Theorem uvcfval 21089

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 3459	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3459	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		df-uvc 21088	. . . . 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 485	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
7		fveq2 6819	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
8		uvcfval.o	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
9	7, 8	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
10		fveq2 6819	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
11		uvcfval.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
12	10, 11	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
13	9, 12	ifeq12d 4493	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
14	13	adantr 481	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
15	6, 14	mpteq12dv 5180	. . . . . 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 5180	. . . . 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 482	. . . 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 483	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝑅 ∈ V)
19		simpr 485	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
20		mptexg 7147	. . . . 5 ⊢ (𝐼 ∈ V → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
21	20	adantl 482	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
22	5, 17, 18, 19, 21	ovmpod 7479	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 unitVec 𝐼) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
23	2, 3, 22	syl2an 596	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 unitVec 𝐼) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
24	1, 23	eqtrid 2788	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  Vcvv 3441  ifcif 4472   ↦ cmpt 5172  ‘cfv 6473  (class class class)co 7329   ∈ cmpo 7331  0_gc0g 17239  1_rcur 19824   unitVec cuvc 21087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-uvc 21088

This theorem is referenced by:  uvcval  21090  uvcff  21096  frlmdim  31933