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Mirrors > Home > MPE Home > Th. List > uvcvvcl | Structured version Visualization version GIF version |
Description: A coordinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
Ref | Expression |
---|---|
uvcfval.u | ⊢ 𝑈 = (𝑅 unitVec 𝐼) |
uvcfval.o | ⊢ 1 = (1r‘𝑅) |
uvcfval.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
uvcvvcl | ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uvcfval.u | . . 3 ⊢ 𝑈 = (𝑅 unitVec 𝐼) | |
2 | uvcfval.o | . . 3 ⊢ 1 = (1r‘𝑅) | |
3 | uvcfval.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | uvcvval 21677 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
5 | 2 | fvexi 6898 | . . . 4 ⊢ 1 ∈ V |
6 | 3 | fvexi 6898 | . . . 4 ⊢ 0 ∈ V |
7 | ifpr 4690 | . . . 4 ⊢ (( 1 ∈ V ∧ 0 ∈ V) → if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 }) | |
8 | 5, 6, 7 | mp2an 689 | . . 3 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 } |
9 | prcom 4731 | . . 3 ⊢ { 1 , 0 } = { 0 , 1 } | |
10 | 8, 9 | eleqtri 2825 | . 2 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 0 , 1 } |
11 | 4, 10 | eqeltrdi 2835 | 1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ifcif 4523 {cpr 4625 ‘cfv 6536 (class class class)co 7404 0gc0g 17392 1rcur 20084 unitVec cuvc 21673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-uvc 21674 |
This theorem is referenced by: (None) |
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