Theorem marrepval0 22885

Description: Second substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)

Hypotheses

Ref	Expression
marrepfval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marrepfval.b	⊢ 𝐵 = (Base‘𝐴)
marrepfval.q	⊢ 𝑄 = (𝑁 matRRep 𝑅)
marrepfval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
marrepval0	⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))

Distinct variable groups:   𝑖,𝑁,𝑗,𝑘,𝑙   𝑅,𝑖,𝑗,𝑘,𝑙   𝑖,𝑀,𝑗,𝑘,𝑙   𝑆,𝑖,𝑗,𝑘,𝑙

Allowed substitution hints:   𝐴(𝑖, 𝑗, 𝑘, 𝑙)   𝐵(𝑖, 𝑗, 𝑘, 𝑙)   𝑄(𝑖, 𝑗, 𝑘, 𝑙)   0 (𝑖, 𝑗, 𝑘, 𝑙)

Proof of Theorem marrepval0

Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marrepfval.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marrepfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
3	1, 2	matrcl 22736	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 500	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
5	4, 4	jca 521	. . . 4 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
6	5	adantr 486	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
7		mpoexga 8095	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V)
8	6, 7	syl 18	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V)
9		ifeq1 4485	. . . . . . 7 ⊢ (𝑠 = 𝑆 → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 ))
10	9	adantl 487	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → if(𝑗 = 𝑙, 𝑠, 0 ) = if(𝑗 = 𝑙, 𝑆, 0 ))
11		oveq 7430	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
12	11	adantr 486	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
13	10, 12	ifeq12d 4503	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)) = if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))
14	13	mpoeq3dv 7503	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗))))
15	14	mpoeq3dv 7503	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑠 = 𝑆) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
16		marrepfval.q	. . . 4 ⊢ 𝑄 = (𝑁 matRRep 𝑅)
17		marrepfval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
18	1, 2, 16, 17	marrepfval 22884	. . 3 ⊢ 𝑄 = (𝑚 ∈ 𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))
19	15, 18	ovmpoga 7578	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅) ∧ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))) ∈ V) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))
20	8, 19	mpd3an3 1491	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ifcif 4481  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  Fincfn 8981  Basecbs 17396  0_gc0g 17619   Mat cmat 22731   matRRep cmarrep 22880

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-1cn 11265  ax-addcl 11267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-nn 12343  df-slot 17369  df-ndx 17381  df-base 17397  df-mat 22732  df-marrep 22882

This theorem is used by:  marrepval  22886  minmar1marrep  22974