Theorem uvcfval 21915

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 3476	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		elex 3476	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V)
4		df-uvc 21914	. . . . 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 489	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑖 = 𝐼)
7		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
8		uvcfval.o	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
9	7, 8	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
10		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
11		uvcfval.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑅)
12	10, 11	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
13	9, 12	ifeq12d 4510	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
14	13	adantr 485	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)) = if(𝑘 = 𝑗, 1 , 0 ))
15	6, 14	mpteq12dv 5199	. . . . . 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 5199	. . . . 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 486	. . . 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 487	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝑅 ∈ V)
19		simpr 489	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → 𝐼 ∈ V)
20		mptexg 7221	. . . . 5 ⊢ (𝐼 ∈ V → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
21	20	adantl 486	. . . 4 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))) ∈ V)
22	5, 17, 18, 19, 21	ovmpod 7564	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 unitVec 𝐼) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
23	2, 3, 22	syl2an 607	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 unitVec 𝐼) = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
24	1, 23	eqtrid 2810	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ifcif 4488   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  0_gc0g 17493  1_rcur 20264   unitVec cuvc 21913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-uvc 21914

This theorem is referenced by:  uvcval  21916  uvcff  21922  frlmdim  33979