . . . 4 ⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐾 = 𝐽) | |
12 | 10, 11 | sylib 217 | . . 3 ⊢ (𝜑 → ¬ 𝐾 = 𝐽) |
13 | 12 | iffalsed 4535 | . 2 ⊢ (𝜑 → if(𝐾 = 𝐽, (1r‘𝑅), 0 ) = 0 ) |
14 | 9, 13 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ifcif 4524 ‘cfv 6535 (class class class)co 7396 0gc0g 17372 1rcur 19987 unitVec cuvc 21310 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-uvc 21311 |
This theorem is referenced by: uvcf1 21320 uvcresum 21321 frlmssuvc1 21322 frlmsslsp 21324 frlmup2 21327 |
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