Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  madjusmdetlem3 Structured version   Visualization version   GIF version

Theorem madjusmdetlem3 33301
Description: Lemma for madjusmdet 33303. (Contributed by Thierry Arnoux, 27-Aug-2020.)
Hypotheses
Ref Expression
madjusmdet.b 𝐵 = (Base‘𝐴)
madjusmdet.a 𝐴 = ((1...𝑁) Mat 𝑅)
madjusmdet.d 𝐷 = ((1...𝑁) maDet 𝑅)
madjusmdet.k 𝐾 = ((1...𝑁) maAdju 𝑅)
madjusmdet.t · = (.r𝑅)
madjusmdet.z 𝑍 = (ℤRHom‘𝑅)
madjusmdet.e 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
madjusmdet.n (𝜑𝑁 ∈ ℕ)
madjusmdet.r (𝜑𝑅 ∈ CRing)
madjusmdet.i (𝜑𝐼 ∈ (1...𝑁))
madjusmdet.j (𝜑𝐽 ∈ (1...𝑁))
madjusmdet.m (𝜑𝑀𝐵)
madjusmdetlem2.p 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
madjusmdetlem2.s 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))
madjusmdetlem4.q 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))
madjusmdetlem4.t 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))
madjusmdetlem3.w 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
madjusmdetlem3.u (𝜑𝑈𝐵)
Assertion
Ref Expression
madjusmdetlem3 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Distinct variable groups:   𝐵,𝑖,𝑗   𝑖,𝐼,𝑗   𝑖,𝐽,𝑗   𝑖,𝑀,𝑗   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗   𝑄,𝑖,𝑗   𝑅,𝑖,𝑗   𝜑,𝑖,𝑗   𝑆,𝑖,𝑗   𝑇,𝑖,𝑗   𝑈,𝑖,𝑗   𝑖,𝑊,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐷(𝑖,𝑗)   · (𝑖,𝑗)   𝐸(𝑖,𝑗)   𝐾(𝑖,𝑗)   𝑍(𝑖,𝑗)

Proof of Theorem madjusmdetlem3
StepHypRef Expression
1 madjusmdet.n . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ)
2 nnuz 12863 . . . . . . . . . . 11 ℕ = (ℤ‘1)
31, 2eleqtrdi 2835 . . . . . . . . . 10 (𝜑𝑁 ∈ (ℤ‘1))
4 fzdif2 32474 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
53, 4syl 17 . . . . . . . . 9 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
6 difss 4124 . . . . . . . . 9 ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
75, 6eqsstrrdi 4030 . . . . . . . 8 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
87adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9 simprl 768 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
108, 9sseldd 3976 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
11 simprr 770 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
128, 11sseldd 3976 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13 ovexd 7437 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ V)
14 madjusmdetlem3.w . . . . . . 7 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
1514ovmpt4g 7548 . . . . . 6 ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁) ∧ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ V) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
1610, 12, 13, 15syl3anc 1368 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
179, 11ovresd 7568 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
18 eqid 2724 . . . . . . 7 (𝐼(subMat1‘𝑈)𝐽) = (𝐼(subMat1‘𝑈)𝐽)
191adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
20 madjusmdet.i . . . . . . . 8 (𝜑𝐼 ∈ (1...𝑁))
2120adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
22 madjusmdet.j . . . . . . . 8 (𝜑𝐽 ∈ (1...𝑁))
2322adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐽 ∈ (1...𝑁))
24 madjusmdetlem3.u . . . . . . . . 9 (𝜑𝑈𝐵)
25 madjusmdet.a . . . . . . . . . 10 𝐴 = ((1...𝑁) Mat 𝑅)
26 eqid 2724 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
27 madjusmdet.b . . . . . . . . . 10 𝐵 = (Base‘𝐴)
2825, 26, 27matbas2i 22248 . . . . . . . . 9 (𝑈𝐵𝑈 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
2924, 28syl 17 . . . . . . . 8 (𝜑𝑈 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
3029adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑈 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
31 fz1ssnn 13530 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3231, 10sselid 3973 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
3331, 12sselid 3973 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
34 eqidd 2725 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
35 eqidd 2725 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)))
3618, 19, 19, 21, 23, 30, 32, 33, 34, 35smatlem 33269 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1))𝑈if(𝑗 < 𝐽, 𝑗, (𝑗 + 1))))
37 madjusmdet.d . . . . . . . . 9 𝐷 = ((1...𝑁) maDet 𝑅)
38 madjusmdet.k . . . . . . . . 9 𝐾 = ((1...𝑁) maAdju 𝑅)
39 madjusmdet.t . . . . . . . . 9 · = (.r𝑅)
40 madjusmdet.z . . . . . . . . 9 𝑍 = (ℤRHom‘𝑅)
41 madjusmdet.e . . . . . . . . 9 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
42 madjusmdet.r . . . . . . . . 9 (𝜑𝑅 ∈ CRing)
43 madjusmdet.m . . . . . . . . 9 (𝜑𝑀𝐵)
44 madjusmdetlem2.p . . . . . . . . 9 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
45 madjusmdetlem2.s . . . . . . . . 9 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))
4627, 25, 37, 38, 39, 40, 41, 1, 42, 20, 20, 43, 44, 45madjusmdetlem2 33300 . . . . . . . 8 ((𝜑𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃𝑆)‘𝑖))
479, 46syldan 590 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃𝑆)‘𝑖))
48 madjusmdetlem4.q . . . . . . . . 9 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))
49 madjusmdetlem4.t . . . . . . . . 9 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))
5027, 25, 37, 38, 39, 40, 41, 1, 42, 22, 22, 43, 48, 49madjusmdetlem2 33300 . . . . . . . 8 ((𝜑𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5111, 50syldan 590 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5247, 51oveq12d 7420 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1))𝑈if(𝑗 < 𝐽, 𝑗, (𝑗 + 1))) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
5336, 52eqtrd 2764 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
5416, 17, 533eqtr4rd 2775 . . . 4 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗))
5554ralrimivva 3192 . . 3 (𝜑 → ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗))
56 eqid 2724 . . . . 5 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
5725, 27, 56, 18, 1, 20, 22, 24smatcl 33274 . . . 4 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
58 fzfid 13936 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
59 eqid 2724 . . . . . . . . . . . . . 14 (1...𝑁) = (1...𝑁)
60 eqid 2724 . . . . . . . . . . . . . 14 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
61 eqid 2724 . . . . . . . . . . . . . 14 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
6259, 44, 60, 61fzto1st 32733 . . . . . . . . . . . . 13 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
6320, 62syl 17 . . . . . . . . . . . 12 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
64 eluzfz2 13507 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
653, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (1...𝑁))
6659, 45, 60, 61fzto1st 32733 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
6765, 66syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
68 eqid 2724 . . . . . . . . . . . . . . 15 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
6960, 61, 68symginv 19314 . . . . . . . . . . . . . 14 (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7067, 69syl 17 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7160symggrp 19312 . . . . . . . . . . . . . . 15 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
7258, 71syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
7361, 68grpinvcl 18909 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7472, 67, 73syl2anc 583 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7570, 74eqeltrrd 2826 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
76 eqid 2724 . . . . . . . . . . . . . 14 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
7760, 61, 76symgov 19295 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
7860, 61, 76symgcl 19296 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7977, 78eqeltrrd 2826 . . . . . . . . . . . 12 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
8063, 75, 79syl2anc 583 . . . . . . . . . . 11 (𝜑 → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
81803ad2ant1 1130 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
82 simp2 1134 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
8360, 61symgfv 19291 . . . . . . . . . 10 (((𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8481, 82, 83syl2anc 583 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8559, 48, 60, 61fzto1st 32733 . . . . . . . . . . . . 13 (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8622, 85syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8759, 49, 60, 61fzto1st 32733 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8865, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8960, 61, 68symginv 19314 . . . . . . . . . . . . . 14 (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) = 𝑇)
9088, 89syl 17 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) = 𝑇)
9161, 68grpinvcl 18909 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9272, 88, 91syl2anc 583 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9390, 92eqeltrrd 2826 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
9460, 61, 76symgov 19295 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
9560, 61, 76symgcl 19296 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9694, 95eqeltrrd 2826 . . . . . . . . . . . 12 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9786, 93, 96syl2anc 583 . . . . . . . . . . 11 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
98973ad2ant1 1130 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
99 simp3 1135 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
10060, 61symgfv 19291 . . . . . . . . . 10 (((𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
10198, 99, 100syl2anc 583 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
102243ad2ant1 1130 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
10325, 26, 27, 84, 101, 102matecld 22252 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ (Base‘𝑅))
10425, 26, 27, 58, 42, 103matbas2d 22249 . . . . . . 7 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗))) ∈ 𝐵)
10514, 104eqeltrid 2829 . . . . . 6 (𝜑𝑊𝐵)
10625, 27submatres 33278 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
1071, 105, 106syl2anc 583 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
108 eqid 2724 . . . . . 6 (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
10925, 27, 56, 108, 1, 65, 65, 105smatcl 33274 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
110107, 109eqeltrrd 2826 . . . 4 (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111 eqid 2724 . . . . 5 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
112111, 56eqmat 22250 . . . 4 (((𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)) ∧ (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) → ((𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗)))
11357, 110, 112syl2anc 583 . . 3 (𝜑 → ((𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗)))
11455, 113mpbird 257 . 2 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
115114, 107eqtr4d 2767 1 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  Vcvv 3466  cdif 3938  wss 3941  ifcif 4521  {csn 4621   class class class wbr 5139  cmpt 5222   × cxp 5665  ccnv 5666  cres 5669  ccom 5671  cfv 6534  (class class class)co 7402  cmpo 7404  m cmap 8817  Fincfn 8936  1c1 11108   + caddc 11110   < clt 11246  cle 11247  cmin 11442  cn 12210  cuz 12820  ...cfz 13482  Basecbs 17145  +gcplusg 17198  .rcmulr 17199  Grpcgrp 18855  invgcminusg 18856  SymGrpcsymg 19278  CRingccrg 20131  ℤRHomczrh 21356   Mat cmat 22231   maDet cmdat 22410   maAdju cmadu 22458  subMat1csmat 33265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-ot 4630  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-supp 8142  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-sup 9434  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-nn 12211  df-2 12273  df-3 12274  df-4 12275  df-5 12276  df-6 12277  df-7 12278  df-8 12279  df-9 12280  df-n0 12471  df-z 12557  df-dec 12676  df-uz 12821  df-rp 12973  df-fz 13483  df-fzo 13626  df-struct 17081  df-sets 17098  df-slot 17116  df-ndx 17128  df-base 17146  df-ress 17175  df-plusg 17211  df-mulr 17212  df-sca 17214  df-vsca 17215  df-ip 17216  df-tset 17217  df-ple 17218  df-ds 17220  df-hom 17222  df-cco 17223  df-0g 17388  df-prds 17394  df-pws 17396  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-efmnd 18786  df-grp 18858  df-minusg 18859  df-symg 19279  df-pmtr 19354  df-sra 21013  df-rgmod 21014  df-dsmm 21597  df-frlm 21612  df-mat 22232  df-subma 22403  df-smat 33266
This theorem is referenced by:  madjusmdetlem4  33302
  Copyright terms: Public domain W3C validator