uvcvv0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > uvcvv0		Structured version Visualization version GIF version

Theorem uvcvv0 20323

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 2752	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
7		uvcvv0.z	. . . 4 ⊢ 0 = (0_g‘𝑅)
8	5, 6, 7	uvcvval 20319	. . 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 1476	. 2 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, (1_r‘𝑅), 0 ))
10		uvcvv0.jk	. . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐾)
11		nesym 2980	. . . 4 ⊢ (𝐽 ≠ 𝐾 ↔ ¬ 𝐾 = 𝐽)
12	10, 11	sylib 208	. . 3 ⊢ (𝜑 → ¬ 𝐾 = 𝐽)
13	12	iffalsed 4233	. 2 ⊢ (𝜑 → if(𝐾 = 𝐽, (1_r‘𝑅), 0 ) = 0 )
14	9, 13	eqtrd 2786	1 ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 ifcif 4222 ‘cfv 6041 (class class class)co 6805 0_gc0g 16294 1_rcur 18693 unitVec cuvc 20315

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pr 5047

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-reu 3049 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-uvc 20316

This theorem is referenced by: uvcf1 20325 uvcresum 20326 frlmssuvc1 20327 frlmsslsp 20329 frlmup2 20332

W3C validator