uvcvv1 - Metamath Proof Explorer

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Theorem uvcvv1 21969

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 2765	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
7	4, 5, 6	uvcvval 21966	. . 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 2765	. . 3 ⊢ 𝐽 = 𝐽
10		iftrue 4495	. . 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 2800	1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ifcif 4489 ‘cfv 6540 (class class class)co 7416 0_gc0g 17510 1_rcur 20287 unitVec cuvc 21962

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pr 5406

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7419 df-oprab 7420 df-mpo 7421 df-uvc 21963

This theorem is used by: uvcf1 21972 uvcresum 21973 frlmssuvc2 21975 frlmup2 21979 uvcn0 43343 0prjspnrel 43392

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