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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 2821 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 4, 5, 6 | uvcvval 20930 | . . 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 1369 | . 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0g‘𝑅))) |
9 | eqid 2821 | . . 3 ⊢ 𝐽 = 𝐽 | |
10 | iftrue 4473 | . . 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 2856 | 1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ifcif 4467 ‘cfv 6355 (class class class)co 7156 0gc0g 16713 1rcur 19251 unitVec cuvc 20926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-uvc 20927 |
This theorem is referenced by: uvcf1 20936 uvcresum 20937 frlmssuvc2 20939 frlmup2 20943 uvcn0 39171 0prjspnrel 39289 |
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