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Theorem 1smat1 30191
Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20596. (Contributed by Thierry Arnoux, 19-Aug-2020.)
Hypotheses
Ref Expression
1smat1.1 1 = (1r‘((1...𝑁) Mat 𝑅))
1smat1.r (𝜑𝑅 ∈ Ring)
1smat1.n (𝜑𝑁 ∈ ℕ)
1smat1.i (𝜑𝐼 ∈ (1...𝑁))
Assertion
Ref Expression
1smat1 (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))

Proof of Theorem 1smat1
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2806 . . . . 5 (𝐼(subMat1‘ 1 )𝐼) = (𝐼(subMat1‘ 1 )𝐼)
2 1smat1.n . . . . . 6 (𝜑𝑁 ∈ ℕ)
32adantr 468 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑁 ∈ ℕ)
4 1smat1.i . . . . . 6 (𝜑𝐼 ∈ (1...𝑁))
54adantr 468 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝐼 ∈ (1...𝑁))
6 1smat1.r . . . . . . . 8 (𝜑𝑅 ∈ Ring)
7 fzfi 12991 . . . . . . . 8 (1...𝑁) ∈ Fin
8 eqid 2806 . . . . . . . . 9 ((1...𝑁) Mat 𝑅) = ((1...𝑁) Mat 𝑅)
9 eqid 2806 . . . . . . . . 9 (Base‘((1...𝑁) Mat 𝑅)) = (Base‘((1...𝑁) Mat 𝑅))
10 1smat1.1 . . . . . . . . 9 1 = (1r‘((1...𝑁) Mat 𝑅))
118, 9, 10mat1bas 20462 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (1...𝑁) ∈ Fin) → 1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
126, 7, 11sylancl 576 . . . . . . 7 (𝜑1 ∈ (Base‘((1...𝑁) Mat 𝑅)))
13 eqid 2806 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
148, 13matbas2 20433 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
157, 6, 14sylancr 577 . . . . . . 7 (𝜑 → ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))) = (Base‘((1...𝑁) Mat 𝑅)))
1612, 15eleqtrrd 2888 . . . . . 6 (𝜑1 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
1716adantr 468 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 1 ∈ ((Base‘𝑅) ↑𝑚 ((1...𝑁) × (1...𝑁))))
18 fz1ssnn 12591 . . . . . 6 (1...(𝑁 − 1)) ⊆ ℕ
19 simprl 778 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...(𝑁 − 1)))
2018, 19sseldi 3796 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℕ)
21 simprr 780 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...(𝑁 − 1)))
2218, 21sseldi 3796 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℕ)
23 eqidd 2807 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)))
24 eqidd 2807 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)))
251, 3, 3, 5, 5, 17, 20, 22, 23, 24smatlem 30184 . . . 4 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) 1 if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))))
26 eqid 2806 . . . . 5 (1r𝑅) = (1r𝑅)
27 eqid 2806 . . . . 5 (0g𝑅) = (0g𝑅)
287a1i 11 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...𝑁) ∈ Fin)
296adantr 468 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑅 ∈ Ring)
30 nnuz 11937 . . . . . . . . 9 ℕ = (ℤ‘1)
3120, 30syl6eleq 2895 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (ℤ‘1))
32 fznatpl1 12614 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑖 ∈ (1...(𝑁 − 1))) → (𝑖 + 1) ∈ (1...𝑁))
333, 19, 32syl2anc 575 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 + 1) ∈ (1...𝑁))
34 peano2fzr 12573 . . . . . . . 8 ((𝑖 ∈ (ℤ‘1) ∧ (𝑖 + 1) ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
3531, 33, 34syl2anc 575 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ (1...𝑁))
3635, 33jca 503 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖 ∈ (1...𝑁) ∧ (𝑖 + 1) ∈ (1...𝑁)))
37 eleq1 2873 . . . . . . 7 (𝑖 = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) → (𝑖 ∈ (1...𝑁) ↔ if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁)))
38 eleq1 2873 . . . . . . 7 ((𝑖 + 1) = if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) → ((𝑖 + 1) ∈ (1...𝑁) ↔ if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁)))
3937, 38ifboth 4317 . . . . . 6 ((𝑖 ∈ (1...𝑁) ∧ (𝑖 + 1) ∈ (1...𝑁)) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁))
4036, 39syl 17 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) ∈ (1...𝑁))
4122, 30syl6eleq 2895 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (ℤ‘1))
42 fznatpl1 12614 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑗 ∈ (1...(𝑁 − 1))) → (𝑗 + 1) ∈ (1...𝑁))
433, 21, 42syl2anc 575 . . . . . . . 8 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 + 1) ∈ (1...𝑁))
44 peano2fzr 12573 . . . . . . . 8 ((𝑗 ∈ (ℤ‘1) ∧ (𝑗 + 1) ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
4541, 43, 44syl2anc 575 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ (1...𝑁))
4645, 43jca 503 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑗 ∈ (1...𝑁) ∧ (𝑗 + 1) ∈ (1...𝑁)))
47 eleq1 2873 . . . . . . 7 (𝑗 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) → (𝑗 ∈ (1...𝑁) ↔ if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁)))
48 eleq1 2873 . . . . . . 7 ((𝑗 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) → ((𝑗 + 1) ∈ (1...𝑁) ↔ if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁)))
4947, 48ifboth 4317 . . . . . 6 ((𝑗 ∈ (1...𝑁) ∧ (𝑗 + 1) ∈ (1...𝑁)) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁))
5046, 49syl 17 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ∈ (1...𝑁))
518, 26, 27, 28, 29, 40, 50, 10mat1ov 20461 . . . 4 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) 1 if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))) = if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1r𝑅), (0g𝑅)))
52 simpr 473 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → 𝑖 < 𝐼)
5352iftrued 4287 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = 𝑖)
5453eqeq1d 2808 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))))
55 simpr 473 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
5655iftrued 4287 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
5756eqeq2d 2816 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
58 simpr 473 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
5958iffalsed 4290 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = (𝑗 + 1))
6059eqeq2d 2816 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = (𝑗 + 1)))
6120nnred 11316 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℝ)
6261ad2antrr 708 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
63 fz1ssnn 12591 . . . . . . . . . . . . . . . . 17 (1...𝑁) ⊆ ℕ
6463, 4sseldi 3796 . . . . . . . . . . . . . . . 16 (𝜑𝐼 ∈ ℕ)
6564nnred 11316 . . . . . . . . . . . . . . 15 (𝜑𝐼 ∈ ℝ)
6665ad3antrrr 712 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
6722nnred 11316 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℝ)
6867ad2antrr 708 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
69 1red 10322 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 1 ∈ ℝ)
7068, 69readdcld 10350 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑗 + 1) ∈ ℝ)
7152adantr 468 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝐼)
7264nnzd 11743 . . . . . . . . . . . . . . . 16 (𝜑𝐼 ∈ ℤ)
7372ad3antrrr 712 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
7422nnzd 11743 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℤ)
7574ad2antrr 708 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑗 ∈ ℤ)
7666, 68, 58nltled 10468 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼𝑗)
77 zleltp1 11690 . . . . . . . . . . . . . . . 16 ((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝐼𝑗𝐼 < (𝑗 + 1)))
7877biimpa 464 . . . . . . . . . . . . . . 15 (((𝐼 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ 𝐼𝑗) → 𝐼 < (𝑗 + 1))
7973, 75, 76, 78syl21anc 857 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝐼 < (𝑗 + 1))
8062, 66, 70, 71, 79lttrd 10479 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < (𝑗 + 1))
8162, 80ltned 10454 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 ≠ (𝑗 + 1))
8281neneqd 2983 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = (𝑗 + 1))
8362, 66, 68, 71, 76ltletrd 10478 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖 < 𝑗)
8462, 83ltned 10454 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → 𝑖𝑗)
8584neneqd 2983 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
8682, 852falsed 367 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = (𝑗 + 1) ↔ 𝑖 = 𝑗))
8760, 86bitrd 270 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
8857, 87pm2.61dan 838 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → (𝑖 = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
8954, 88bitrd 270 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
90 simpr 473 . . . . . . . . . 10 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → ¬ 𝑖 < 𝐼)
9190iffalsed 4290 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = (𝑖 + 1))
9291eqeq1d 2808 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ (𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1))))
93 simpr 473 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝐼)
9493iftrued 4287 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = 𝑗)
9594eqeq2d 2816 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ (𝑖 + 1) = 𝑗))
9667ad2antrr 708 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ∈ ℝ)
9765ad3antrrr 712 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℝ)
9861ad2antrr 708 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ∈ ℝ)
99 1red 10322 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 1 ∈ ℝ)
10098, 99readdcld 10350 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ∈ ℝ)
10172ad3antrrr 712 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 ∈ ℤ)
10220nnzd 11743 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℤ)
103102ad2antrr 708 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖 ∈ ℤ)
10490adantr 468 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 < 𝐼)
10597, 98, 104nltled 10468 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼𝑖)
106 zleltp1 11690 . . . . . . . . . . . . . . . . 17 ((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝐼𝑖𝐼 < (𝑖 + 1)))
107106biimpa 464 . . . . . . . . . . . . . . . 16 (((𝐼 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝐼𝑖) → 𝐼 < (𝑖 + 1))
108101, 103, 105, 107syl21anc 857 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝐼 < (𝑖 + 1))
10996, 97, 100, 93, 108lttrd 10479 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < (𝑖 + 1))
11096, 109ltned 10454 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 ≠ (𝑖 + 1))
111110necomd 3033 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → (𝑖 + 1) ≠ 𝑗)
112111neneqd 2983 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ (𝑖 + 1) = 𝑗)
11396, 97, 98, 93, 105ltletrd 10478 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗 < 𝑖)
11496, 113ltned 10454 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑗𝑖)
115114necomd 3033 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → 𝑖𝑗)
116115neneqd 2983 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ¬ 𝑖 = 𝑗)
117112, 1162falsed 367 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = 𝑗𝑖 = 𝑗))
11895, 117bitrd 270 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
119 simpr 473 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ¬ 𝑗 < 𝐼)
120119iffalsed 4290 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) = (𝑗 + 1))
121120eqeq2d 2816 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ (𝑖 + 1) = (𝑗 + 1)))
12220nncnd 11317 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑖 ∈ ℂ)
123122ad3antrrr 712 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 ∈ ℂ)
12422nncnd 11317 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → 𝑗 ∈ ℂ)
125124ad3antrrr 712 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑗 ∈ ℂ)
126 1cnd 10316 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 1 ∈ ℂ)
127 simpr 473 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → (𝑖 + 1) = (𝑗 + 1))
128123, 125, 126, 127addcan2ad 10523 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ (𝑖 + 1) = (𝑗 + 1)) → 𝑖 = 𝑗)
129 simpr 473 . . . . . . . . . . . 12 (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → 𝑖 = 𝑗)
130129oveq1d 6885 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) ∧ 𝑖 = 𝑗) → (𝑖 + 1) = (𝑗 + 1))
131128, 130impbida 826 . . . . . . . . . 10 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = (𝑗 + 1) ↔ 𝑖 = 𝑗))
132121, 131bitrd 270 . . . . . . . . 9 ((((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) ∧ ¬ 𝑗 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
133118, 132pm2.61dan 838 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → ((𝑖 + 1) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
13492, 133bitrd 270 . . . . . . 7 (((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) ∧ ¬ 𝑖 < 𝐼) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
13589, 134pm2.61dan 838 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)) ↔ 𝑖 = 𝑗))
136135ifbid 4301 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1r𝑅), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
137 eqid 2806 . . . . . 6 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
138 fzfid 12992 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (1...(𝑁 − 1)) ∈ Fin)
139 eqid 2806 . . . . . 6 (1r‘((1...(𝑁 − 1)) Mat 𝑅)) = (1r‘((1...(𝑁 − 1)) Mat 𝑅))
140137, 26, 27, 138, 29, 19, 21, 139mat1ov 20461 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
141136, 140eqtr4d 2843 . . . 4 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → if(if(𝑖 < 𝐼, 𝑖, (𝑖 + 1)) = if(𝑗 < 𝐼, 𝑗, (𝑗 + 1)), (1r𝑅), (0g𝑅)) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
14225, 51, 1413eqtrd 2844 . . 3 ((𝜑 ∧ (𝑖 ∈ (1...(𝑁 − 1)) ∧ 𝑗 ∈ (1...(𝑁 − 1)))) → (𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
143142ralrimivva 3159 . 2 (𝜑 → ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗))
1441, 2, 2, 4, 4, 16smatrcl 30183 . . . 4 (𝜑 → (𝐼(subMat1‘ 1 )𝐼) ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
145 elmapfn 8111 . . . 4 ((𝐼(subMat1‘ 1 )𝐼) ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → (𝐼(subMat1‘ 1 )𝐼) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
146144, 145syl 17 . . 3 (𝜑 → (𝐼(subMat1‘ 1 )𝐼) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
147 fzfi 12991 . . . . . 6 (1...(𝑁 − 1)) ∈ Fin
148 eqid 2806 . . . . . . 7 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149137, 148, 139mat1bas 20462 . . . . . 6 ((𝑅 ∈ Ring ∧ (1...(𝑁 − 1)) ∈ Fin) → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
1506, 147, 149sylancl 576 . . . . 5 (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
151137, 13matbas2 20433 . . . . . 6 (((1...(𝑁 − 1)) ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
152147, 6, 151sylancr 577 . . . . 5 (𝜑 → ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) = (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
153150, 152eleqtrrd 2888 . . . 4 (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
154 elmapfn 8111 . . . 4 ((1r‘((1...(𝑁 − 1)) Mat 𝑅)) ∈ ((Base‘𝑅) ↑𝑚 ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
155153, 154syl 17 . . 3 (𝜑 → (1r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))
156 eqfnov2 6993 . . 3 (((𝐼(subMat1‘ 1 )𝐼) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1))) ∧ (1r‘((1...(𝑁 − 1)) Mat 𝑅)) Fn ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))) → ((𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗)))
157146, 155, 156syl2anc 575 . 2 (𝜑 → ((𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)) ↔ ∀𝑖 ∈ (1...(𝑁 − 1))∀𝑗 ∈ (1...(𝑁 − 1))(𝑖(𝐼(subMat1‘ 1 )𝐼)𝑗) = (𝑖(1r‘((1...(𝑁 − 1)) Mat 𝑅))𝑗)))
158143, 157mpbird 248 1 (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  ifcif 4279   class class class wbr 4844   × cxp 5309   Fn wfn 6092  cfv 6097  (class class class)co 6870  𝑚 cmap 8088  Fincfn 8188  cc 10215  cr 10216  1c1 10218   + caddc 10220   < clt 10355  cle 10356  cmin 10547  cn 11301  cz 11639  cuz 11900  ...cfz 12545  Basecbs 16064  0gc0g 16301  1rcur 18699  Ringcrg 18745   Mat cmat 20419  subMat1csmat 30180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-inf2 8781  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-ot 4379  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-of 7123  df-om 7292  df-1st 7394  df-2nd 7395  df-supp 7526  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-er 7975  df-map 8090  df-ixp 8142  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-fsupp 8511  df-sup 8583  df-oi 8650  df-card 9044  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-nn 11302  df-2 11360  df-3 11361  df-4 11362  df-5 11363  df-6 11364  df-7 11365  df-8 11366  df-9 11367  df-n0 11556  df-z 11640  df-dec 11756  df-uz 11901  df-fz 12546  df-fzo 12686  df-seq 13021  df-hash 13334  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16171  df-hom 16173  df-cco 16174  df-0g 16303  df-gsum 16304  df-prds 16309  df-pws 16311  df-mre 16447  df-mrc 16448  df-acs 16450  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17789  df-ghm 17856  df-cntz 17947  df-cmn 18392  df-abl 18393  df-mgp 18688  df-ur 18700  df-ring 18747  df-subrg 18978  df-lmod 19065  df-lss 19133  df-sra 19377  df-rgmod 19378  df-dsmm 20282  df-frlm 20297  df-mamu 20396  df-mat 20420  df-smat 30181
This theorem is referenced by: (None)
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