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Theorem madjusmdetlem1 30340
Description: Lemma for madjusmdet 30344. (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 20736 . . . 4 ((𝑀𝐵𝐽 ∈ (1...𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐽(𝐾𝑀)𝐼) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)))
91, 2, 3, 8syl3anc 1490 . . 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 6378 . . 3 (𝐷𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
129, 11syl6eqr 2817 . 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 12980 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
20 crngring 18825 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2118, 20syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
224, 5minmar1cl 20735 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2321, 1, 3, 2, 22syl22anc 867 . . . 4 (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2410, 23syl5eqel 2848 . . 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 30338 . 2 (𝜑 → (𝐷𝑈) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)))
28 simplr 785 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
2928fveq2d 6379 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
30 madjusmdetlem1.1 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃𝑁) = 𝐼)
31303ad2ant1 1163 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑁) = 𝐼)
3231ad2antrr 717 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
3329, 32eqtrd 2799 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
34 simpr 477 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑗 = 𝑁)
3534fveq2d 6379 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = (𝑄𝑁))
36 madjusmdetlem1.2 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄𝑁) = 𝐽)
37363ad2ant1 1163 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑁) = 𝐽)
3837ad2antrr 717 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑁) = 𝐽)
3935, 38eqtrd 2799 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = 𝐽)
4033, 39oveq12d 6860 . . . . . . . . . . 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 1163 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑀𝐵)
4241ad2antrr 717 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑀𝐵)
4333ad2ant1 1163 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐼 ∈ (1...𝑁))
4443ad2antrr 717 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
4523ad2ant1 1163 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐽 ∈ (1...𝑁))
4645ad2antrr 717 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47 eqid 2765 . . . . . . . . . . . . 13 ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48 eqid 2765 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
49 eqid 2765 . . . . . . . . . . . . 13 (0g𝑅) = (0g𝑅)
504, 5, 47, 48, 49minmar1eval 20732 . . . . . . . . . . . 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 1498 . . . . . . . . . . 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 2765 . . . . . . . . . . . . . 14 𝐼 = 𝐼
5352iftruei 4250 . . . . . . . . . . . . 13 if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r𝑅), (0g𝑅))
54 eqid 2765 . . . . . . . . . . . . . 14 𝐽 = 𝐽
5554iftruei 4250 . . . . . . . . . . . . 13 if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)) = (1r𝑅)
5653, 55eqtri 2787 . . . . . . . . . . . 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 2804 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (1r𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
59 simplr 785 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
6059fveq2d 6379 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
6131ad2antrr 717 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
6260, 61eqtrd 2799 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
6362oveq1d 6857 . . . . . . . . . . 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 717 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀𝐵)
6543ad2antrr 717 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
6645ad2antrr 717 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
67263ad2ant1 1163 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄𝐺)
68 simp3 1168 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69 eqid 2765 . . . . . . . . . . . . . . 15 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
7069, 13symgfv 18070 . . . . . . . . . . . . . 14 ((𝑄𝐺𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7167, 68, 70syl2anc 579 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7271ad2antrr 717 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄𝑗) ∈ (1...𝑁))
734, 5, 47, 48, 49minmar1eval 20732 . . . . . . . . . . . 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 1498 . . . . . . . . . . 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 4251 . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝑗) = 𝐽)
7877fveq2d 6379 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = (𝑄𝐽))
7969, 13symgbasf1o 18066 . . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
8268ad2antrr 717 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83 f1ocnvfv1 6724 . . . . . . . . . . . . . . . . . 18 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑗)) = 𝑗)
8481, 82, 83syl2anc 579 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = 𝑗)
8536fveq2d 6379 . . . . . . . . . . . . . . . . . . . 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 11923 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
8987, 88syl6eleq 2854 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘1))
90 eluzfz2 12556 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
9189, 90syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (1...𝑁))
92 f1ocnvfv1 6724 . . . . . . . . . . . . . . . . . . . . 21 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑁)) = 𝑁)
9386, 91, 92syl2anc 579 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑄‘(𝑄𝑁)) = 𝑁)
9485, 93eqtr3d 2801 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑄𝐽) = 𝑁)
95943ad2ant1 1163 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝐽) = 𝑁)
9695ad2antrr 717 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝐽) = 𝑁)
9778, 84, 963eqtr3d 2807 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 = 𝑁)
9897ex 401 . . . . . . . . . . . . . . 15 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → ((𝑄𝑗) = 𝐽𝑗 = 𝑁))
9998con3d 149 . . . . . . . . . . . . . 14 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) → (¬ 𝑗 = 𝑁 → ¬ (𝑄𝑗) = 𝐽))
10099imp 395 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ¬ (𝑄𝑗) = 𝐽)
101100iffalsed 4254 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)) = (0g𝑅))
10276, 101eqtrd 2799 . . . . . . . . . . 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 2804 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (0g𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
10458, 103ifeqda 4278 . . . . . . . . 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 1167 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
106105adantr 472 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑖 ∈ (1...𝑁))
10768adantr 472 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑗 ∈ (1...𝑁))
108 ovexd 6876 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) ∈ V)
10910oveqi 6855 . . . . . . . . . . . . . 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)))
111110mpt2eq3ia 6918 . . . . . . . . . . . 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 2787 . . . . . . . . . . 11 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
113112ovmpt4g 6981 . . . . . . . . . 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 1490 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → (𝑖𝑊𝑗) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
115104, 114ifeqda 4278 . . . . . . . 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)))
116115mpt2eq3dva 6917 . . . . . . 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 2765 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
118253ad2ant1 1163 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃𝐺)
11969, 13symgfv 18070 . . . . . . . . . . . 12 ((𝑃𝐺𝑖 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
120118, 105, 119syl2anc 579 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
121243ad2ant1 1163 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
1224, 117, 5, 120, 71, 121matecld 20508 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑖)𝑈(𝑄𝑗)) ∈ (Base‘𝑅))
1234, 117, 5, 19, 18, 122matbas2d 20505 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗))) ∈ 𝐵)
12417, 123syl5eqel 2848 . . . . . . . 8 (𝜑𝑊𝐵)
125117, 48ringidcl 18835 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
12621, 125syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
127 eqid 2765 . . . . . . . . 9 ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
1284, 5, 127, 49marrepval 20645 . . . . . . . 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 867 . . . . . . 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 2809 . . . . . 6 (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁) = 𝑊)
132131fveq2d 6379 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐷𝑊))
133 eqid 2765 . . . . . . . . . . . 12 ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
1344, 133, 5submaval 20664 . . . . . . . . . . 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 1490 . . . . . . . . . 10 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136 fzdif2 30000 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
13789, 136syl 17 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
138 mpt2eq12 6913 . . . . . . . . . . 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 579 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140135, 139eqtrd 2799 . . . . . . . . 9 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141 difssd 3900 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142137, 141eqsstr3d 3800 . . . . . . . . . 10 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
1434, 5submabas 20661 . . . . . . . . . 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 579 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145140, 144eqeltrd 2844 . . . . . . . 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 2765 . . . . . . . . 9 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148 eqid 2765 . . . . . . . . 9 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149146, 147, 148, 117mdetcl 20679 . . . . . . . 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 579 . . . . . . 7 (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151117, 16, 48ringlidm 18838 . . . . . . 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 579 . . . . . 6 (𝜑 → ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
1534fveq2i 6378 . . . . . . . . . . 11 (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
1545, 153eqtri 2787 . . . . . . . . . 10 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155124, 154syl6eleq 2854 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156 smadiadetr 20759 . . . . . . . . 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 867 . . . . . . . 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 6376 . . . . . . . . 9 (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁))
15916oveqi 6855 . . . . . . . . 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 2779 . . . . . . . 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 225 . . . . . . 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 6857 . . . . . . . . . 10 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163162, 146syl6eqr 2817 . . . . . . . . 9 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164163fveq1d 6377 . . . . . . . 8 (𝜑 → ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
165164oveq2d 6858 . . . . . . 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 2799 . . . . . 6 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
1674, 5submat1n 30318 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
16887, 124, 167syl2anc 579 . . . . . . 7 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169168fveq2d 6379 . . . . . 6 (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170152, 166, 1693eqtr4d 2809 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171132, 170eqtr3d 2801 . . . 4 (𝜑 → (𝐷𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
1724, 5, 87, 3, 2, 21, 1, 10submatminr1 30323 . . . . . 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 2799 . . . . 5 (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175174fveq2d 6379 . . . 4 (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176171, 175eqtr4d 2802 . . 3 (𝜑 → (𝐷𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177176oveq2d 6858 . 2 (𝜑 → ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
17812, 27, 1773eqtrd 2803 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 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cdif 3729  wss 3732  ifcif 4243  {csn 4334  ccnv 5276  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cmpt2 6844  1c1 10190   · cmul 10194  cmin 10520  cn 11274  cuz 11886  ...cfz 12533  Basecbs 16130  .rcmulr 16215  0gc0g 16366  SymGrpcsymg 18060  pmSgncpsgn 18172  1rcur 18768  Ringcrg 18814  CRingccrg 18815  ℤRHomczrh 20121   Mat cmat 20489   matRRep cmarrep 20639   subMat csubma 20659   maDet cmdat 20667   maAdju cmadu 20715   minMatR1 cminmar1 20716  subMat1csmat 30306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-xor 1634  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-dec 11741  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-word 13487  df-lsw 13534  df-concat 13542  df-s1 13567  df-substr 13617  df-pfx 13662  df-splice 13765  df-reverse 13783  df-s2 13877  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-starv 16229  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-unif 16237  df-hom 16238  df-cco 16239  df-0g 16368  df-gsum 16369  df-prds 16374  df-pws 16376  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-mulg 17808  df-subg 17855  df-ghm 17922  df-gim 17965  df-cntz 18013  df-oppg 18039  df-symg 18061  df-pmtr 18125  df-psgn 18174  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-cring 18817  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-dvr 18950  df-rnghom 18984  df-drng 19018  df-subrg 19047  df-sra 19446  df-rgmod 19447  df-cnfld 20020  df-zring 20092  df-zrh 20125  df-dsmm 20352  df-frlm 20367  df-mat 20490  df-marrep 20641  df-subma 20660  df-mdet 20668  df-madu 20717  df-minmar1 20718  df-smat 30307
This theorem is referenced by:  madjusmdetlem4  30343
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