Theorem minmar1val 22876

Description: Third substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)

Hypotheses

Ref	Expression
minmar1fval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
minmar1fval.b	⊢ 𝐵 = (Base‘𝐴)
minmar1fval.q	⊢ 𝑄 = (𝑁 minMatR1 𝑅)
minmar1fval.o	⊢ 1 = (1r‘𝑅)
minmar1fval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
minmar1val	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))

Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝐾,𝑗   𝑖,𝐿,𝑗

Allowed substitution hints:   𝐴(𝑖, 𝑗)   𝐵(𝑖, 𝑗)   𝑄(𝑖, 𝑗)   1 (𝑖, 𝑗)   0 (𝑖, 𝑗)

Proof of Theorem minmar1val

Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		minmar1fval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		minmar1fval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		minmar1fval.q	. . . 4 ⊢ 𝑄 = (𝑁 minMatR1 𝑅)
4		minmar1fval.o	. . . 4 ⊢ 1 = (1r‘𝑅)
5		minmar1fval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6	1, 2, 3, 4, 5	minmar1val0 22875	. . 3 ⊢ (𝑀 ∈ 𝐵 → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))
7	6	3ad2ant1 1151	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))
8		simp2 1155	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → 𝐾 ∈ 𝑁)
9		simpl3 1212	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑘 = 𝐾) → 𝐿 ∈ 𝑁)
10	1, 2	matrcl 22640	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
11	10	simpld 500	. . . . . . 7 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
12	11, 11	jca 521	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
13	12	3ad2ant1 1151	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
14	13	adantr 486	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
15		mpoexga 8080	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) ∈ V)
16	14, 15	syl 18	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) ∈ V)
17		eqeq2 2774	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
18	17	adantr 486	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
19		eqeq2 2774	. . . . . . . 8 ⊢ (𝑙 = 𝐿 → (𝑗 = 𝑙 ↔ 𝑗 = 𝐿))
20	19	ifbid 4509	. . . . . . 7 ⊢ (𝑙 = 𝐿 → if(𝑗 = 𝑙, 1 , 0 ) = if(𝑗 = 𝐿, 1 , 0 ))
21	20	adantl 487	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → if(𝑗 = 𝑙, 1 , 0 ) = if(𝑗 = 𝐿, 1 , 0 ))
22	18, 21	ifbieq1d 4510	. . . . 5 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)) = if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))
23	22	mpoeq3dv 7496	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
24	23	adantl 487	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
25	8, 9, 16, 24	ovmpodv2 7575	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → ((𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))))
26	7, 25	mpd 16	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ifcif 4485  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  Fincfn 8956  Basecbs 17307  0_gc0g 17530  1_rcur 20326   Mat cmat 22635   minMatR1 cminmar1 22861

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-minmar1 22863

This theorem is used by:  minmar1eval  22877  maducoevalmin1  22880  smadiadet  22898  smadiadetglem2  22900