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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 2737 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 7 | uvcvv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
| 8 | 5, 6, 7 | uvcvval 21806 | . . 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 2997 | . . . 4 ⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐾 = 𝐽) | |
| 12 | 10, 11 | sylib 218 | . . 3 ⊢ (𝜑 → ¬ 𝐾 = 𝐽) |
| 13 | 12 | iffalsed 4536 | . 2 ⊢ (𝜑 → if(𝐾 = 𝐽, (1r‘𝑅), 0 ) = 0 ) |
| 14 | 9, 13 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ifcif 4525 ‘cfv 6561 (class class class)co 7431 0gc0g 17484 1rcur 20178 unitVec cuvc 21802 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-uvc 21803 |
| This theorem is referenced by: uvcf1 21812 uvcresum 21813 frlmssuvc1 21814 frlmsslsp 21816 frlmup2 21819 |
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