uvcvv0 - Metamath Proof Explorer

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Theorem uvcvv0 22101

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 2760	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
7		uvcvv0.z	. . . 4 ⊢ 0 = (0_g‘𝑅)
8	5, 6, 7	uvcvval 22097	. . 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 1400	. 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1_r‘𝑅), 0 ))
10		uvcvv0.jk	. . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾)
11		nesym 3011	. . . 4 ⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐾 = 𝐽)
12	10, 11	sylib 221	. . 3 ⊢ (𝜑 → ¬ 𝐾 = 𝐽)
13	12	iffalsed 4492	. 2 ⊢ (𝜑 → if(𝐾 = 𝐽, (1_r‘𝑅), 0 ) = 0 )
14	9, 13	eqtrd 2795	1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ifcif 4481 ‘cfv 6538 (class class class)co 7420 0_gc0g 17615 1_rcur 20412 unitVec cuvc 22093

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pr 5390

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 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 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-uvc 22094

This theorem is used by: uvcf1 22103 uvcresum 22104 frlmssuvc1 22105 frlmsslsp 22107 frlmup2 22110

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