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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 2622 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | uvcvv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
8 | 5, 6, 7 | uvcvval 20125 | . . 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 1329 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1r‘𝑅), 0 )) |
10 | uvcvv0.jk | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾) | |
11 | nesym 2850 | . . . 4 ⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐾 = 𝐽) | |
12 | 10, 11 | sylib 208 | . . 3 ⊢ (𝜑 → ¬ 𝐾 = 𝐽) |
13 | 12 | iffalsed 4097 | . 2 ⊢ (𝜑 → if(𝐾 = 𝐽, (1r‘𝑅), 0 ) = 0 ) |
14 | 9, 13 | eqtrd 2656 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ifcif 4086 ‘cfv 5888 (class class class)co 6650 0gc0g 16100 1rcur 18501 unitVec cuvc 20121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-uvc 20122 |
This theorem is referenced by: uvcf1 20131 uvcresum 20132 frlmssuvc1 20133 frlmsslsp 20135 frlmup2 20138 |
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