Theorem marepvval0 21175

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 21021	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 497	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
5	4	adantr 483	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝑁 ∈ Fin)
6	5	mptexd 6987	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))) ∈ V)
7		fveq1 6669	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
8	7	adantl 484	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑐‘𝑖) = (𝐶‘𝑖))
9		oveq 7162	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
10	9	adantr 483	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
11	8, 10	ifeq12d 4487	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))
12	11	mpoeq3dv 7233	. . . 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 5162	. . 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 21174	. . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑐 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)))))
17	13, 16	ovmpoga 7304	. 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 1458	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑀𝑄𝐶) = (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ifcif 4467   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158   ↑_m cmap 8406  Fincfn 8509  Basecbs 16483   Mat cmat 21016   matRepV cmatrepV 21166

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-slot 16487  df-base 16489  df-mat 21017  df-marepv 21168

This theorem is referenced by:  marepvval  21176