Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2818 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 4, 5, 6 | uvcvval 20858 | . . 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 1365 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g‘𝑅))) |
9 | eqid 2818 | . . 3 ⊢ 𝐽 = 𝐽 | |
10 | iftrue 4469 | . . 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 2853 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ifcif 4463 ‘cfv 6348 (class class class)co 7145 0gc0g 16701 1rcur 19180 unitVec cuvc 20854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-uvc 20855 |
This theorem is referenced by: uvcf1 20864 uvcresum 20865 frlmssuvc2 20867 frlmup2 20871 uvcn0 39029 0prjspnrel 39147 |
Copyright terms: Public domain | W3C validator |