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Theorem madjusmdetlem1 30991
Description: Lemma for madjusmdet 30995. (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 21189 . . . 4 ((𝑀𝐵𝐽 ∈ (1...𝑁) ∧ 𝐼 ∈ (1...𝑁)) → (𝐽(𝐾𝑀)𝐼) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)))
91, 2, 3, 8syl3anc 1363 . . 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 6666 . . 3 (𝐷𝑈) = (𝐷‘(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽))
129, 11syl6eqr 2871 . 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 13329 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
20 crngring 19237 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2118, 20syl 17 . . . . 5 (𝜑𝑅 ∈ Ring)
224, 5minmar1cl 21188 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼 ∈ (1...𝑁) ∧ 𝐽 ∈ (1...𝑁))) → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2321, 1, 3, 2, 22syl22anc 834 . . . 4 (𝜑 → (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ∈ 𝐵)
2410, 23eqeltrid 2914 . . 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 30989 . 2 (𝜑 → (𝐷𝑈) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)))
28 simplr 765 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑖 = 𝑁)
2928fveq2d 6667 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
30 madjusmdetlem1.1 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃𝑁) = 𝐼)
31303ad2ant1 1125 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑁) = 𝐼)
3231ad2antrr 722 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
3329, 32eqtrd 2853 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
34 simpr 485 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑗 = 𝑁)
3534fveq2d 6667 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = (𝑄𝑁))
36 madjusmdetlem1.2 . . . . . . . . . . . . . . 15 (𝜑 → (𝑄𝑁) = 𝐽)
37363ad2ant1 1125 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑁) = 𝐽)
3837ad2antrr 722 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑁) = 𝐽)
3935, 38eqtrd 2853 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (𝑄𝑗) = 𝐽)
4033, 39oveq12d 7163 . . . . . . . . . . 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 1125 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑀𝐵)
4241ad2antrr 722 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝑀𝐵)
4333ad2ant1 1125 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐼 ∈ (1...𝑁))
4443ad2antrr 722 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
4523ad2ant1 1125 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝐽 ∈ (1...𝑁))
4645ad2antrr 722 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
47 eqid 2818 . . . . . . . . . . . . 13 ((1...𝑁) minMatR1 𝑅) = ((1...𝑁) minMatR1 𝑅)
48 eqid 2818 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
49 eqid 2818 . . . . . . . . . . . . 13 (0g𝑅) = (0g𝑅)
504, 5, 47, 48, 49minmar1eval 21186 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 2818 . . . . . . . . . . . . . 14 𝐼 = 𝐼
5352iftruei 4470 . . . . . . . . . . . . 13 if(𝐼 = 𝐼, if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)), (𝐼𝑀𝐽)) = if(𝐽 = 𝐽, (1r𝑅), (0g𝑅))
54 eqid 2818 . . . . . . . . . . . . . 14 𝐽 = 𝐽
5554iftruei 4470 . . . . . . . . . . . . 13 if(𝐽 = 𝐽, (1r𝑅), (0g𝑅)) = (1r𝑅)
5653, 55eqtri 2841 . . . . . . . . . . . 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 2858 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ 𝑗 = 𝑁) → (1r𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
59 simplr 765 . . . . . . . . . . . . . 14 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑖 = 𝑁)
6059fveq2d 6667 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = (𝑃𝑁))
6131ad2antrr 722 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑁) = 𝐼)
6260, 61eqtrd 2853 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑃𝑖) = 𝐼)
6362oveq1d 7160 . . . . . . . . . . 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 722 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝑀𝐵)
6543ad2antrr 722 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐼 ∈ (1...𝑁))
6645ad2antrr 722 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → 𝐽 ∈ (1...𝑁))
67263ad2ant1 1125 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑄𝐺)
68 simp3 1130 . . . . . . . . . . . . . 14 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑗 ∈ (1...𝑁))
69 eqid 2818 . . . . . . . . . . . . . . 15 (SymGrp‘(1...𝑁)) = (SymGrp‘(1...𝑁))
7069, 13symgfv 18443 . . . . . . . . . . . . . 14 ((𝑄𝐺𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7167, 68, 70syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝑗) ∈ (1...𝑁))
7271ad2antrr 722 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (𝑄𝑗) ∈ (1...𝑁))
734, 5, 47, 48, 49minmar1eval 21186 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 4471 . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝑗) = 𝐽)
7877fveq2d 6667 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = (𝑄𝐽))
7969, 13symgbasf1o 18439 . . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑄:(1...𝑁)–1-1-onto→(1...𝑁))
8268ad2antrr 722 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 ∈ (1...𝑁))
83 f1ocnvfv1 7024 . . . . . . . . . . . . . . . . . 18 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑗)) = 𝑗)
8481, 82, 83syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄‘(𝑄𝑗)) = 𝑗)
8536fveq2d 6667 . . . . . . . . . . . . . . . . . . . 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 12269 . . . . . . . . . . . . . . . . . . . . . . 23 ℕ = (ℤ‘1)
8987, 88eleqtrdi 2920 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑁 ∈ (ℤ‘1))
90 eluzfz2 12903 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘1) → 𝑁 ∈ (1...𝑁))
9189, 90syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (1...𝑁))
92 f1ocnvfv1 7024 . . . . . . . . . . . . . . . . . . . . 21 ((𝑄:(1...𝑁)–1-1-onto→(1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → (𝑄‘(𝑄𝑁)) = 𝑁)
9386, 91, 92syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑄‘(𝑄𝑁)) = 𝑁)
9485, 93eqtr3d 2855 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑄𝐽) = 𝑁)
95943ad2ant1 1125 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑄𝐽) = 𝑁)
9695ad2antrr 722 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → (𝑄𝐽) = 𝑁)
9778, 84, 963eqtr3d 2861 . . . . . . . . . . . . . . . 16 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ (𝑄𝑗) = 𝐽) → 𝑗 = 𝑁)
9897ex 413 . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → ¬ (𝑄𝑗) = 𝐽)
101100iffalsed 4474 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → if((𝑄𝑗) = 𝐽, (1r𝑅), (0g𝑅)) = (0g𝑅))
10276, 101eqtrd 2853 . . . . . . . . . . 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 2858 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ 𝑖 = 𝑁) ∧ ¬ 𝑗 = 𝑁) → (0g𝑅) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
10458, 103ifeqda 4498 . . . . . . . . 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 1129 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁))
106105adantr 481 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑖 ∈ (1...𝑁))
10768adantr 481 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → 𝑗 ∈ (1...𝑁))
108 ovexd 7180 . . . . . . . . . 10 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)) ∈ V)
10910oveqi 7158 . . . . . . . . . . . . . 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 7221 . . . . . . . . . . . 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 2841 . . . . . . . . . . 11 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
113112ovmpt4g 7286 . . . . . . . . . 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 1363 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) ∧ ¬ 𝑖 = 𝑁) → (𝑖𝑊𝑗) = ((𝑃𝑖)(𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)(𝑄𝑗)))
115104, 114ifeqda 4498 . . . . . . . 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 7220 . . . . . . 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 2818 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
118253ad2ant1 1125 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑃𝐺)
11969, 13symgfv 18443 . . . . . . . . . . . 12 ((𝑃𝐺𝑖 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
120118, 105, 119syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → (𝑃𝑖) ∈ (1...𝑁))
121243ad2ant1 1125 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → 𝑈𝐵)
1224, 117, 5, 120, 71, 121matecld 20963 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑁) ∧ 𝑗 ∈ (1...𝑁)) → ((𝑃𝑖)𝑈(𝑄𝑗)) ∈ (Base‘𝑅))
1234, 117, 5, 19, 18, 122matbas2d 20960 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃𝑖)𝑈(𝑄𝑗))) ∈ 𝐵)
12417, 123eqeltrid 2914 . . . . . . . 8 (𝜑𝑊𝐵)
125117, 48ringidcl 19247 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
12621, 125syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
127 eqid 2818 . . . . . . . . 9 ((1...𝑁) matRRep 𝑅) = ((1...𝑁) matRRep 𝑅)
1284, 5, 127, 49marrepval 21099 . . . . . . . 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 834 . . . . . . 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 2863 . . . . . 6 (𝜑 → (𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁) = 𝑊)
132131fveq2d 6667 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐷𝑊))
133 eqid 2818 . . . . . . . . . . . 12 ((1...𝑁) subMat 𝑅) = ((1...𝑁) subMat 𝑅)
1344, 133, 5submaval 21118 . . . . . . . . . . 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 1363 . . . . . . . . . 10 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)))
136 fzdif2 30440 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘1) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
13789, 136syl 17 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
138 mpoeq12 7216 . . . . . . . . . . 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 584 . . . . . . . . . 10 (𝜑 → (𝑖 ∈ ((1...𝑁) ∖ {𝑁}), 𝑗 ∈ ((1...𝑁) ∖ {𝑁}) ↦ (𝑖𝑊𝑗)) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
140135, 139eqtrd 2853 . . . . . . . . 9 (𝜑 → (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁) = (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)))
141 difssd 4106 . . . . . . . . . . 11 (𝜑 → ((1...𝑁) ∖ {𝑁}) ⊆ (1...𝑁))
142137, 141eqsstrrd 4003 . . . . . . . . . 10 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
1434, 5submabas 21115 . . . . . . . . . 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 584 . . . . . . . . 9 (𝜑 → (𝑖 ∈ (1...(𝑁 − 1)), 𝑗 ∈ (1...(𝑁 − 1)) ↦ (𝑖𝑊𝑗)) ∈ (Base‘((1...(𝑁 − 1)) Mat 𝑅)))
145140, 144eqeltrd 2910 . . . . . . . 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 2818 . . . . . . . . 9 ((1...(𝑁 − 1)) Mat 𝑅) = ((1...(𝑁 − 1)) Mat 𝑅)
148 eqid 2818 . . . . . . . . 9 (Base‘((1...(𝑁 − 1)) Mat 𝑅)) = (Base‘((1...(𝑁 − 1)) Mat 𝑅))
149146, 147, 148, 117mdetcl 21133 . . . . . . . 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 584 . . . . . . 7 (𝜑 → (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) ∈ (Base‘𝑅))
151117, 16, 48ringlidm 19250 . . . . . . 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 584 . . . . . 6 (𝜑 → ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
1534fveq2i 6666 . . . . . . . . . . 11 (Base‘𝐴) = (Base‘((1...𝑁) Mat 𝑅))
1545, 153eqtri 2841 . . . . . . . . . 10 𝐵 = (Base‘((1...𝑁) Mat 𝑅))
155124, 154eleqtrdi 2920 . . . . . . . . 9 (𝜑𝑊 ∈ (Base‘((1...𝑁) Mat 𝑅)))
156 smadiadetr 21212 . . . . . . . . 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 834 . . . . . . . 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 6664 . . . . . . . . 9 (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (((1...𝑁) maDet 𝑅)‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁))
15916oveqi 7158 . . . . . . . . 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 2833 . . . . . . . 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 235 . . . . . . 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 7160 . . . . . . . . . 10 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = ((1...(𝑁 − 1)) maDet 𝑅))
163162, 146syl6eqr 2871 . . . . . . . . 9 (𝜑 → (((1...𝑁) ∖ {𝑁}) maDet 𝑅) = 𝐸)
164163fveq1d 6665 . . . . . . . 8 (𝜑 → ((((1...𝑁) ∖ {𝑁}) maDet 𝑅)‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
165164oveq2d 7161 . . . . . . 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 2853 . . . . . 6 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = ((1r𝑅) · (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))))
1674, 5submat1n 30969 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑊𝐵) → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
16887, 124, 167syl2anc 584 . . . . . . 7 (𝜑 → (𝑁(subMat1‘𝑊)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁))
169168fveq2d 6667 . . . . . 6 (𝜑 → (𝐸‘(𝑁(subMat1‘𝑊)𝑁)) = (𝐸‘(𝑁(((1...𝑁) subMat 𝑅)‘𝑊)𝑁)))
170152, 166, 1693eqtr4d 2863 . . . . 5 (𝜑 → (𝐷‘(𝑁(𝑊((1...𝑁) matRRep 𝑅)(1r𝑅))𝑁)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
171132, 170eqtr3d 2855 . . . 4 (𝜑 → (𝐷𝑊) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
1724, 5, 87, 3, 2, 21, 1, 10submatminr1 30974 . . . . . 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 2853 . . . . 5 (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
175174fveq2d 6667 . . . 4 (𝜑 → (𝐸‘(𝐼(subMat1‘𝑀)𝐽)) = (𝐸‘(𝑁(subMat1‘𝑊)𝑁)))
176171, 175eqtr4d 2856 . . 3 (𝜑 → (𝐷𝑊) = (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))
177176oveq2d 7161 . 2 (𝜑 → ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐷𝑊)) = ((𝑍‘((𝑆𝑃) · (𝑆𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
17812, 27, 1773eqtrd 2857 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 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  wss 3933  ifcif 4463  {csn 4557  ccnv 5547  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  1c1 10526   · cmul 10530  cmin 10858  cn 11626  cuz 12231  ...cfz 12880  Basecbs 16471  .rcmulr 16554  0gc0g 16701  SymGrpcsymg 18433  pmSgncpsgn 18546  1rcur 19180  Ringcrg 19226  CRingccrg 19227  ℤRHomczrh 20575   Mat cmat 20944   matRRep cmarrep 21093   subMat csubma 21113   maDet cmdat 21121   maAdju cmadu 21169   minMatR1 cminmar1 21170  subMat1csmat 30957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-xor 1496  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-ot 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-rp 12378  df-fz 12881  df-fzo 13022  df-seq 13358  df-exp 13418  df-hash 13679  df-word 13850  df-lsw 13903  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-splice 14100  df-reverse 14109  df-s2 14198  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-0g 16703  df-gsum 16704  df-prds 16709  df-pws 16711  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-mulg 18163  df-subg 18214  df-ghm 18294  df-gim 18337  df-cntz 18385  df-oppg 18412  df-symg 18434  df-pmtr 18499  df-psgn 18548  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-cring 19229  df-oppr 19302  df-dvdsr 19320  df-unit 19321  df-invr 19351  df-dvr 19362  df-rnghom 19396  df-drng 19433  df-subrg 19462  df-sra 19873  df-rgmod 19874  df-cnfld 20474  df-zring 20546  df-zrh 20579  df-dsmm 20804  df-frlm 20819  df-mat 20945  df-marrep 21095  df-subma 21114  df-mdet 21122  df-madu 21171  df-minmar1 21172  df-smat 30958
This theorem is referenced by:  madjusmdetlem4  30994
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