uvcvv0 - Metamath Proof Explorer

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Theorem uvcvv0 21681

Description: The unit vector is zero 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)
uvcvv0.k	⊢ (𝜑 → 𝐾 ∈ 𝐼)
uvcvv0.jk	⊢ (𝜑 → 𝐽 ≠ 𝐾)
uvcvv0.z	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
uvcvv0	⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 )

**Proof of Theorem uvcvv0**
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		uvcvv0.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝐼)
5		uvcvv.u	. . . 4 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
6		eqid 2726	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
7		uvcvv0.z	. . . 4 ⊢ 0 = (0_g‘𝑅)
8	5, 6, 7	uvcvval 21677	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1_r‘𝑅), 0 ))
9	1, 2, 3, 4, 8	syl31anc 1370	. 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1_r‘𝑅), 0 ))
10		uvcvv0.jk	. . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾)
11		nesym 2991	. . . 4 ⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐾 = 𝐽)
12	10, 11	sylib 217	. . 3 ⊢ (𝜑 → ¬ 𝐾 = 𝐽)
13	12	iffalsed 4534	. 2 ⊢ (𝜑 → if(𝐾 = 𝐽, (1_r‘𝑅), 0 ) = 0 )
14	9, 13	eqtrd 2766	1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ifcif 4523 ‘cfv 6536 (class class class)co 7404 0_gc0g 17392 1_rcur 20084 unitVec cuvc 21673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-uvc 21674

This theorem is referenced by: uvcf1 21683 uvcresum 21684 frlmssuvc1 21685 frlmsslsp 21687 frlmup2 21690

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