Theorem uvcval 19943

Description: Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)

Hypotheses

Ref	Expression
uvcfval.u	⊢ 𝑈 = (𝑅 unitVec 𝐼)
uvcfval.o	⊢ 1 = (1r‘𝑅)
uvcfval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
uvcval	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))

Distinct variable groups:   1 ,𝑘   𝑅,𝑘   𝑘,𝐼   0 ,𝑘   𝑘,𝐽

Allowed substitution hints:   𝑈(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem uvcval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvcfval.u	. . . . 5 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
2		uvcfval.o	. . . . 5 ⊢ 1 = (1r‘𝑅)
3		uvcfval.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	uvcfval 19942	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
5	4	fveq1d 6105	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑈‘𝐽) = ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
6	5	3adant3 1074	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
7		simp3 1056	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐽 ∈ 𝐼)
8		mptexg 6389	. . . 4 ⊢ (𝐼 ∈ 𝑊 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
9	8	3ad2ant2 1076	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
10		eqeq2 2621	. . . . . 6 ⊢ (𝑗 = 𝐽 → (𝑘 = 𝑗 ↔ 𝑘 = 𝐽))
11	10	ifbid 4058	. . . . 5 ⊢ (𝑗 = 𝐽 → if(𝑘 = 𝑗, 1 , 0 ) = if(𝑘 = 𝐽, 1 , 0 ))
12	11	mpteq2dv 4673	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
13		eqid 2610	. . . 4 ⊢ (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
14	12, 13	fvmptg 6189	. . 3 ⊢ ((𝐽 ∈ 𝐼 ∧ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V) → ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
15	7, 9, 14	syl2anc 691	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
16	6, 15	eqtrd 2644	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  0_gc0g 15923  1_rcur 18324   unitVec cuvc 19940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-uvc 19941

This theorem is referenced by:  uvcvval  19944