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Theorem pm2mpmhmlem2 22794

Description: Lemma 2 for pm2mpmhm 22795. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)

**Hypotheses**
Ref	Expression
pm2mpmhm.p	⊢ 𝑃 = (Poly₁‘𝑅)
pm2mpmhm.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pm2mpmhm.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
pm2mpmhm.q	⊢ 𝑄 = (Poly₁‘𝐴)
pm2mpmhm.t	⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
pm2mpmhm.b	⊢ 𝐵 = (Base‘𝐶)

**Assertion**
Ref	Expression
pm2mpmhmlem2	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)))

Distinct variable groups: 𝑥,𝐵,𝑦 𝑥,𝑁,𝑦 𝑥,𝑅,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦) 𝐶(𝑥,𝑦) 𝑃(𝑥,𝑦) 𝑄(𝑥,𝑦) 𝑇(𝑥,𝑦)

Proof of Theorem pm2mpmhmlem2 Dummy variables 𝑘 𝑙 𝑛 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 767	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
2		simplr 769	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ Ring)
3		pm2mpmhm.p	. . . . . . . 8 ⊢ 𝑃 = (Poly₁‘𝑅)
4		pm2mpmhm.c	. . . . . . . 8 ⊢ 𝐶 = (𝑁 Mat 𝑃)
5	3, 4	pmatring 22667	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
6	5	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ Ring)
7		simpl 482	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	adantl 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
9		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10	9	adantl 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
11		pm2mpmhm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
12		eqid 2737	. . . . . . 7 ⊢ (._r‘𝐶) = (._r‘𝐶)
13	11, 12	ringcl 20222	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
14	6, 8, 10, 13	syl3anc 1374	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
15		eqid 2737	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑄) = ( ·_𝑠 ‘𝑄)
16		eqid 2737	. . . . . 6 ⊢ (._g‘(mulGrp‘𝑄)) = (._g‘(mulGrp‘𝑄))
17		eqid 2737	. . . . . 6 ⊢ (var₁‘𝐴) = (var₁‘𝐴)
18		pm2mpmhm.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
19		pm2mpmhm.q	. . . . . 6 ⊢ 𝑄 = (Poly₁‘𝐴)
20		pm2mpmhm.t	. . . . . 6 ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅)
21	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 22771	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(._r‘𝐶)𝑦) ∈ 𝐵) → (𝑇‘(𝑥(._r‘𝐶)𝑦)) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ (((𝑥(._r‘𝐶)𝑦) decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
22	1, 2, 14, 21	syl3anc 1374	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(._r‘𝐶)𝑦)) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ (((𝑥(._r‘𝐶)𝑦) decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
23	3, 4, 11, 18	decpmatmul 22747	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑥(._r‘𝐶)𝑦) decompPMat 𝑘) = (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))))
24	23	ad4ant234 1177	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → ((𝑥(._r‘𝐶)𝑦) decompPMat 𝑘) = (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))))
25	24	oveq1d 7375	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (((𝑥(._r‘𝐶)𝑦) decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) = ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))
26	25	mpteq2dva 5179	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ (((𝑥(._r‘𝐶)𝑦) decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) = (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))
27	26	oveq2d 7376	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ (((𝑥(._r‘𝐶)𝑦) decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
28		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘𝑄)
29	18	matring 22418	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
30	29	ad2antrr 727	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝐴 ∈ Ring)
31		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		eqid 2737	. . . . . . . 8 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
33		ringcmn 20254	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
34	29, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
35	34	ad3antrrr 731	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
36		fzfid 13926	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
37	30	ad2antrr 727	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
38		simp-5r 786	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
39	8	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
40		elfznn0 13565	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ₀)
41	40	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ₀)
42	3, 4, 11, 18, 31	decpmatcl 22742	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ ℕ₀) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
43	38, 39, 41, 42	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
44	10	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
45		fznn0sub 13501	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → (𝑘 − 𝑧) ∈ ℕ₀)
46	45	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘 − 𝑧) ∈ ℕ₀)
47	3, 4, 11, 18, 31	decpmatcl 22742	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ (𝑘 − 𝑧) ∈ ℕ₀) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
48	38, 44, 46, 47	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
49		eqid 2737	. . . . . . . . . . . . 13 ⊢ (._r‘𝐴) = (._r‘𝐴)
50	31, 49	ringcl 20222	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Ring ∧ (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴) ∧ (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴)) → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))) ∈ (Base‘𝐴))
51	37, 43, 48, 50	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))) ∈ (Base‘𝐴))
52	51	ralrimiva 3130	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))) ∈ (Base‘𝐴))
53	31, 35, 36, 52	gsummptcl 19933	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) ∈ (Base‘𝐴))
54	53	ralrimiva 3130	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ∀𝑘 ∈ ℕ₀ (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) ∈ (Base‘𝐴))
55	3, 4, 11, 18, 49, 32	decpmatmulsumfsupp 22748	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))) finSupp (0_g‘𝐴))
56	55	adantr 480	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))) finSupp (0_g‘𝐴))
57		simpr 484	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
58	19, 28, 17, 16, 30, 31, 15, 32, 54, 56, 57	gsummoncoe1 22283	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑛) = ⦋𝑛 / 𝑘⦌(𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))))
59		csbov2g 7408	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ⦋𝑛 / 𝑘⦌(𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) = (𝐴 Σ_g ⦋𝑛 / 𝑘⦌(𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))))
60		id 22	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
61		oveq2 7368	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
62		oveq1 7367	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑧) = (𝑛 − 𝑧))
63	62	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑧)))
64	63	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))) = ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))))
65	61, 64	mpteq12dv 5173	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧)))))
66	65	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ₀ ∧ 𝑘 = 𝑛) → (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧)))))
67	60, 66	csbied 3874	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → ⦋𝑛 / 𝑘⦌(𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))) = (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧)))))
68	67	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → (𝐴 Σ_g ⦋𝑛 / 𝑘⦌(𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) = (𝐴 Σ_g (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))))))
69	59, 68	eqtrd 2772	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ → ⦋𝑛 / 𝑘⦌(𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) = (𝐴 Σ_g (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))))))
70	69	adantl 481	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ⦋𝑛 / 𝑘⦌(𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) = (𝐴 Σ_g (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))))))
71		eqidd 2738	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑟 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙)))))) = (𝑟 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙)))))))
72		oveq2 7368	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
73		fvoveq1 7383	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑛 → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙)) = ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙)))
74	73	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙))) = (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙))))
75	72, 74	mpteq12dv 5173	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑛 → (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙)))))
76	75	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑟 = 𝑛 → (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙))))) = (𝐴 Σ_g (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙))))))
77	76	adantl 481	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑟 = 𝑛) → (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙))))) = (𝐴 Σ_g (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙))))))
78		ovexd 7395	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝐴 Σ_g (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙))))) ∈ V)
79	71, 77, 57, 78	fvmptd 6949	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ((𝑟 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙))))))‘𝑛) = (𝐴 Σ_g (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙))))))
80		eqid 2737	. . . . . . . . . 10 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
81	19	ply1ring 22221	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
82	29, 81	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
83		ringcmn 20254	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
85	84	ad2antrr 727	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝑄 ∈ CMnd)
86		nn0ex 12434	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
87	86	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ℕ₀ ∈ V)
88	7	anim2i 618	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
89		df-3an 1089	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
90	88, 89	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
91	90	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
92	3, 4, 11, 15, 16, 17, 18, 19, 28	pm2mpghmlem1 22788	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
93	91, 92	sylan 581	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
94	93	fmpttd 7061	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))):ℕ₀⟶(Base‘𝑄))
95	3, 4, 11, 15, 16, 17, 18, 19	pm2mpghmlem2 22787	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
96	91, 95	syl 17	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
97	28, 80, 85, 87, 94, 96	gsumcl 19881	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄))
98	9	anim2i 618	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
99		df-3an 1089	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
100	98, 99	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
101	100	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
102	3, 4, 11, 15, 16, 17, 18, 19, 28	pm2mpghmlem1 22788	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
103	101, 102	sylan 581	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
104	103	fmpttd 7061	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))):ℕ₀⟶(Base‘𝑄))
105	1, 2, 10	3jca 1129	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
106	105	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
107	3, 4, 11, 15, 16, 17, 18, 19	pm2mpghmlem2 22787	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
108	106, 107	syl 17	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
109	28, 80, 85, 87, 104, 108	gsumcl 19881	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄))
110		eqid 2737	. . . . . . . . . . 11 ⊢ (._r‘𝑄) = (._r‘𝑄)
111	19, 110, 49, 28	coe1mul 22245	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ring ∧ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄) ∧ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄)) → (coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))) = (𝑟 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙)))))))
112	111	fveq1d 6836	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ring ∧ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄) ∧ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄)) → ((coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))‘𝑛) = ((𝑟 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙))))))‘𝑛))
113	30, 97, 109, 112	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ((coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))‘𝑛) = ((𝑟 ∈ ℕ₀ ↦ (𝐴 Σ_g (𝑙 ∈ (0...𝑟) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑟 − 𝑙))))))‘𝑛))
114		oveq2 7368	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
115		oveq2 7368	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑙 → (𝑛 − 𝑧) = (𝑛 − 𝑙))
116	115	oveq2d 7376	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑙)))
117	114, 116	oveq12d 7378	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))) = ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙))))
118	117	cbvmptv 5190	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙))))
119	29	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
120		simp-5r 786	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
121	8	ad3antrrr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
122		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
123	3, 4, 11, 18, 31	decpmatcl 22742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
124	120, 121, 122, 123	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
125	124	ralrimiva 3130	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ₀ (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
126	2, 8	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
127	126	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
128	3, 4, 11, 18, 32	decpmatfsupp 22744	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑥 decompPMat 𝑘)) finSupp (0_g‘𝐴))
129	127, 128	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ₀ ↦ (𝑥 decompPMat 𝑘)) finSupp (0_g‘𝐴))
130		elfznn0 13565	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ₀)
131	130	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℕ₀)
132	19, 28, 17, 16, 119, 31, 15, 32, 125, 129, 131	gsummoncoe1 22283	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙) = ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘))
133		csbov2g 7408	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘))
134		csbvarg 4375	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌𝑘 = 𝑙)
135	134	oveq2d 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘) = (𝑥 decompPMat 𝑙))
136	133, 135	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
137	136	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
138	132, 137	eqtr2d 2773	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑥 decompPMat 𝑙) = ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙))
139	10	ad3antrrr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑦 ∈ 𝐵)
140	3, 4, 11, 18, 31	decpmatcl 22742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
141	120, 139, 122, 140	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
142	141	ralrimiva 3130	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ₀ (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
143	2, 10	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
144	143	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
145	3, 4, 11, 18, 32	decpmatfsupp 22744	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑦 decompPMat 𝑘)) finSupp (0_g‘𝐴))
146	144, 145	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑘 ∈ ℕ₀ ↦ (𝑦 decompPMat 𝑘)) finSupp (0_g‘𝐴))
147		fznn0sub 13501	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ₀)
148	147	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛 − 𝑙) ∈ ℕ₀)
149	19, 28, 17, 16, 119, 31, 15, 32, 142, 146, 148	gsummoncoe1 22283	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙)) = ⦋(𝑛 − 𝑙) / 𝑘⦌(𝑦 decompPMat 𝑘))
150		ovex 7393	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 𝑙) ∈ V
151		csbov2g 7408	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 𝑙) ∈ V → ⦋(𝑛 − 𝑙) / 𝑘⦌(𝑦 decompPMat 𝑘) = (𝑦 decompPMat ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘))
152	150, 151	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ⦋(𝑛 − 𝑙) / 𝑘⦌(𝑦 decompPMat 𝑘) = (𝑦 decompPMat ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘))
153		csbvarg 4375	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 𝑙) ∈ V → ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘 = (𝑛 − 𝑙))
154	150, 153	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘 = (𝑛 − 𝑙))
155	154	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘) = (𝑦 decompPMat (𝑛 − 𝑙)))
156	149, 152, 155	3eqtrrd 2777	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat (𝑛 − 𝑙)) = ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙)))
157	138, 156	oveq12d 7378	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙))) = (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙))))
158	157	mpteq2dva 5179	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑙 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙)))) = (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙)))))
159	118, 158	eqtrid 2784	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧)))) = (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙)))))
160	159	oveq2d 7376	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝐴 Σ_g (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))))) = (𝐴 Σ_g (𝑙 ∈ (0...𝑛) ↦ (((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑙)(._r‘𝐴)((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘(𝑛 − 𝑙))))))
161	79, 113, 160	3eqtr4rd 2783	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝐴 Σ_g (𝑧 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))))) = ((coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))‘𝑛))
162	58, 70, 161	3eqtrd 2776	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑛) = ((coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))‘𝑛))
163	162	ralrimiva 3130	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑛 ∈ ℕ₀ ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑛) = ((coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))‘𝑛))
164	29	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring)
165	84	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ CMnd)
166	86	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ℕ₀ ∈ V)
167	19	ply1lmod 22225	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
168	29, 167	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
169	168	ad2antrr 727	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑄 ∈ LMod)
170	34	ad2antrr 727	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
171		fzfid 13926	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
172	29	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
173		simp-4r 784	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
174		simplrl 777	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
175	174	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
176	40	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ₀)
177	173, 175, 176, 42	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
178		simplrr 778	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑦 ∈ 𝐵)
179	178	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
180	45	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘 − 𝑧) ∈ ℕ₀)
181	173, 179, 180, 47	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
182	172, 177, 181, 50	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))) ∈ (Base‘𝐴))
183	182	ralrimiva 3130	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → ∀𝑧 ∈ (0...𝑘)((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))) ∈ (Base‘𝐴))
184	31, 170, 171, 183	gsummptcl 19933	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) ∈ (Base‘𝐴))
185	29	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
186	19	ply1sca 22226	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
187	185, 186	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 = (Scalar‘𝑄))
188	187	eqcomd 2743	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Scalar‘𝑄) = 𝐴)
189	188	fveq2d 6838	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
190	184, 189	eleqtrrd 2840	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) ∈ (Base‘(Scalar‘𝑄)))
191		eqid 2737	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
192	19, 17, 191, 16, 28	ply1moncl 22246	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
193	185, 192	sylancom 589	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
194		eqid 2737	. . . . . . . . . 10 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
195		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
196	28, 194, 15, 195	lmodvscl 20864	. . . . . . . . 9 ⊢ ((𝑄 ∈ LMod ∧ (𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))))) ∈ (Base‘(Scalar‘𝑄)) ∧ (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄)) → ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
197	169, 190, 193, 196	syl3anc 1374	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
198	197	fmpttd 7061	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))):ℕ₀⟶(Base‘𝑄))
199	3, 4, 11, 15, 16, 17, 18, 19, 28, 20	pm2mpmhmlem1 22793	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
200	28, 80, 165, 166, 198, 199	gsumcl 19881	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄))
201	82	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ Ring)
202	90, 92	sylan 581	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
203	202	fmpttd 7061	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))):ℕ₀⟶(Base‘𝑄))
204	90, 95	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
205	28, 80, 165, 166, 203, 204	gsumcl 19881	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄))
206	100, 102	sylan 581	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
207	206	fmpttd 7061	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))):ℕ₀⟶(Base‘𝑄))
208	1, 2, 10, 107	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
209	28, 80, 165, 166, 207, 208	gsumcl 19881	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄))
210	28, 110	ringcl 20222	. . . . . . 7 ⊢ ((𝑄 ∈ Ring ∧ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄) ∧ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄)) → ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))) ∈ (Base‘𝑄))
211	201, 205, 209, 210	syl3anc 1374	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))) ∈ (Base‘𝑄))
212		eqid 2737	. . . . . . 7 ⊢ (coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))) = (coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
213		eqid 2737	. . . . . . 7 ⊢ (coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))) = (coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))
214	19, 28, 212, 213	ply1coe1eq 22275	. . . . . 6 ⊢ ((𝐴 ∈ Ring ∧ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) ∈ (Base‘𝑄) ∧ ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))) ∈ (Base‘𝑄)) → (∀𝑛 ∈ ℕ₀ ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑛) = ((coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))‘𝑛) ↔ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) = ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))))
215	164, 200, 211, 214	syl3anc 1374	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∀𝑛 ∈ ℕ₀ ((coe₁‘(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))‘𝑛) = ((coe₁‘((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))‘𝑛) ↔ (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) = ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))))
216	163, 215	mpbid 232	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑧 ∈ (0...𝑘) ↦ ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧)))))( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))) = ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))
217	22, 27, 216	3eqtrd 2776	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))
218	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 22771	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑇‘𝑥) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
219	1, 2, 8, 218	syl3anc 1374	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘𝑥) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
220	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 22771	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑇‘𝑦) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
221	1, 2, 10, 220	syl3anc 1374	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘𝑦) = (𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))))
222	219, 221	oveq12d 7378	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)) = ((𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))(._r‘𝑄)(𝑄 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))))))
223	217, 222	eqtr4d 2775	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)))
224	223	ralrimivva 3181	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ⦋csb 3838 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6492 (class class class)co 7360 Fincfn 8886 finSupp cfsupp 9267 0cc0 11029 − cmin 11368 ℕ₀cn0 12428 ...cfz 13452 Basecbs 17170 ._rcmulr 17212 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 Σ_g cgsu 17394 ._gcmg 19034 CMndccmn 19746 mulGrpcmgp 20112 Ringcrg 20205 LModclmod 20846 var₁cv1 22149 Poly₁cpl1 22150 coe₁cco1 22151 Mat cmat 22382 decompPMat cdecpmat 22737 pMatToMatPoly cpm2mp 22767

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-srg 20159 df-ring 20207 df-subrng 20514 df-subrg 20538 df-lmod 20848 df-lss 20918 df-sra 21160 df-rgmod 21161 df-dsmm 21722 df-frlm 21737 df-psr 21899 df-mvr 21900 df-mpl 21901 df-opsr 21903 df-psr1 22153 df-vr1 22154 df-ply1 22155 df-coe1 22156 df-mamu 22366 df-mat 22383 df-decpmat 22738 df-pm2mp 22768

This theorem is referenced by: pm2mpmhm 22795

W3C validator