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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 21208 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
5 | 2 | fvexi 6857 | . . . 4 ⊢ 1 ∈ V |
6 | 3 | fvexi 6857 | . . . 4 ⊢ 0 ∈ V |
7 | ifpr 4653 | . . . 4 ⊢ (( 1 ∈ V ∧ 0 ∈ V) → if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 }) | |
8 | 5, 6, 7 | mp2an 691 | . . 3 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 } |
9 | prcom 4694 | . . 3 ⊢ { 1 , 0 } = { 0 , 1 } | |
10 | 8, 9 | eleqtri 2832 | . 2 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 0 , 1 } |
11 | 4, 10 | eqeltrdi 2842 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ifcif 4487 {cpr 4589 ‘cfv 6497 (class class class)co 7358 0gc0g 17326 1rcur 19918 unitVec cuvc 21204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-uvc 21205 |
This theorem is referenced by: (None) |
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