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Theorem madjusmdetlem3 29674
Description: Lemma for madjusmdet 29676. (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 11667 . . . . . . . . . . 11 ℕ = (ℤ‘1)
31, 2syl6eleq 2708 . . . . . . . . . 10 (𝜑𝑁 ∈ (ℤ‘1))
4 fzdif2 29390 . . . . . . . . . 10 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
53, 4syl 17 . . . . . . . . 9 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
6 difss 3715 . . . . . . . . 9 ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁)
75, 6syl6eqssr 3635 . . . . . . . 8 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
87adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
9 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
108, 9sseldd 3584 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
11 simprr 795 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
128, 11sseldd 3584 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
13 ovex 6632 . . . . . . 7 (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ V
1413a1i 11 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ V)
15 madjusmdetlem3.w . . . . . . 7 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
1615ovmpt4g 6736 . . . . . 6 ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁) ∧ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ V) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
1710, 12, 14, 16syl3anc 1323 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖𝑊𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
189, 11ovresd 6754 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))𝑗) = (𝑖𝑊𝑗))
19 eqid 2621 . . . . . . 7 (𝐼(subMat1‘𝑈)𝐽) = (𝐼(subMat1‘𝑈)𝐽)
201adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
21 madjusmdet.i . . . . . . . 8 (𝜑𝐼 ∈ (1...𝑁))
2221adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
23 madjusmdet.j . . . . . . . 8 (𝜑𝐽 ∈ (1...𝑁))
2423adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐽 ∈ (1...𝑁))
25 madjusmdetlem3.u . . . . . . . . 9 (𝜑𝑈𝐵)
26 madjusmdet.a . . . . . . . . . 10 𝐴 = ((1...𝑁) Mat 𝑅)
27 eqid 2621 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
28 madjusmdet.b . . . . . . . . . 10 𝐵 = (Base‘𝐴)
2926, 27, 28matbas2i 20147 . . . . . . . . 9 (𝑈𝐵𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
3025, 29syl 17 . . . . . . . 8 (𝜑𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
3130adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑈 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
32 fz1ssnn 12314 . . . . . . . 8 (1...𝑁) ⊆ ℕ
3332, 10sseldi 3581 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
3432, 12sseldi 3581 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
35 eqidd 2622 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
36 eqidd 2622 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)))
3719, 20, 20, 22, 24, 31, 33, 34, 35, 36smatlem 29642 . . . . . 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))))
38 madjusmdet.d . . . . . . . . 9 𝐷 = ((1...𝑁) maDet 𝑅)
39 madjusmdet.k . . . . . . . . 9 𝐾 = ((1...𝑁) maAdju 𝑅)
40 madjusmdet.t . . . . . . . . 9 · = (.r𝑅)
41 madjusmdet.z . . . . . . . . 9 𝑍 = (ℤRHom‘𝑅)
42 madjusmdet.e . . . . . . . . 9 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
43 madjusmdet.r . . . . . . . . 9 (𝜑𝑅 ∈ CRing)
44 madjusmdet.m . . . . . . . . 9 (𝜑𝑀𝐵)
45 madjusmdetlem2.p . . . . . . . . 9 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖𝐼, (𝑖 − 1), 𝑖)))
46 madjusmdetlem2.s . . . . . . . . 9 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖𝑁, (𝑖 − 1), 𝑖)))
4728, 26, 38, 39, 40, 41, 42, 1, 43, 21, 21, 44, 45, 46madjusmdetlem2 29673 . . . . . . . 8 ((𝜑𝑖 ∈ (1...(𝑁 − 1))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃𝑆)‘𝑖))
489, 47syldan 487 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = ((𝑃𝑆)‘𝑖))
49 madjusmdetlem4.q . . . . . . . . 9 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗𝐽, (𝑗 − 1), 𝑗)))
50 madjusmdetlem4.t . . . . . . . . 9 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗𝑁, (𝑗 − 1), 𝑗)))
5128, 26, 38, 39, 40, 41, 42, 1, 43, 23, 23, 44, 49, 50madjusmdetlem2 29673 . . . . . . . 8 ((𝜑𝑗 ∈ (1...(𝑁 − 1))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5211, 51syldan 487 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐽, 𝑗, (𝑗 + 1)) = ((𝑄𝑇)‘𝑗))
5348, 52oveq12d 6622 . . . . . 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)))
5437, 53eqtrd 2655 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘𝑈)𝐽)𝑗) = (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)))
5517, 18, 543eqtr4rd 2666 . . . 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))
5655ralrimivva 2965 . . 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))
57 eqid 2621 . . . . 5 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
5826, 28, 57, 19, 1, 21, 23, 25smatcl 29647 . . . 4 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
59 fzfid 12712 . . . . . . . 8 (𝜑 → (1...𝑁) ∈ Fin)
60 eqid 2621 . . . . . . . . . . . . . 14 (1...𝑁) = (1...𝑁)
61 eqid 2621 . . . . . . . . . . . . . 14 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
62 eqid 2621 . . . . . . . . . . . . . 14 (Base‘(SymGrp‘(1...𝑁))) = (Base‘(SymGrp‘(1...𝑁)))
6360, 45, 61, 62fzto1st 29635 . . . . . . . . . . . . 13 (𝐼 ∈ (1...𝑁) → 𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
6421, 63syl 17 . . . . . . . . . . . 12 (𝜑𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))))
65 eluzfz2 12291 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
663, 65syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ (1...𝑁))
6760, 46, 61, 62fzto1st 29635 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
6866, 67syl 17 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
69 eqid 2621 . . . . . . . . . . . . . . 15 (invg‘(SymGrp‘(1...𝑁))) = (invg‘(SymGrp‘(1...𝑁)))
7061, 62, 69symginv 17743 . . . . . . . . . . . . . 14 (𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7168, 70syl 17 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) = 𝑆)
7261symggrp 17741 . . . . . . . . . . . . . . 15 ((1...𝑁) ∈ Fin → (SymGrp‘(1...𝑁)) ∈ Grp)
7359, 72syl 17 . . . . . . . . . . . . . 14 (𝜑 → (SymGrp‘(1...𝑁)) ∈ Grp)
7462, 69grpinvcl 17388 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7573, 68, 74syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
7671, 75eqeltrrd 2699 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (Base‘(SymGrp‘(1...𝑁))))
77 eqid 2621 . . . . . . . . . . . . . 14 (+g‘(SymGrp‘(1...𝑁))) = (+g‘(SymGrp‘(1...𝑁)))
7861, 62, 77symgov 17731 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) = (𝑃𝑆))
7961, 62, 77symgcl 17732 . . . . . . . . . . . . 13 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃(+g‘(SymGrp‘(1...𝑁)))𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
8078, 79eqeltrrd 2699 . . . . . . . . . . . 12 ((𝑃 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑆 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
8164, 76, 80syl2anc 692 . . . . . . . . . . 11 (𝜑 → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
82813ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))))
83 simp2 1060 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
8461, 62symgfv 17728 . . . . . . . . . 10 (((𝑃𝑆) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8582, 83, 84syl2anc 692 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑆)‘𝑖) ∈ (1...𝑁))
8660, 49, 61, 62fzto1st 29635 . . . . . . . . . . . . 13 (𝐽 ∈ (1...𝑁) → 𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8723, 86syl 17 . . . . . . . . . . . 12 (𝜑𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))))
8860, 50, 61, 62fzto1st 29635 . . . . . . . . . . . . . . 15 (𝑁 ∈ (1...𝑁) → 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
8966, 88syl 17 . . . . . . . . . . . . . 14 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
9061, 62, 69symginv 17743 . . . . . . . . . . . . . 14 (𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) = 𝑇)
9189, 90syl 17 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) = 𝑇)
9262, 69grpinvcl 17388 . . . . . . . . . . . . . 14 (((SymGrp‘(1...𝑁)) ∈ Grp ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9373, 89, 92syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → ((invg‘(SymGrp‘(1...𝑁)))‘𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9491, 93eqeltrrd 2699 . . . . . . . . . . . 12 (𝜑𝑇 ∈ (Base‘(SymGrp‘(1...𝑁))))
9561, 62, 77symgov 17731 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) = (𝑄𝑇))
9661, 62, 77symgcl 17732 . . . . . . . . . . . . 13 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄(+g‘(SymGrp‘(1...𝑁)))𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9795, 96eqeltrrd 2699 . . . . . . . . . . . 12 ((𝑄 ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑇 ∈ (Base‘(SymGrp‘(1...𝑁)))) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
9887, 94, 97syl2anc 692 . . . . . . . . . . 11 (𝜑 → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
99983ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))))
100 simp3 1061 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
10161, 62symgfv 17728 . . . . . . . . . 10 (((𝑄𝑇) ∈ (Base‘(SymGrp‘(1...𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
10299, 100, 101syl2anc 692 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑄𝑇)‘𝑗) ∈ (1...𝑁))
103253ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
10426, 27, 28, 85, 102, 103matecld 20151 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗)) ∈ (Base‘𝑅))
10526, 27, 28, 59, 43, 104matbas2d 20148 . . . . . . 7 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃𝑆)‘𝑖)𝑈((𝑄𝑇)‘𝑗))) ∈ 𝐵)
10615, 105syl5eqel 2702 . . . . . 6 (𝜑𝑊𝐵)
10726, 28submatres 29651 . . . . . 6 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
1081, 106, 107syl2anc 692 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
109 eqid 2621 . . . . . 6 (𝑁(subMat1‘𝑊)𝑁) = (𝑁(subMat1‘𝑊)𝑁)
11026, 28, 57, 109, 1, 66, 66, 106smatcl 29647 . . . . 5 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
111108, 110eqeltrrd 2699 . . . 4 (𝜑 → (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
112 eqid 2621 . . . . 5 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
113112, 57eqmat 20149 . . . 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)))
11458, 111, 113syl2anc 692 . . 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)))
11556, 114mpbird 247 . 2 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑊 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
116115, 108eqtr4d 2658 1 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cdif 3552  wss 3555  ifcif 4058  {csn 4148   class class class wbr 4613  cmpt 4673   × cxp 5072  ccnv 5073  cres 5076  ccom 5078  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802  Fincfn 7899  1c1 9881   + caddc 9883   < clt 10018  cle 10019  cmin 10210  cn 10964  cuz 11631  ...cfz 12268  Basecbs 15781  +gcplusg 15862  .rcmulr 15863  Grpcgrp 17343  invgcminusg 17344  SymGrpcsymg 17718  CRingccrg 18469  ℤRHomczrh 19767   Mat cmat 20132   maDet cmdat 20309   maAdju cmadu 20357  subMat1csmat 29638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-0g 16023  df-prds 16029  df-pws 16031  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-symg 17719  df-pmtr 17783  df-sra 19091  df-rgmod 19092  df-dsmm 19995  df-frlm 20010  df-mat 20133  df-subma 20302  df-smat 29639
This theorem is referenced by:  madjusmdetlem4  29675
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