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Theorem madjusmdetlem3 31151
Description: Lemma for madjusmdet 31153. (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 12269 . . . . . . . . . . 11 ℕ = (ℤ‘1)
31, 2eleqtrdi 2924 . . . . . . . . . 10 (𝜑𝑁 ∈ (ℤ‘1))
4 fzdif2 30524 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
53, 4syl 17 . . . . . . . . 9 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
6 difss 4083 . . . . . . . . 9 ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
75, 6eqsstrrdi 3997 . . . . . . . 8 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
87adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
108, 9sseldd 3943 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
11 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
128, 11sseldd 3943 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13 ovexd 7175 . . . . . 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 7281 . . . . . 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 7300 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
18 eqid 2822 . . . . . . 7 (𝐼(subMat1‘𝑈)𝐽) = (𝐼(subMat1‘𝑈)𝐽)
191adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
20 madjusmdet.i . . . . . . . 8 (𝜑𝐼 ∈ (1...𝑁))
2120adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
22 madjusmdet.j . . . . . . . 8 (𝜑𝐽 ∈ (1...𝑁))
2322adantr 484 . . . . . . 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 2822 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
27 madjusmdet.b . . . . . . . . . 10 𝐵 = (Base‘𝐴)
2825, 26, 27matbas2i 21025 . . . . . . . . 9 (𝑈𝐵𝑈 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
2924, 28syl 17 . . . . . . . 8 (𝜑𝑈 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
3029adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑈 ∈ ((Base‘𝑅) ↑m ((1...𝑁) × (1...𝑁))))
31 fz1ssnn 12933 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3231, 10sseldi 3940 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
3331, 12sseldi 3940 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
34 eqidd 2823 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
35 eqidd 2823 . . . . . . 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 31119 . . . . . 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 31150 . . . . . . . 8 ((𝜑𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃𝑆)‘𝑖))
479, 46syldan 594 . . . . . . 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 31150 . . . . . . . 8 ((𝜑𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5111, 50syldan 594 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5247, 51oveq12d 7158 . . . . . 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 2857 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
5416, 17, 533eqtr4rd 2868 . . . 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 3181 . . 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 2822 . . . . 5 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
5725, 27, 56, 18, 1, 20, 22, 24smatcl 31124 . . . 4 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
58 fzfid 13336 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
59 eqid 2822 . . . . . . . . . . . . . 14 (1...𝑁) = (1...𝑁)
60 eqid 2822 . . . . . . . . . . . . . 14 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
61 eqid 2822 . . . . . . . . . . . . . 14 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
6259, 44, 60, 61fzto1st 30776 . . . . . . . . . . . . 13 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
6320, 62syl 17 . . . . . . . . . . . 12 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
64 eluzfz2 12910 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
653, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (1...𝑁))
6659, 45, 60, 61fzto1st 30776 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
6765, 66syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
68 eqid 2822 . . . . . . . . . . . . . . 15 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
6960, 61, 68symginv 18521 . . . . . . . . . . . . . 14 (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7067, 69syl 17 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7160symggrp 18519 . . . . . . . . . . . . . . 15 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
7258, 71syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
7361, 68grpinvcl 18142 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7472, 67, 73syl2anc 587 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7570, 74eqeltrrd 2915 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
76 eqid 2822 . . . . . . . . . . . . . 14 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
7760, 61, 76symgov 18503 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
7860, 61, 76symgcl 18504 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7977, 78eqeltrrd 2915 . . . . . . . . . . . 12 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
8063, 75, 79syl2anc 587 . . . . . . . . . . 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 18499 . . . . . . . . . 10 (((𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8481, 82, 83syl2anc 587 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8559, 48, 60, 61fzto1st 30776 . . . . . . . . . . . . 13 (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8622, 85syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8759, 49, 60, 61fzto1st 30776 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8865, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8960, 61, 68symginv 18521 . . . . . . . . . . . . . 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 18142 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9272, 88, 91syl2anc 587 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9390, 92eqeltrrd 2915 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
9460, 61, 76symgov 18503 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
9560, 61, 76symgcl 18504 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9694, 95eqeltrrd 2915 . . . . . . . . . . . 12 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9786, 93, 96syl2anc 587 . . . . . . . . . . 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 18499 . . . . . . . . . 10 (((𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
10198, 99, 100syl2anc 587 . . . . . . . . 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 21029 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ (Base‘𝑅))
10425, 26, 27, 58, 42, 103matbas2d 21026 . . . . . . 7 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗))) ∈ 𝐵)
10514, 104eqeltrid 2918 . . . . . 6 (𝜑𝑊𝐵)
10625, 27submatres 31128 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
1071, 105, 106syl2anc 587 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
108 eqid 2822 . . . . . 6 (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
10925, 27, 56, 108, 1, 65, 65, 105smatcl 31124 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
110107, 109eqeltrrd 2915 . . . 4 (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111 eqid 2822 . . . . 5 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
112111, 56eqmat 21027 . . . 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 587 . . 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 260 . 2 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
115114, 107eqtr4d 2860 1 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  Vcvv 3469  cdif 3905  wss 3908  ifcif 4439  {csn 4539   class class class wbr 5042  cmpt 5122   × cxp 5530  ccnv 5531  cres 5534  ccom 5536  cfv 6334  (class class class)co 7140  cmpo 7142  m cmap 8393  Fincfn 8496  1c1 10527   + caddc 10529   < clt 10664  cle 10665  cmin 10859  cn 11625  cuz 12231  ...cfz 12885  Basecbs 16474  +gcplusg 16556  .rcmulr 16557  Grpcgrp 18094  invgcminusg 18095  SymGrpcsymg 18486  CRingccrg 19289  ℤRHomczrh 20191   Mat cmat 21010   maDet cmdat 21187   maAdju cmadu 21235  subMat1csmat 31115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-ot 4548  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-struct 16476  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-mulr 16570  df-sca 16572  df-vsca 16573  df-ip 16574  df-tset 16575  df-ple 16576  df-ds 16578  df-hom 16580  df-cco 16581  df-0g 16706  df-prds 16712  df-pws 16714  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-submnd 17948  df-efmnd 18025  df-grp 18097  df-minusg 18098  df-symg 18487  df-pmtr 18561  df-sra 19935  df-rgmod 19936  df-dsmm 20419  df-frlm 20434  df-mat 21011  df-subma 21180  df-smat 31116
This theorem is referenced by:  madjusmdetlem4  31152
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