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Theorem madjusmdetlem3 30436
Description: Lemma for madjusmdet 30438. (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 12012 . . . . . . . . . . 11 ℕ = (ℤ‘1)
31, 2syl6eleq 2916 . . . . . . . . . 10 (𝜑𝑁 ∈ (ℤ‘1))
4 fzdif2 30094 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
53, 4syl 17 . . . . . . . . 9 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
6 difss 3966 . . . . . . . . 9 ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
75, 6syl6eqssr 3881 . . . . . . . 8 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
87adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9 simprl 787 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
108, 9sseldd 3828 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
11 simprr 789 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
128, 11sseldd 3828 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13 ovexd 6944 . . . . . 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 7048 . . . . . 6 ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁) ∧ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ V) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
1610, 12, 13, 15syl3anc 1494 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
179, 11ovresd 7066 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
18 eqid 2825 . . . . . . 7 (𝐼(subMat1‘𝑈)𝐽) = (𝐼(subMat1‘𝑈)𝐽)
191adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
20 madjusmdet.i . . . . . . . 8 (𝜑𝐼 ∈ (1...𝑁))
2120adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
22 madjusmdet.j . . . . . . . 8 (𝜑𝐽 ∈ (1...𝑁))
2322adantr 474 . . . . . . 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 2825 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
27 madjusmdet.b . . . . . . . . . 10 𝐵 = (Base‘𝐴)
2825, 26, 27matbas2i 20602 . . . . . . . . 9 (𝑈𝐵𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
2924, 28syl 17 . . . . . . . 8 (𝜑𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
3029adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
31 fz1ssnn 12672 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3231, 10sseldi 3825 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
3331, 12sseldi 3825 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
34 eqidd 2826 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
35 eqidd 2826 . . . . . . 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 30404 . . . . . 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 30435 . . . . . . . 8 ((𝜑𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃𝑆)‘𝑖))
479, 46syldan 585 . . . . . . 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 30435 . . . . . . . 8 ((𝜑𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5111, 50syldan 585 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5247, 51oveq12d 6928 . . . . . 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 2861 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
5416, 17, 533eqtr4rd 2872 . . . 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 3180 . . 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 2825 . . . . 5 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
5725, 27, 56, 18, 1, 20, 22, 24smatcl 30409 . . . 4 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
58 fzfid 13074 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
59 eqid 2825 . . . . . . . . . . . . . 14 (1...𝑁) = (1...𝑁)
60 eqid 2825 . . . . . . . . . . . . . 14 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
61 eqid 2825 . . . . . . . . . . . . . 14 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
6259, 44, 60, 61fzto1st 30394 . . . . . . . . . . . . 13 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
6320, 62syl 17 . . . . . . . . . . . 12 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
64 eluzfz2 12649 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
653, 64syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (1...𝑁))
6659, 45, 60, 61fzto1st 30394 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
6765, 66syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
68 eqid 2825 . . . . . . . . . . . . . . 15 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
6960, 61, 68symginv 18179 . . . . . . . . . . . . . 14 (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7067, 69syl 17 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7160symggrp 18177 . . . . . . . . . . . . . . 15 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
7258, 71syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
7361, 68grpinvcl 17828 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7472, 67, 73syl2anc 579 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7570, 74eqeltrrd 2907 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
76 eqid 2825 . . . . . . . . . . . . . 14 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
7760, 61, 76symgov 18167 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
7860, 61, 76symgcl 18168 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7977, 78eqeltrrd 2907 . . . . . . . . . . . 12 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
8063, 75, 79syl2anc 579 . . . . . . . . . . 11 (𝜑 → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
81803ad2ant1 1167 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
82 simp2 1171 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
8360, 61symgfv 18164 . . . . . . . . . 10 (((𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8481, 82, 83syl2anc 579 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8559, 48, 60, 61fzto1st 30394 . . . . . . . . . . . . 13 (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8622, 85syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8759, 49, 60, 61fzto1st 30394 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8865, 87syl 17 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8960, 61, 68symginv 18179 . . . . . . . . . . . . . 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 17828 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9272, 88, 91syl2anc 579 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9390, 92eqeltrrd 2907 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
9460, 61, 76symgov 18167 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
9560, 61, 76symgcl 18168 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9694, 95eqeltrrd 2907 . . . . . . . . . . . 12 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9786, 93, 96syl2anc 579 . . . . . . . . . . 11 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
98973ad2ant1 1167 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
99 simp3 1172 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
10060, 61symgfv 18164 . . . . . . . . . 10 (((𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
10198, 99, 100syl2anc 579 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
102243ad2ant1 1167 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
10325, 26, 27, 84, 101, 102matecld 20606 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ (Base‘𝑅))
10425, 26, 27, 58, 42, 103matbas2d 20603 . . . . . . 7 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗))) ∈ 𝐵)
10514, 104syl5eqel 2910 . . . . . 6 (𝜑𝑊𝐵)
10625, 27submatres 30413 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
1071, 105, 106syl2anc 579 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
108 eqid 2825 . . . . . 6 (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
10925, 27, 56, 108, 1, 65, 65, 105smatcl 30409 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
110107, 109eqeltrrd 2907 . . . 4 (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111 eqid 2825 . . . . 5 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
112111, 56eqmat 20604 . . . 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 579 . . 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 249 . 2 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
115114, 107eqtr4d 2864 1 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  wral 3117  Vcvv 3414  cdif 3795  wss 3798  ifcif 4308  {csn 4399   class class class wbr 4875  cmpt 4954   × cxp 5344  ccnv 5345  cres 5348  ccom 5350  cfv 6127  (class class class)co 6910  cmpt2 6912  𝑚 cmap 8127  Fincfn 8228  1c1 10260   + caddc 10262   < clt 10398  cle 10399  cmin 10592  cn 11357  cuz 11975  ...cfz 12626  Basecbs 16229  +gcplusg 16312  .rcmulr 16313  Grpcgrp 17783  invgcminusg 17784  SymGrpcsymg 18154  CRingccrg 18909  ℤRHomczrh 20215   Mat cmat 20587   maDet cmdat 20765   maAdju cmadu 20813  subMat1csmat 30400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-ot 4408  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-supp 7565  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-ixp 8182  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-fsupp 8551  df-sup 8623  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-uz 11976  df-rp 12120  df-fz 12627  df-fzo 12768  df-struct 16231  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-ress 16237  df-plusg 16325  df-mulr 16326  df-sca 16328  df-vsca 16329  df-ip 16330  df-tset 16331  df-ple 16332  df-ds 16334  df-hom 16336  df-cco 16337  df-0g 16462  df-prds 16468  df-pws 16470  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-grp 17786  df-minusg 17787  df-symg 18155  df-pmtr 18219  df-sra 19540  df-rgmod 19541  df-dsmm 20446  df-frlm 20461  df-mat 20588  df-subma 20758  df-smat 30401
This theorem is referenced by:  madjusmdetlem4  30437
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