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Theorem madjusmdetlem1 31094
Description: Lemma for madjusmdet 31098. (Contributed by Thierry Arnoux, 22-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)
madjusmdetlem1.g 𝐺 = (Base‘(SymGrp‘(1...𝑁)))
madjusmdetlem1.s 𝑆 = (pmSgn‘(1...𝑁))
madjusmdetlem1.u 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
madjusmdetlem1.w 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗)))
madjusmdetlem1.p (𝜑𝑃𝐺)
madjusmdetlem1.q (𝜑𝑄𝐺)
madjusmdetlem1.1 (𝜑 → (𝑃𝑁) = 𝐼)
madjusmdetlem1.2 (𝜑 → (𝑄𝑁) = 𝐽)
madjusmdetlem1.3 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Assertion
Ref Expression
madjusmdetlem1 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
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   𝑖,𝐺,𝑗   𝑖,𝑊,𝑗   𝑈,𝑖,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐷(𝑖,𝑗)   𝑆(𝑖,𝑗)   · (𝑖,𝑗)   𝐸(𝑖,𝑗)   𝐾(𝑖,𝑗)   𝑍(𝑖,𝑗)

Proof of Theorem madjusmdetlem1
StepHypRef Expression
1 madjusmdet.m . . . 4 (𝜑𝑀𝐵)
2 madjusmdet.j . . . 4 (𝜑𝐽 ∈ (1...𝑁))
3 madjusmdet.i . . . 4 (𝜑𝐼 ∈ (1...𝑁))
4 madjusmdet.a . . . . 5 𝐴 = ((1...𝑁) Mat 𝑅)
5 madjusmdet.b . . . . 5 𝐵 = (Base‘𝐴)
6 madjusmdet.d . . . . 5 𝐷 = ((1...𝑁) maDet 𝑅)
7 madjusmdet.k . . . . 5 𝐾 = ((1...𝑁) maAdju 𝑅)
84, 5, 6, 7maducoevalmin1 21263 . . . 4 ((𝑀𝐵𝐽 ∈ (1...𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐽(𝐾𝑀)𝐼) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)))
91, 2, 3, 8syl3anc 1367 . . 3 (𝜑 → (𝐽(𝐾𝑀)𝐼) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)))
10 madjusmdetlem1.u . . . 4 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)
1110fveq2i 6675 . . 3 (𝐷𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
129, 11syl6eqr 2876 . 2 (𝜑 → (𝐽(𝐾𝑀)𝐼) = (𝐷𝑈))
13 madjusmdetlem1.g . . 3 𝐺 = (Base‘(SymGrp‘(1...𝑁)))
14 madjusmdetlem1.s . . 3 𝑆 = (pmSgn‘(1...𝑁))
15 madjusmdet.z . . 3 𝑍 = (ℤRHom‘𝑅)
16 madjusmdet.t . . 3 · = (.r𝑅)
17 madjusmdetlem1.w . . 3 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗)))
18 madjusmdet.r . . 3 (𝜑𝑅 ∈ CRing)
19 fzfid 13344 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
20 crngring 19310 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2118, 20syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
224, 5minmar1cl 21262 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2321, 1, 3, 2, 22syl22anc 836 . . . 4 (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2410, 23eqeltrid 2919 . . 3 (𝜑𝑈𝐵)
25 madjusmdetlem1.p . . 3 (𝜑𝑃𝐺)
26 madjusmdetlem1.q . . 3 (𝜑𝑄𝐺)
274, 5, 6, 13, 14, 15, 16, 17, 18, 19, 24, 25, 26mdetpmtr12 31092 . 2 (𝜑 → (𝐷𝑈) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)))
28 simplr 767 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
2928fveq2d 6676 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
30 madjusmdetlem1.1 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃𝑁) = 𝐼)
31303ad2ant1 1129 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑁) = 𝐼)
3231ad2antrr 724 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
3329, 32eqtrd 2858 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
34 simpr 487 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑗 = 𝑁)
3534fveq2d 6676 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = (𝑄𝑁))
36 madjusmdetlem1.2 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄𝑁) = 𝐽)
37363ad2ant1 1129 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑁) = 𝐽)
3837ad2antrr 724 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑁) = 𝐽)
3935, 38eqtrd 2858 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = 𝐽)
4033, 39oveq12d 7176 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) = (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)𝐽))
4113ad2ant1 1129 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑀𝐵)
4241ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑀𝐵)
4333ad2ant1 1129 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐼 ∈ (1...𝑁))
4443ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
4523ad2ant1 1129 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐽 ∈ (1...𝑁))
4645ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47 eqid 2823 . . . . . . . . . . . . 13 ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48 eqid 2823 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
49 eqid 2823 . . . . . . . . . . . . 13 (0g𝑅) = (0g𝑅)
504, 5, 47, 48, 49minmar1eval 21260 . . . . . . . . . . . 12 ((𝑀𝐵 ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁)) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)𝐽) = if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)))
5142, 44, 46, 44, 46, 50syl122anc 1375 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)𝐽) = if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)))
52 eqid 2823 . . . . . . . . . . . . . 14 𝐼 = 𝐼
5352iftruei 4476 . . . . . . . . . . . . 13 if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r𝑅), (0g𝑅))
54 eqid 2823 . . . . . . . . . . . . . 14 𝐽 = 𝐽
5554iftruei 4476 . . . . . . . . . . . . 13 if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)) = (1r𝑅)
5653, 55eqtri 2846 . . . . . . . . . . . 12 if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)) = (1r𝑅)
5756a1i 11 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)) = (1r𝑅))
5840, 51, 573eqtrrd 2863 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (1r𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
59 simplr 767 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
6059fveq2d 6676 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
6131ad2antrr 724 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
6260, 61eqtrd 2858 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
6362oveq1d 7173 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) = (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
6441ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀𝐵)
6543ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
6645ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
67263ad2ant1 1129 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄𝐺)
68 simp3 1134 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69 eqid 2823 . . . . . . . . . . . . . . 15 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
7069, 13symgfv 18510 . . . . . . . . . . . . . 14 ((𝑄𝐺𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7167, 68, 70syl2anc 586 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7271ad2antrr 724 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄𝑗) ∈ (1...𝑁))
734, 5, 47, 48, 49minmar1eval 21260 . . . . . . . . . . . 12 ((𝑀𝐵 ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁)) ∧ (𝐼 ∈ (1...𝑁) ∧ (𝑄𝑗) ∈ (1...𝑁))) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) = if(𝐼 = 𝐼, if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀(𝑄𝑗))))
7464, 65, 66, 65, 72, 73syl122anc 1375 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝐼(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) = if(𝐼 = 𝐼, if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀(𝑄𝑗))))
7552a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 = 𝐼)
7675iftrued 4477 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀(𝑄𝑗))) = if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)))
77 simpr 487 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝑗) = 𝐽)
7877fveq2d 6676 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = (𝑄𝐽))
7969, 13symgbasf1o 18505 . . . . . . . . . . . . . . . . . . . 20 (𝑄𝐺𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
8067, 79syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
8180ad2antrr 724 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
8268ad2antrr 724 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83 f1ocnvfv1 7035 . . . . . . . . . . . . . . . . . 18 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑗)) = 𝑗)
8481, 82, 83syl2anc 586 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = 𝑗)
8536fveq2d 6676 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑄‘(𝑄𝑁)) = (𝑄𝐽))
8626, 79syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
87 madjusmdet.n . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℕ)
88 nnuz 12284 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
8987, 88eleqtrdi 2925 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘1))
90 eluzfz2 12918 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
9189, 90syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (1...𝑁))
92 f1ocnvfv1 7035 . . . . . . . . . . . . . . . . . . . . 21 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑁)) = 𝑁)
9386, 91, 92syl2anc 586 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑄‘(𝑄𝑁)) = 𝑁)
9485, 93eqtr3d 2860 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑄𝐽) = 𝑁)
95943ad2ant1 1129 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝐽) = 𝑁)
9695ad2antrr 724 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝐽) = 𝑁)
9778, 84, 963eqtr3d 2866 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 = 𝑁)
9897ex 415 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → ((𝑄𝑗) = 𝐽𝑗 = 𝑁))
9998con3d 155 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → (¬ 𝑗 = 𝑁 → ¬ (𝑄𝑗) = 𝐽))
10099imp 409 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ¬ (𝑄𝑗) = 𝐽)
101100iffalsed 4480 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)) = (0g𝑅))
10276, 101eqtrd 2858 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if(𝐼 = 𝐼, if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀(𝑄𝑗))) = (0g𝑅))
10363, 74, 1023eqtrrd 2863 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (0g𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
10458, 103ifeqda 4504 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → if(𝑗 = 𝑁, (1r𝑅), (0g𝑅)) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
105 simp2 1133 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
106105adantr 483 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑖 ∈ (1...𝑁))
10768adantr 483 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑗 ∈ (1...𝑁))
108 ovexd 7193 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) ∈ V)
10910oveqi 7171 . . . . . . . . . . . . . 14 ((𝑃𝑖)𝑈(𝑄𝑗)) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗))
110109a1i 11 . . . . . . . . . . . . 13 ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑖)𝑈(𝑄𝑗)) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
111110mpoeq3ia 7234 . . . . . . . . . . . 12 (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗))) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
11217, 111eqtri 2846 . . . . . . . . . . 11 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
113112ovmpt4g 7299 . . . . . . . . . 10 ((𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁) ∧ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) ∈ V) → (𝑖𝑊𝑗) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
114106, 107, 108, 113syl3anc 1367 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → (𝑖𝑊𝑗) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
115104, 114ifeqda 4504 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r𝑅), (0g𝑅)), (𝑖𝑊𝑗)) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
116115mpoeq3dva 7233 . . . . . . 7 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r𝑅), (0g𝑅)), (𝑖𝑊𝑗))) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗))))
117 eqid 2823 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
118253ad2ant1 1129 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃𝐺)
11969, 13symgfv 18510 . . . . . . . . . . . 12 ((𝑃𝐺𝑖 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
120118, 105, 119syl2anc 586 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
121243ad2ant1 1129 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
1224, 117, 5, 120, 71, 121matecld 21037 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑖)𝑈(𝑄𝑗)) ∈ (Base‘𝑅))
1234, 117, 5, 19, 18, 122matbas2d 21034 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗))) ∈ 𝐵)
12417, 123eqeltrid 2919 . . . . . . . 8 (𝜑𝑊𝐵)
125117, 48ringidcl 19320 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
12621, 125syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
127 eqid 2823 . . . . . . . . 9 ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
1284, 5, 127, 49marrepval 21173 . . . . . . . 8 (((𝑊𝐵 ∧ (1r𝑅) ∈ (Base‘𝑅)) ∧ (𝑁 ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r𝑅), (0g𝑅)), (𝑖𝑊𝑗))))
129124, 126, 91, 91, 128syl22anc 836 . . . . . . 7 (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁) = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ if(𝑖 = 𝑁, if(𝑗 = 𝑁, (1r𝑅), (0g𝑅)), (𝑖𝑊𝑗))))
130112a1i 11 . . . . . . 7 (𝜑𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗))))
131116, 129, 1303eqtr4d 2868 . . . . . 6 (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁) = 𝑊)
132131fveq2d 6676 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐷𝑊))
133 eqid 2823 . . . . . . . . . . . 12 ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
1344, 133, 5submaval 21192 . . . . . . . . . . 11 ((𝑊𝐵𝑁 ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
135124, 91, 91, 134syl3anc 1367 . . . . . . . . . 10 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136 fzdif2 30516 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
13789, 136syl 17 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
138 mpoeq12 7229 . . . . . . . . . . 11 ((((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)) ∧ ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1))) → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
139137, 137, 138syl2anc 586 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140135, 139eqtrd 2858 . . . . . . . . 9 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141 difssd 4111 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142137, 141eqsstrrd 4008 . . . . . . . . . 10 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
1434, 5submabas 21189 . . . . . . . . . 10 ((𝑊𝐵 ∧ (1...(𝑁 − 1)) ⊆ (1...𝑁)) → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
144124, 142, 143syl2anc 586 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145140, 144eqeltrd 2915 . . . . . . . 8 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
146 madjusmdet.e . . . . . . . . 9 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)
147 eqid 2823 . . . . . . . . 9 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148 eqid 2823 . . . . . . . . 9 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149146, 147, 148, 117mdetcl 21207 . . . . . . . 8 ((𝑅 ∈ CRing ∧ (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅))) → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
15018, 145, 149syl2anc 586 . . . . . . 7 (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151117, 16, 48ringlidm 19323 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅)) → ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
15221, 150, 151syl2anc 586 . . . . . 6 (𝜑 → ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
1534fveq2i 6675 . . . . . . . . . . 11 (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
1545, 153eqtri 2846 . . . . . . . . . 10 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155124, 154eleqtrdi 2925 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156 smadiadetr 21286 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅))) ∧ (𝑁 ∈ (1...𝑁) ∧ (1r𝑅) ∈ (Base‘𝑅))) → (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅)(.r𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
15718, 155, 91, 126, 156syl22anc 836 . . . . . . . 8 (𝜑 → (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅)(.r𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
1586fveq1i 6673 . . . . . . . . 9 (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁))
15916oveqi 7171 . . . . . . . . 9 ((1r𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = ((1r𝑅)(.r𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
160158, 159eqeq12i 2838 . . . . . . . 8 ((𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) ↔ (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅)(.r𝑅)((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
161157, 160sylibr 236 . . . . . . 7 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
162137oveq1d 7173 . . . . . . . . . 10 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163162, 146syl6eqr 2876 . . . . . . . . 9 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164163fveq1d 6674 . . . . . . . 8 (𝜑 → ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
165164oveq2d 7174 . . . . . . 7 (𝜑 → ((1r𝑅) · ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
166161, 165eqtrd 2858 . . . . . 6 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
1674, 5submat1n 31072 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
16887, 124, 167syl2anc 586 . . . . . . 7 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169168fveq2d 6676 . . . . . 6 (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170152, 166, 1693eqtr4d 2868 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171132, 170eqtr3d 2860 . . . 4 (𝜑 → (𝐷𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
1724, 5, 87, 3, 2, 21, 1, 10submatminr1 31077 . . . . . 6 (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝑈)𝐽))
173 madjusmdetlem1.3 . . . . . 6 (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
174172, 173eqtrd 2858 . . . . 5 (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175174fveq2d 6676 . . . 4 (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176171, 175eqtr4d 2861 . . 3 (𝜑 → (𝐷𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177176oveq2d 7174 . 2 (𝜑 → ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
17812, 27, 1773eqtrd 2862 1 (𝜑 → (𝐽(𝐾𝑀)𝐼) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  w3a 1083   = wceq 1537  wcel 2114  Vcvv 3496  cdif 3935  wss 3938  ifcif 4469  {csn 4569  ccnv 5556  1-1-ontowf1o 6356  cfv 6357  (class class class)co 7158  cmpo 7160  1c1 10540   · cmul 10544  cmin 10872  cn 11640  cuz 12246  ...cfz 12895  Basecbs 16485  .rcmulr 16568  0gc0g 16715  SymGrpcsymg 18497  pmSgncpsgn 18619  1rcur 19253  Ringcrg 19299  CRingccrg 19300  ℤRHomczrh 20649   Mat cmat 21018   matRRep cmarrep 21167   subMat csubma 21187   maDet cmdat 21195   maAdju cmadu 21243   minMatR1 cminmar1 21244  subMat1csmat 31060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-addf 10618  ax-mulf 10619
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-xor 1502  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-ot 4578  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-tpos 7894  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-dec 12102  df-uz 12247  df-rp 12393  df-fz 12896  df-fzo 13037  df-seq 13373  df-exp 13433  df-hash 13694  df-word 13865  df-lsw 13917  df-concat 13925  df-s1 13952  df-substr 14005  df-pfx 14035  df-splice 14114  df-reverse 14123  df-s2 14212  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-hom 16591  df-cco 16592  df-0g 16717  df-gsum 16718  df-prds 16723  df-pws 16725  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-efmnd 18036  df-grp 18108  df-minusg 18109  df-mulg 18227  df-subg 18278  df-ghm 18358  df-gim 18401  df-cntz 18449  df-oppg 18476  df-symg 18498  df-pmtr 18572  df-psgn 18621  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-ring 19301  df-cring 19302  df-oppr 19375  df-dvdsr 19393  df-unit 19394  df-invr 19424  df-dvr 19435  df-rnghom 19469  df-drng 19506  df-subrg 19535  df-sra 19946  df-rgmod 19947  df-cnfld 20548  df-zring 20620  df-zrh 20653  df-dsmm 20878  df-frlm 20893  df-mat 21019  df-marrep 21169  df-subma 21188  df-mdet 21196  df-madu 21245  df-minmar1 21246  df-smat 31061
This theorem is referenced by:  madjusmdetlem4  31097
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