uvcvv1 - Metamath Proof Explorer

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Theorem uvcvv1 22002

Description: The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)

**Hypotheses**
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 = (1_r‘𝑅)

**Assertion**
Ref	Expression
uvcvv1	⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 )

**Proof of Theorem uvcvv1**
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 = (1_r‘𝑅)
6		eqid 2760	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7	4, 5, 6	uvcvval 21999	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐽 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0_g‘𝑅)))
8	1, 2, 3, 3, 7	syl31anc 1400	. 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = if(𝐽 = 𝐽, 1 , (0_g‘𝑅)))
9		eqid 2760	. . 3 ⊢ 𝐽 = 𝐽
10		iftrue 4488	. . 3 ⊢ (𝐽 = 𝐽 → if(𝐽 = 𝐽, 1 , (0_g‘𝑅)) = 1 )
11	9, 10	mp1i 14	. 2 ⊢ (𝜑 → if(𝐽 = 𝐽, 1 , (0_g‘𝑅)) = 1 )
12	8, 11	eqtrd 2795	1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ifcif 4482 ‘cfv 6533 (class class class)co 7413 0_gc0g 17524 1_rcur 20320 unitVec cuvc 21995

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418 df-uvc 21996

This theorem is used by: uvcf1 22005 uvcresum 22006 frlmssuvc2 22008 frlmup2 22012 uvcn0 43424 0prjspnrel 43473

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