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Mirrors > Home > MPE Home > Th. List > uvcvvcl | Structured version Visualization version GIF version |
Description: A coodinate 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 19944 | . 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) |
5 | fvex 6113 | . . . . 5 ⊢ (1r‘𝑅) ∈ V | |
6 | 2, 5 | eqeltri 2684 | . . . 4 ⊢ 1 ∈ V |
7 | fvex 6113 | . . . . 5 ⊢ (0g‘𝑅) ∈ V | |
8 | 3, 7 | eqeltri 2684 | . . . 4 ⊢ 0 ∈ V |
9 | ifpr 4180 | . . . 4 ⊢ (( 1 ∈ V ∧ 0 ∈ V) → if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 }) | |
10 | 6, 8, 9 | mp2an 704 | . . 3 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 1 , 0 } |
11 | prcom 4211 | . . 3 ⊢ { 1 , 0 } = { 0 , 1 } | |
12 | 10, 11 | eleqtri 2686 | . 2 ⊢ if(𝐾 = 𝐽, 1 , 0 ) ∈ { 0 , 1 } |
13 | 4, 12 | syl6eqel 2696 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 {cpr 4127 ‘cfv 5804 (class class class)co 6549 0gc0g 15923 1rcur 18324 unitVec cuvc 19940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-uvc 19941 |
This theorem is referenced by: (None) |
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