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Theorem madjusmdetlem3 30204
Description: Lemma for madjusmdet 30206. (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 11916 . . . . . . . . . . 11 ℕ = (ℤ‘1)
31, 2syl6eleq 2849 . . . . . . . . . 10 (𝜑𝑁 ∈ (ℤ‘1))
4 fzdif2 29860 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
53, 4syl 17 . . . . . . . . 9 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
6 difss 3880 . . . . . . . . 9 ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
75, 6syl6eqssr 3797 . . . . . . . 8 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
87adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
108, 9sseldd 3745 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
11 simprr 813 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
128, 11sseldd 3745 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13 ovexd 6843 . . . . . 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 6948 . . . . . 6 ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁) ∧ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ V) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
1610, 12, 13, 15syl3anc 1477 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
179, 11ovresd 6966 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
18 eqid 2760 . . . . . . 7 (𝐼(subMat1‘𝑈)𝐽) = (𝐼(subMat1‘𝑈)𝐽)
191adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
20 madjusmdet.i . . . . . . . 8 (𝜑𝐼 ∈ (1...𝑁))
2120adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
22 madjusmdet.j . . . . . . . 8 (𝜑𝐽 ∈ (1...𝑁))
2322adantr 472 . . . . . . 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 2760 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
27 madjusmdet.b . . . . . . . . . 10 𝐵 = (Base‘𝐴)
2825, 26, 27matbas2i 20430 . . . . . . . . 9 (𝑈𝐵𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
2924, 28syl 17 . . . . . . . 8 (𝜑𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
3029adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
31 fz1ssnn 12565 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3231, 10sseldi 3742 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
3331, 12sseldi 3742 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
34 eqidd 2761 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
35 eqidd 2761 . . . . . . 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 30172 . . . . . 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 30203 . . . . . . . 8 ((𝜑𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃𝑆)‘𝑖))
479, 46syldan 488 . . . . . . 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 30203 . . . . . . . 8 ((𝜑𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5111, 50syldan 488 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5247, 51oveq12d 6831 . . . . . 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 2794 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
5416, 17, 533eqtr4rd 2805 . . . 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 3109 . . 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 2760 . . . . 5 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
5725, 27, 56, 18, 1, 20, 22, 24smatcl 30177 . . . 4 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
58 fzfid 12966 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
59 eqid 2760 . . . . . . . . . . . . . 14 (1...𝑁) = (1...𝑁)
60 eqid 2760 . . . . . . . . . . . . . 14 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
61 eqid 2760 . . . . . . . . . . . . . 14 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
6259, 44, 60, 61fzto1st 30162 . . . . . . . . . . . . 13 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
6320, 62syl 17 . . . . . . . . . . . 12 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
64 eluzfz2 12542 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
653, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (1...𝑁))
6659, 45, 60, 61fzto1st 30162 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
6765, 66syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
68 eqid 2760 . . . . . . . . . . . . . . 15 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
6960, 61, 68symginv 18022 . . . . . . . . . . . . . 14 (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7067, 69syl 17 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7160symggrp 18020 . . . . . . . . . . . . . . 15 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
7258, 71syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
7361, 68grpinvcl 17668 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7472, 67, 73syl2anc 696 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7570, 74eqeltrrd 2840 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
76 eqid 2760 . . . . . . . . . . . . . 14 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
7760, 61, 76symgov 18010 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
7860, 61, 76symgcl 18011 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7977, 78eqeltrrd 2840 . . . . . . . . . . . 12 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
8063, 75, 79syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
81803ad2ant1 1128 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
82 simp2 1132 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
8360, 61symgfv 18007 . . . . . . . . . 10 (((𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8481, 82, 83syl2anc 696 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8559, 48, 60, 61fzto1st 30162 . . . . . . . . . . . . 13 (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8622, 85syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8759, 49, 60, 61fzto1st 30162 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8865, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8960, 61, 68symginv 18022 . . . . . . . . . . . . . 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 17668 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9272, 88, 91syl2anc 696 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9390, 92eqeltrrd 2840 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
9460, 61, 76symgov 18010 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
9560, 61, 76symgcl 18011 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9694, 95eqeltrrd 2840 . . . . . . . . . . . 12 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9786, 93, 96syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
98973ad2ant1 1128 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
99 simp3 1133 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
10060, 61symgfv 18007 . . . . . . . . . 10 (((𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
10198, 99, 100syl2anc 696 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
102243ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
10325, 26, 27, 84, 101, 102matecld 20434 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ (Base‘𝑅))
10425, 26, 27, 58, 42, 103matbas2d 20431 . . . . . . 7 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗))) ∈ 𝐵)
10514, 104syl5eqel 2843 . . . . . 6 (𝜑𝑊𝐵)
10625, 27submatres 30181 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
1071, 105, 106syl2anc 696 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
108 eqid 2760 . . . . . 6 (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
10925, 27, 56, 108, 1, 65, 65, 105smatcl 30177 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
110107, 109eqeltrrd 2840 . . . 4 (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111 eqid 2760 . . . . 5 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
112111, 56eqmat 20432 . . . 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 696 . . 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 247 . 2 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
115114, 107eqtr4d 2797 1 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cdif 3712  wss 3715  ifcif 4230  {csn 4321   class class class wbr 4804  cmpt 4881   × cxp 5264  ccnv 5265  cres 5268  ccom 5270  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  Fincfn 8121  1c1 10129   + caddc 10131   < clt 10266  cle 10267  cmin 10458  cn 11212  cuz 11879  ...cfz 12519  Basecbs 16059  +gcplusg 16143  .rcmulr 16144  Grpcgrp 17623  invgcminusg 17624  SymGrpcsymg 17997  CRingccrg 18748  ℤRHomczrh 20050   Mat cmat 20415   maDet cmdat 20592   maAdju cmadu 20640  subMat1csmat 30168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-rp 12026  df-fz 12520  df-fzo 12660  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-prds 16310  df-pws 16312  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627  df-symg 17998  df-pmtr 18062  df-sra 19374  df-rgmod 19375  df-dsmm 20278  df-frlm 20293  df-mat 20416  df-subma 20585  df-smat 30169
This theorem is referenced by:  madjusmdetlem4  30205
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