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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 2736 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | uvcvv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
8 | 5, 6, 7 | uvcvval 20702 | . . 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 1375 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1r‘𝑅), 0 )) |
10 | uvcvv0.jk | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
11 | nesym 2988 | . . . 4 ⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐾 = 𝐽) | |
12 | 10, 11 | sylib 221 | . . 3 ⊢ (𝜑 → ¬ 𝐾 = 𝐽) |
13 | 12 | iffalsed 4436 | . 2 ⊢ (𝜑 → if(𝐾 = 𝐽, (1r‘𝑅), 0 ) = 0 ) |
14 | 9, 13 | eqtrd 2771 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ifcif 4425 ‘cfv 6358 (class class class)co 7191 0gc0g 16898 1rcur 19470 unitVec cuvc 20698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-uvc 20699 |
This theorem is referenced by: uvcf1 20708 uvcresum 20709 frlmssuvc1 20710 frlmsslsp 20712 frlmup2 20715 |
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