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Theorem pm2mpmhmlem2 22695

Description: Lemma 2 for pm2mpmhm 22696. (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 766	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
2		simplr 768	. . . . 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 22568	. . . . . . 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 2727	. . . . . . 7 ⊢ (._r‘𝐶) = (._r‘𝐶)
13	11, 12	ringcl 20174	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
14	6, 8, 10, 13	syl3anc 1369	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
15		eqid 2727	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑄) = ( ·_𝑠 ‘𝑄)
16		eqid 2727	. . . . . 6 ⊢ (._g‘(mulGrp‘𝑄)) = (._g‘(mulGrp‘𝑄))
17		eqid 2727	. . . . . 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 22672	. . . . 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 1369	. . . 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 22648	. . . . . . . 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 1173	. . . . . . 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 7429	. . . . . 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 5242	. . . . 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 7430	. . . 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 2727	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘𝑄)
29	18	matring 22319	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
30	29	ad2antrr 725	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝐴 ∈ Ring)
31		eqid 2727	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		eqid 2727	. . . . . . . 8 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
33		ringcmn 20200	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
34	29, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
35	34	ad3antrrr 729	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
36		fzfid 13956	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
37	30	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
38		simp-5r 785	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
39	8	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
40		elfznn0 13612	. . . . . . . . . . . . . 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 22643	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ ℕ₀) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
43	38, 39, 41, 42	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
44	10	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
45		fznn0sub 13551	. . . . . . . . . . . . . 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 22643	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ (𝑘 − 𝑧) ∈ ℕ₀) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
48	38, 44, 46, 47	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
49		eqid 2727	. . . . . . . . . . . . 13 ⊢ (._r‘𝐴) = (._r‘𝐴)
50	31, 49	ringcl 20174	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 3141	. . . . . . . . . 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 19906	. . . . . . . . 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 3141	. . . . . . . 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 22649	. . . . . . . . 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 22201	. . . . . . 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 7460	. . . . . . . . 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 7422	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
62		oveq1 7421	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑧) = (𝑛 − 𝑧))
63	62	oveq2d 7430	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑧)))
64	63	oveq2d 7430	. . . . . . . . . . . . 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 5233	. . . . . . . . . . . 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 3927	. . . . . . . . . 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 7430	. . . . . . . . 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 2767	. . . . . . . 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 2728	. . . . . . . . 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 7422	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
73		fvoveq1 7437	. . . . . . . . . . . . 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 7430	. . . . . . . . . . . 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 5233	. . . . . . . . . . 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 7430	. . . . . . . . . 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 7449	. . . . . . . . 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 7006	. . . . . . . 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 2727	. . . . . . . . . 10 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
81	19	ply1ring 22140	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
82	29, 81	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
83		ringcmn 20200	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
85	84	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝑄 ∈ CMnd)
86		nn0ex 12494	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
87	86	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ℕ₀ ∈ V)
88	7	anim2i 616	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
89		df-3an 1087	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
90	88, 89	sylibr 233	. . . . . . . . . . . . 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 22689	. . . . . . . . . . . 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 579	. . . . . . . . . . 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 7119	. . . . . . . . . 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 22688	. . . . . . . . . . 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 19854	. . . . . . . . 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 616	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
99		df-3an 1087	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
100	98, 99	sylibr 233	. . . . . . . . . . . . 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 22689	. . . . . . . . . . . 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 579	. . . . . . . . . . 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 7119	. . . . . . . . . 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 1126	. . . . . . . . . . . 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 22688	. . . . . . . . . . 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 19854	. . . . . . . . 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 2727	. . . . . . . . . . 11 ⊢ (._r‘𝑄) = (._r‘𝑄)
111	19, 110, 49, 28	coe1mul 22163	. . . . . . . . . 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 6893	. . . . . . . . 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 1369	. . . . . . . 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 7422	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
115		oveq2 7422	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑙 → (𝑛 − 𝑧) = (𝑛 − 𝑙))
116	115	oveq2d 7430	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑙)))
117	114, 116	oveq12d 7432	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))) = ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙))))
118	117	cbvmptv 5255	. . . . . . . . . 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 729	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
120		simp-5r 785	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
121	8	ad3antrrr 729	. . . . . . . . . . . . . . . 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 22643	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
124	120, 121, 122, 123	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
125	124	ralrimiva 3141	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 22645	. . . . . . . . . . . . . . 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 13612	. . . . . . . . . . . . . . 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 22201	. . . . . . . . . . . . 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 7460	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘))
134		csbvarg 4427	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌𝑘 = 𝑙)
135	134	oveq2d 7430	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘) = (𝑥 decompPMat 𝑙))
136	133, 135	eqtrd 2767	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . 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 22643	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
141	120, 139, 122, 140	syl3anc 1369	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
142	141	ralrimiva 3141	. . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 22645	. . . . . . . . . . . . . . 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 13551	. . . . . . . . . . . . . . 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 22201	. . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 𝑙) ∈ V
151		csbov2g 7460	. . . . . . . . . . . . . 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 4427	. . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 7432	. . . . . . . . . . 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 5242	. . . . . . . . . 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 2779	. . . . . . . . 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 7430	. . . . . . . 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 2778	. . . . . . 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 2771	. . . . . 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 3141	. . . . 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 22144	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
168	29, 167	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
169	168	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑄 ∈ LMod)
170	34	ad2antrr 725	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
171		fzfid 13956	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
172	29	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
173		simp-4r 783	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
174		simplrl 776	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
178		simplrr 777	. . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 1369	. . . . . . . . . . . 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 3141	. . . . . . . . . . 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 19906	. . . . . . . . . 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 725	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
186	19	ply1sca 22145	. . . . . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Scalar‘𝑄) = 𝐴)
189	188	fveq2d 6895	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
190	184, 189	eleqtrrd 2831	. . . . . . . . 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 2727	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
192	19, 17, 191, 16, 28	ply1moncl 22164	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
193	185, 192	sylancom 587	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
194		eqid 2727	. . . . . . . . . 10 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
195		eqid 2727	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
196	28, 194, 15, 195	lmodvscl 20743	. . . . . . . . 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 1369	. . . . . . . 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 7119	. . . . . . 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 22694	. . . . . . 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 19854	. . . . . 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 579	. . . . . . . . 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 7119	. . . . . . . 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 19854	. . . . . . 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 579	. . . . . . . . 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 7119	. . . . . . . 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 1369	. . . . . . . 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 19854	. . . . . . 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 20174	. . . . . . 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 1369	. . . . . 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 2727	. . . . . . 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 2727	. . . . . . 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 22193	. . . . . 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 1369	. . . . 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 231	. . . 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 2771	. . 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 22672	. . . . 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 1369	. . . 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 22672	. . . . 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 1369	. . . 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 7432	. . 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 2770	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)))
224	223	ralrimivva 3195	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 ⦋csb 3889 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 Fincfn 8953 finSupp cfsupp 9375 0cc0 11124 − cmin 11460 ℕ₀cn0 12488 ...cfz 13502 Basecbs 17165 ._rcmulr 17219 Scalarcsca 17221 ·_𝑠 cvsca 17222 0_gc0g 17406 Σ_g cgsu 17407 ._gcmg 19007 CMndccmn 19719 mulGrpcmgp 20058 Ringcrg 20157 LModclmod 20725 var₁cv1 22069 Poly₁cpl1 22070 coe₁cco1 22071 Mat cmat 22281 decompPMat cdecpmat 22638 pMatToMatPoly cpm2mp 22668

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-srg 20111 df-ring 20159 df-subrng 20465 df-subrg 20490 df-lmod 20727 df-lss 20798 df-sra 21040 df-rgmod 21041 df-dsmm 21646 df-frlm 21661 df-psr 21822 df-mvr 21823 df-mpl 21824 df-opsr 21826 df-psr1 22073 df-vr1 22074 df-ply1 22075 df-coe1 22076 df-mamu 22260 df-mat 22282 df-decpmat 22639 df-pm2mp 22669

This theorem is referenced by: pm2mpmhm 22696

W3C validator