Theorem uvcfval 20901

Description: Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem uvcfval

Dummy variables 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uvcfval.u	. 2 ⊢ 𝑈 = (𝑅 unitVec 𝐼)
2		elex 3440	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3440	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		df-uvc 20900	. . . . 5 ⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))
5	4	a1i 11	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))))))
6		simpr 484	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
7		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
8		uvcfval.o	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
9	7, 8	eqtr4di 2797	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
10		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
11		uvcfval.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
12	10, 11	eqtr4di 2797	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
13	9, 12	ifeq12d 4477	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
14	13	adantr 480	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
15	6, 14	mpteq12dv 5161	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))
16	6, 15	mpteq12dv 5161	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
17	16	adantl 481	. . . 4 ⊢ (((𝑅 ∈ V ∧ 𝐼 ∈ V) ∧ (𝑟 = 𝑅 ∧ 𝑖 = 𝐼)) → (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
18		simpl 482	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝑅 ∈ V)
19		simpr 484	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
20		mptexg 7079	. . . . 5 ⊢ (𝐼 ∈ V → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
21	20	adantl 481	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
22	5, 17, 18, 19, 21	ovmpod 7403	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 unitVec 𝐼) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
23	2, 3, 22	syl2an 595	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 unitVec 𝐼) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
24	1, 23	eqtrid 2790	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ifcif 4456   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  0_gc0g 17067  1_rcur 19652   unitVec cuvc 20899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-uvc 20900

This theorem is referenced by:  uvcval  20902  uvcff  20908  frlmdim  31596