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Theorem pm2mpmhmlem2 22312

Description: Lemma 2 for pm2mpmhm 22313. (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 765	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
2		simplr 767	. . . . 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 22185	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
6	5	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ Ring)
7		simpl 483	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	adantl 482	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
9		simpr 485	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10	9	adantl 482	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
11		pm2mpmhm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
12		eqid 2732	. . . . . . 7 ⊢ (._r‘𝐶) = (._r‘𝐶)
13	11, 12	ringcl 20066	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
14	6, 8, 10, 13	syl3anc 1371	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
15		eqid 2732	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑄) = ( ·_𝑠 ‘𝑄)
16		eqid 2732	. . . . . 6 ⊢ (._g‘(mulGrp‘𝑄)) = (._g‘(mulGrp‘𝑄))
17		eqid 2732	. . . . . 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 22289	. . . . 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 1371	. . . 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 22265	. . . . . . . 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 1175	. . . . . . 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 7420	. . . . . 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 5247	. . . . 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 7421	. . . 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 2732	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘𝑄)
29	18	matring 21936	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
30	29	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝐴 ∈ Ring)
31		eqid 2732	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		eqid 2732	. . . . . . . 8 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
33		ringcmn 20092	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
34	29, 33	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
35	34	ad3antrrr 728	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
36		fzfid 13934	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
37	30	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
38		simp-5r 784	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
39	8	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
40		elfznn0 13590	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ₀)
41	40	adantl 482	. . . . . . . . . . . . 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 22260	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ ℕ₀) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
43	38, 39, 41, 42	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
44	10	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
45		fznn0sub 13529	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → (𝑘 − 𝑧) ∈ ℕ₀)
46	45	adantl 482	. . . . . . . . . . . . 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 22260	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ (𝑘 − 𝑧) ∈ ℕ₀) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
48	38, 44, 46, 47	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
49		eqid 2732	. . . . . . . . . . . . 13 ⊢ (._r‘𝐴) = (._r‘𝐴)
50	31, 49	ringcl 20066	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 3146	. . . . . . . . . 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 19829	. . . . . . . . 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 3146	. . . . . . . 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 22266	. . . . . . . . 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 481	. . . . . . . 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 485	. . . . . . . 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 21819	. . . . . . 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 7451	. . . . . . . . 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 7413	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
62		oveq1 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑧) = (𝑛 − 𝑧))
63	62	oveq2d 7421	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑧)))
64	63	oveq2d 7421	. . . . . . . . . . . . 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 5238	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 3930	. . . . . . . . . 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 7421	. . . . . . . . 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 482	. . . . . . 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 2733	. . . . . . . . 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 7413	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
73		fvoveq1 7428	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . 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 5238	. . . . . . . . . . 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 7421	. . . . . . . . . 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 482	. . . . . . . . 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 7440	. . . . . . . . 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 7002	. . . . . . . 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 2732	. . . . . . . . . 10 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
81	19	ply1ring 21761	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
82	29, 81	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
83		ringcmn 20092	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
84	82, 83	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
85	84	ad2antrr 724	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝑄 ∈ CMnd)
86		nn0ex 12474	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
87	86	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ℕ₀ ∈ V)
88	7	anim2i 617	. . . . . . . . . . . . . 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 233	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
91	90	adantr 481	. . . . . . . . . . . 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 22306	. . . . . . . . . . . 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 580	. . . . . . . . . . 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 7111	. . . . . . . . . 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 22305	. . . . . . . . . . 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 19777	. . . . . . . . 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 617	. . . . . . . . . . . . . 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 233	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
101	100	adantr 481	. . . . . . . . . . . 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 22306	. . . . . . . . . . . 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 580	. . . . . . . . . . 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 7111	. . . . . . . . . 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 1128	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
106	105	adantr 481	. . . . . . . . . . 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 22305	. . . . . . . . . . 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 19777	. . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (._r‘𝑄) = (._r‘𝑄)
111	19, 110, 49, 28	coe1mul 21783	. . . . . . . . . 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 6890	. . . . . . . . 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 1371	. . . . . . . 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 7413	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
115		oveq2 7413	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑙 → (𝑛 − 𝑧) = (𝑛 − 𝑙))
116	115	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑙)))
117	114, 116	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))) = ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙))))
118	117	cbvmptv 5260	. . . . . . . . . 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 728	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
120		simp-5r 784	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
121	8	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
122		simpr 485	. . . . . . . . . . . . . . . 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 22260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
124	120, 121, 122, 123	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
125	124	ralrimiva 3146	. . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
127	126	ad2antrr 724	. . . . . . . . . . . . . . 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 22262	. . . . . . . . . . . . . . 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 13590	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ₀)
131	130	adantl 482	. . . . . . . . . . . . . 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 21819	. . . . . . . . . . . . 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 7451	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘))
134		csbvarg 4430	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌𝑘 = 𝑙)
135	134	oveq2d 7421	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . 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 22260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
141	120, 139, 122, 140	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
142	141	ralrimiva 3146	. . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
144	143	ad2antrr 724	. . . . . . . . . . . . . . 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 22262	. . . . . . . . . . . . . . 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 13529	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ₀)
148	147	adantl 482	. . . . . . . . . . . . . 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 21819	. . . . . . . . . . . . 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 7438	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 𝑙) ∈ V
151		csbov2g 7451	. . . . . . . . . . . . . 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 4430	. . . . . . . . . . . . . . 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 7421	. . . . . . . . . . . . 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 7423	. . . . . . . . . . 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 5247	. . . . . . . . . 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 7421	. . . . . . . 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 3146	. . . . 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 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring)
165	84	adantr 481	. . . . . . 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 21765	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
168	29, 167	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
169	168	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑄 ∈ LMod)
170	34	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
171		fzfid 13934	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
172	29	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
173		simp-4r 782	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
174		simplrl 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
175	174	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
176	40	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ₀)
177	173, 175, 176, 42	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
178		simplrr 776	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑦 ∈ 𝐵)
179	178	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
180	45	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘 − 𝑧) ∈ ℕ₀)
181	173, 179, 180, 47	syl3anc 1371	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 3146	. . . . . . . . . . 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 19829	. . . . . . . . . 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 724	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
186	19	ply1sca 21766	. . . . . . . . . . . . 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 2738	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Scalar‘𝑄) = 𝐴)
189	188	fveq2d 6892	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
190	184, 189	eleqtrrd 2836	. . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
192	19, 17, 191, 16, 28	ply1moncl 21784	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
193	185, 192	sylancom 588	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
194		eqid 2732	. . . . . . . . . 10 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
195		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
196	28, 194, 15, 195	lmodvscl 20481	. . . . . . . . 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 1371	. . . . . . . 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 7111	. . . . . . 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 22311	. . . . . . 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 19777	. . . . . 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 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ Ring)
202	90, 92	sylan 580	. . . . . . . . 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 7111	. . . . . . . 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 19777	. . . . . . 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 580	. . . . . . . . 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 7111	. . . . . . . 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 1371	. . . . . . . 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 19777	. . . . . . 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 20066	. . . . . . 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 1371	. . . . . 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 2732	. . . . . . 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 2732	. . . . . . 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 21813	. . . . . 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 1371	. . . . 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 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 22289	. . . . 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 1371	. . . 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 22289	. . . . 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 1371	. . . 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 7423	. . 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 3200	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⦋csb 3892 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 finSupp cfsupp 9357 0cc0 11106 − cmin 11440 ℕ₀cn0 12468 ...cfz 13480 Basecbs 17140 ._rcmulr 17194 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 Σ_g cgsu 17382 ._gcmg 18944 CMndccmn 19642 mulGrpcmgp 19981 Ringcrg 20049 LModclmod 20463 var₁cv1 21691 Poly₁cpl1 21692 coe₁cco1 21693 Mat cmat 21898 decompPMat cdecpmat 22255 pMatToMatPoly cpm2mp 22285

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-srg 20003 df-ring 20051 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-dsmm 21278 df-frlm 21293 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-coe1 21698 df-mamu 21877 df-mat 21899 df-decpmat 22256 df-pm2mp 22286

This theorem is referenced by: pm2mpmhm 22313

W3C validator