Theorem minmar1val 22969

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 22968	. . 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 22733	. . . . . . . 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 8091	. . . 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 2772	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
18	17	adantr 486	. . . . . 6 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → (𝑖 = 𝑘 ↔ 𝑖 = 𝐾))
19		eqeq2 2772	. . . . . . . 8 ⊢ (𝑙 = 𝐿 → (𝑗 = 𝑙 ↔ 𝑗 = 𝐿))
20	19	ifbid 4505	. . . . . . 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 4506	. . . . 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 7500	. . . 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 7579	. 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 1569   ∈ wcel 2144  Vcvv 3450  ifcif 4481  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  Fincfn 8977  Basecbs 17393  0_gc0g 17616  1_rcur 20413   Mat cmat 22728   minMatR1 cminmar1 22954

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 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-1cn 11262  ax-addcl 11264

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-nn 12340  df-slot 17366  df-ndx 17378  df-base 17394  df-mat 22729  df-minmar1 22956

This theorem is used by:  minmar1eval  22970  maducoevalmin1  22973  smadiadet  22991  smadiadetglem2  22993