Theorem marepvval0 22794

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 22640	. . . . 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 7227	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))) ∈ V)
7		fveq1 6881	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑖) = (𝐶‘𝑖))
8	7	adantl 487	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑐‘𝑖) = (𝐶‘𝑖))
9		oveq 7423	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
10	9	adantr 486	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
11	8, 10	ifeq12d 4507	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑐 = 𝐶) → if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)) = if(𝑗 = 𝑘, (𝐶‘𝑖), (𝑖𝑀𝑗)))
12	11	mpoeq3dv 7496	. . . 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 5203	. . 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 22793	. . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑐 ∈ 𝑉 ↦ (𝑘 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑗 = 𝑘, (𝑐‘𝑖), (𝑖𝑚𝑗)))))
17	13, 16	ovmpoga 7571	. 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 3453  ifcif 4485   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419   ↑_m cmap 8830  Fincfn 8956  Basecbs 17307   Mat cmat 22635   matRepV cmatrepV 22785

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-1cn 11186  ax-addcl 11188

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-nn 12262  df-slot 17280  df-ndx 17292  df-base 17308  df-mat 22636  df-marepv 22787

This theorem is used by:  marepvval  22795