Theorem marepvval 22594

Description: Third 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
marepvval	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))))

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

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

Proof of Theorem marepvval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		marepvfval.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marepvfval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3		marepvfval.q	. . . . 5 ⊢ 𝑄 = (𝑁 matRepV 𝑅)
4		marepvfval.v	. . . . 5 ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)
5	1, 2, 3, 4	marepvval0 22593	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))
6	5	3adant3 1132	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))
7	6	fveq1d 6922	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑀𝑄𝐶)‘𝐾) = ((𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗))))‘𝐾))
8		eqid 2740	. . 3 ⊢ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗))))
9		eqeq2 2752	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑗 = 𝑘 ↔ 𝑗 = 𝐾))
10	9	ifbid 4571	. . . 4 ⊢ (𝑘 = 𝐾 → if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)) = if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗)))
11	10	mpoeq3dv 7529	. . 3 ⊢ (𝑘 = 𝐾 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))))
12		simp3 1138	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → 𝐾 ∈ 𝑁)
13	1, 2	matrcl 22437	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
14	13	simpld 494	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
15	14, 14	jca 511	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
16	15	3ad2ant1 1133	. . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
17		mpoexga 8118	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))) ∈ V)
18	16, 17	syl 17	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))) ∈ V)
19	8, 11, 12, 18	fvmptd3 7052	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗))))‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))))
20	7, 19	eqtrd 2780	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉 ∧ 𝐾 ∈ 𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝐾, (𝐶‘𝑖), (𝑖𝑀𝑗))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ifcif 4548   ↦ cmpt 5249  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   ↑_m cmap 8884  Fincfn 9003  Basecbs 17258   Mat cmat 22432   matRepV cmatrepV 22584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-1cn 11242  ax-addcl 11244

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-nn 12294  df-slot 17229  df-ndx 17241  df-base 17259  df-mat 22433  df-marepv 22586

This theorem is referenced by:  marepveval  22595  marepvcl  22596  1marepvmarrepid  22602  cramerimplem2  22711