Theorem uvcval 21700

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 21699	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
5	4	fveq1d 6862	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑈‘𝐽) = ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
6	5	3adant3 1132	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽))
7		eqid 2730	. . 3 ⊢ (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
8		eqeq2 2742	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑘 = 𝑗 ↔ 𝑘 = 𝐽))
9	8	ifbid 4514	. . . 4 ⊢ (𝑗 = 𝐽 → if(𝑘 = 𝑗, 1 , 0 ) = if(𝑘 = 𝐽, 1 , 0 ))
10	9	mpteq2dv 5203	. . 3 ⊢ (𝑗 = 𝐽 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
11		simp3 1138	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → 𝐽 ∈ 𝐼)
12		mptexg 7197	. . . 4 ⊢ (𝐼 ∈ 𝑊 → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
13	12	3ad2ant2 1134	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )) ∈ V)
14	7, 10, 11, 13	fvmptd3 6993	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → ((𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
15	6, 14	eqtrd 2765	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ifcif 4490   ↦ cmpt 5190  ‘cfv 6513  (class class class)co 7389  0_gc0g 17408  1_rcur 20096   unitVec cuvc 21697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-uvc 21698

This theorem is referenced by:  uvcvval  21701