Theorem marepvval0 22453

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 22299	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 494	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
5	4	adantr 480	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝑁 ∈ Fin)
6	5	mptexd 7198	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))) ∈ V)
7		fveq1 6857	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
8	7	adantl 481	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑐‘𝑖) = (𝐶‘𝑖))
9		oveq 7393	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
10	9	adantr 480	. . . . . 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 7468	. . . 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 5201	. . 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 22452	. . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑐 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)))))
17	13, 16	ovmpoga 7543	. 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 1464	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ifcif 4488   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389   ↑_m cmap 8799  Fincfn 8918  Basecbs 17179   Mat cmat 22294   matRepV cmatrepV 22444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-1cn 11126  ax-addcl 11128

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-nn 12187  df-slot 17152  df-ndx 17164  df-base 17180  df-mat 22295  df-marepv 22446

This theorem is referenced by:  marepvval  22454