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Theorem madjusmdetlem1 30227
Description: Lemma for madjusmdet 30231. (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 20676 . . . 4 ((𝑀𝐵𝐽 ∈ (1...𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐽(𝐾𝑀)𝐼) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)))
91, 2, 3, 8syl3anc 1475 . . 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 6335 . . 3 (𝐷𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
129, 11syl6eqr 2822 . 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 12979 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
20 crngring 18765 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2118, 20syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
224, 5minmar1cl 20675 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2321, 1, 3, 2, 22syl22anc 1476 . . . 4 (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2410, 23syl5eqel 2853 . . 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 30225 . 2 (𝜑 → (𝐷𝑈) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)))
28 simplr 744 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
2928fveq2d 6336 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
30 madjusmdetlem1.1 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃𝑁) = 𝐼)
31303ad2ant1 1126 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑁) = 𝐼)
3231ad2antrr 697 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
3329, 32eqtrd 2804 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
34 simpr 471 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑗 = 𝑁)
3534fveq2d 6336 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = (𝑄𝑁))
36 madjusmdetlem1.2 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄𝑁) = 𝐽)
37363ad2ant1 1126 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑁) = 𝐽)
3837ad2antrr 697 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑁) = 𝐽)
3935, 38eqtrd 2804 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = 𝐽)
4033, 39oveq12d 6810 . . . . . . . . . . 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 1126 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑀𝐵)
4241ad2antrr 697 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑀𝐵)
4333ad2ant1 1126 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐼 ∈ (1...𝑁))
4443ad2antrr 697 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
4523ad2ant1 1126 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐽 ∈ (1...𝑁))
4645ad2antrr 697 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47 eqid 2770 . . . . . . . . . . . . 13 ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48 eqid 2770 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
49 eqid 2770 . . . . . . . . . . . . 13 (0g𝑅) = (0g𝑅)
504, 5, 47, 48, 49minmar1eval 20672 . . . . . . . . . . . 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 1484 . . . . . . . . . . 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 2770 . . . . . . . . . . . . . 14 𝐼 = 𝐼
5352iftruei 4230 . . . . . . . . . . . . 13 if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r𝑅), (0g𝑅))
54 eqid 2770 . . . . . . . . . . . . . 14 𝐽 = 𝐽
5554iftruei 4230 . . . . . . . . . . . . 13 if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)) = (1r𝑅)
5653, 55eqtri 2792 . . . . . . . . . . . 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 2809 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (1r𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
59 simplr 744 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
6059fveq2d 6336 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
6131ad2antrr 697 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
6260, 61eqtrd 2804 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
6362oveq1d 6807 . . . . . . . . . . 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 697 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀𝐵)
6543ad2antrr 697 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
6645ad2antrr 697 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
67263ad2ant1 1126 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄𝐺)
68 simp3 1131 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69 eqid 2770 . . . . . . . . . . . . . . 15 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
7069, 13symgfv 18013 . . . . . . . . . . . . . 14 ((𝑄𝐺𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7167, 68, 70syl2anc 565 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7271ad2antrr 697 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄𝑗) ∈ (1...𝑁))
734, 5, 47, 48, 49minmar1eval 20672 . . . . . . . . . . . 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 1484 . . . . . . . . . . 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 4231 . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝑗) = 𝐽)
7877fveq2d 6336 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = (𝑄𝐽))
7969, 13symgbasf1o 18009 . . . . . . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
8268ad2antrr 697 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83 f1ocnvfv1 6674 . . . . . . . . . . . . . . . . . 18 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑗)) = 𝑗)
8481, 82, 83syl2anc 565 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = 𝑗)
8536fveq2d 6336 . . . . . . . . . . . . . . . . . . . 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 11924 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
8987, 88syl6eleq 2859 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘1))
90 eluzfz2 12555 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
9189, 90syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (1...𝑁))
92 f1ocnvfv1 6674 . . . . . . . . . . . . . . . . . . . . 21 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑁)) = 𝑁)
9386, 91, 92syl2anc 565 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑄‘(𝑄𝑁)) = 𝑁)
9485, 93eqtr3d 2806 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑄𝐽) = 𝑁)
95943ad2ant1 1126 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝐽) = 𝑁)
9695ad2antrr 697 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝐽) = 𝑁)
9778, 84, 963eqtr3d 2812 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 = 𝑁)
9897ex 397 . . . . . . . . . . . . . . 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 393 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ¬ (𝑄𝑗) = 𝐽)
101100iffalsed 4234 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)) = (0g𝑅))
10276, 101eqtrd 2804 . . . . . . . . . . 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 2809 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (0g𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
10458, 103ifeqda 4258 . . . . . . . . 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 1130 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
106105adantr 466 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑖 ∈ (1...𝑁))
10768adantr 466 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑗 ∈ (1...𝑁))
108 ovexd 6824 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) ∈ V)
10910oveqi 6805 . . . . . . . . . . . . . 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 6866 . . . . . . . . . . . 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 2792 . . . . . . . . . . 11 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
113112ovmpt4g 6929 . . . . . . . . . 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 1475 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → (𝑖𝑊𝑗) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
115104, 114ifeqda 4258 . . . . . . . 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 6865 . . . . . . 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 2770 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
118253ad2ant1 1126 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃𝐺)
11969, 13symgfv 18013 . . . . . . . . . . . 12 ((𝑃𝐺𝑖 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
120118, 105, 119syl2anc 565 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
121243ad2ant1 1126 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
1224, 117, 5, 120, 71, 121matecld 20448 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑖)𝑈(𝑄𝑗)) ∈ (Base‘𝑅))
1234, 117, 5, 19, 18, 122matbas2d 20445 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗))) ∈ 𝐵)
12417, 123syl5eqel 2853 . . . . . . . 8 (𝜑𝑊𝐵)
125117, 48ringidcl 18775 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
12621, 125syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
127 eqid 2770 . . . . . . . . 9 ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
1284, 5, 127, 49marrepval 20585 . . . . . . . 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 1476 . . . . . . 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 2814 . . . . . 6 (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁) = 𝑊)
132131fveq2d 6336 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐷𝑊))
133 eqid 2770 . . . . . . . . . . . 12 ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
1344, 133, 5submaval 20604 . . . . . . . . . . 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 1475 . . . . . . . . . 10 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136 fzdif2 29885 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
13789, 136syl 17 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
138 mpt2eq12 6861 . . . . . . . . . . 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 565 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140135, 139eqtrd 2804 . . . . . . . . 9 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141 difssd 3887 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142137, 141eqsstr3d 3787 . . . . . . . . . 10 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
1434, 5submabas 20601 . . . . . . . . . 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 565 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145140, 144eqeltrd 2849 . . . . . . . 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 2770 . . . . . . . . 9 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148 eqid 2770 . . . . . . . . 9 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149146, 147, 148, 117mdetcl 20619 . . . . . . . 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 565 . . . . . . 7 (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151117, 16, 48ringlidm 18778 . . . . . . 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 565 . . . . . 6 (𝜑 → ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
1534fveq2i 6335 . . . . . . . . . . 11 (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
1545, 153eqtri 2792 . . . . . . . . . 10 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155124, 154syl6eleq 2859 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156 smadiadetr 20699 . . . . . . . . 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 1476 . . . . . . . 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 6333 . . . . . . . . 9 (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁))
15916oveqi 6805 . . . . . . . . 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 2784 . . . . . . . 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 224 . . . . . . 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 6807 . . . . . . . . . 10 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163162, 146syl6eqr 2822 . . . . . . . . 9 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164163fveq1d 6334 . . . . . . . 8 (𝜑 → ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
165164oveq2d 6808 . . . . . . 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 2804 . . . . . 6 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
1674, 5submat1n 30205 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
16887, 124, 167syl2anc 565 . . . . . . 7 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169168fveq2d 6336 . . . . . 6 (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170152, 166, 1693eqtr4d 2814 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171132, 170eqtr3d 2806 . . . 4 (𝜑 → (𝐷𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
1724, 5, 87, 3, 2, 21, 1, 10submatminr1 30210 . . . . . 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 2804 . . . . 5 (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175174fveq2d 6336 . . . 4 (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176171, 175eqtr4d 2807 . . 3 (𝜑 → (𝐷𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177176oveq2d 6808 . 2 (𝜑 → ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
17812, 27, 1773eqtrd 2808 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 382  w3a 1070   = wceq 1630  wcel 2144  Vcvv 3349  cdif 3718  wss 3721  ifcif 4223  {csn 4314  ccnv 5248  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  cmpt2 6794  1c1 10138   · cmul 10142  cmin 10467  cn 11221  cuz 11887  ...cfz 12532  Basecbs 16063  .rcmulr 16149  0gc0g 16307  SymGrpcsymg 18003  pmSgncpsgn 18115  1rcur 18708  Ringcrg 18754  CRingccrg 18755  ℤRHomczrh 20062   Mat cmat 20429   matRRep cmarrep 20579   subMat csubma 20599   maDet cmdat 20607   maAdju cmadu 20655   minMatR1 cminmar1 20656  subMat1csmat 30193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-addf 10216  ax-mulf 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-xor 1612  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-ot 4323  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-tpos 7503  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-sup 8503  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-xnn0 11565  df-z 11579  df-dec 11695  df-uz 11888  df-rp 12035  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-word 13494  df-lsw 13495  df-concat 13496  df-s1 13497  df-substr 13498  df-splice 13499  df-reverse 13500  df-s2 13801  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-starv 16163  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-unif 16172  df-hom 16173  df-cco 16174  df-0g 16309  df-gsum 16310  df-prds 16315  df-pws 16317  df-mre 16453  df-mrc 16454  df-acs 16456  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-submnd 17543  df-grp 17632  df-minusg 17633  df-mulg 17748  df-subg 17798  df-ghm 17865  df-gim 17908  df-cntz 17956  df-oppg 17982  df-symg 18004  df-pmtr 18068  df-psgn 18117  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-cring 18757  df-oppr 18830  df-dvdsr 18848  df-unit 18849  df-invr 18879  df-dvr 18890  df-rnghom 18924  df-drng 18958  df-subrg 18987  df-sra 19386  df-rgmod 19387  df-cnfld 19961  df-zring 20033  df-zrh 20066  df-dsmm 20292  df-frlm 20307  df-mat 20430  df-marrep 20581  df-subma 20600  df-mdet 20608  df-madu 20657  df-minmar1 20658  df-smat 30194
This theorem is referenced by:  madjusmdetlem4  30230
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