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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 20858 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
5 | 2 | fvexi 6677 | . . . 4 ⊢ 1 ∈ V |
6 | 3 | fvexi 6677 | . . . 4 ⊢ 0 ∈ V |
7 | ifpr 4621 | . . . 4 ⊢ (( 1 ∈ V ∧ 0 ∈ V) → if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 }) | |
8 | 5, 6, 7 | mp2an 688 | . . 3 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 } |
9 | prcom 4660 | . . 3 ⊢ { 1 , 0 } = { 0 , 1 } | |
10 | 8, 9 | eleqtri 2908 | . 2 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 0 , 1 } |
11 | 4, 10 | syl6eqel 2918 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ifcif 4463 {cpr 4559 ‘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: (None) |
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