Theorem marepvval0 22861

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 22707	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 500	. . . 4 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
5	4	adantr 486	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → 𝑁 ∈ Fin)
6	5	mptexd 7222	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))) ∈ V)
7		fveq1 6876	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
8	7	adantl 487	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑐‘𝑖) = (𝐶‘𝑖))
9		oveq 7418	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
10	9	adantr 486	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
11	8, 10	ifeq12d 4504	. . . . 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 5199	. . 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 22860	. . 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)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ifcif 4482   ↦ cmpt 5186  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8831  Fincfn 8957  Basecbs 17367   Mat cmat 22702   matRepV cmatrepV 22852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-1cn 11239  ax-addcl 11241

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-nn 12317  df-slot 17340  df-ndx 17352  df-base 17368  df-mat 22703  df-marepv 22854

This theorem is used by:  marepvval  22862