Theorem marepvval0 22704

Description: Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)

Hypotheses

Ref	Expression
marepvfval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marepvfval.b	⊢ 𝐵 = (Base‘𝐴)
marepvfval.q	⊢ 𝑄 = (𝑁 matRepV 𝑅)
marepvfval.v	⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)

Assertion

Ref	Expression
marepvval0	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))

Distinct variable groups:   𝑖,𝑁,𝑗,𝑘   𝑅,𝑖,𝑗,𝑘   𝐶,𝑖,𝑗,𝑘   𝑖,𝑀,𝑗,𝑘

Allowed substitution hints:   𝐴(𝑖,𝑗,𝑘)   𝐵(𝑖,𝑗,𝑘)   𝑄(𝑖,𝑗,𝑘)   𝑉(𝑖,𝑗,𝑘)

Proof of Theorem marepvval0

Dummy variables 𝑚 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marepvfval.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marepvfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
3	1, 2	matrcl 22550	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 499	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
5	4	adantr 485	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝑁 ∈ Fin)
6	5	mptexd 7224	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))) ∈ V)
7		fveq1 6882	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
8	7	adantl 486	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑐‘𝑖) = (𝐶‘𝑖))
9		oveq 7418	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
10	9	adantr 485	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
11	8, 10	ifeq12d 4510	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))
12	11	mpoeq3dv 7491	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗))))
13	12	mpteq2dv 5206	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))
14		marepvfval.q	. . . 4 ⊢ 𝑄 = (𝑁 matRepV 𝑅)
15		marepvfval.v	. . . 4 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
16	1, 2, 14, 15	marepvfval 22703	. . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑐 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)))))
17	13, 16	ovmpoga 7566	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))
18	6, 17	mpd3an3 1491	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))

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   ↑_m cmap 8825  Fincfn 8944  Basecbs 17270   Mat cmat 22545   matRepV cmatrepV 22695

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-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-1cn 11159  ax-addcl 11161

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  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-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  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-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  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-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  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-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-nn 12235  df-slot 17243  df-ndx 17255  df-base 17271  df-mat 22546  df-marepv 22697

This theorem is referenced by:  marepvval  22705