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Theorem pm2mpmhmlem2 22941

Description: Lemma 2 for pm2mpmhm 22942. (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 778	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
2		simplr 780	. . . . 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 22814	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
6	5	adantr 485	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ Ring)
7		simpl 487	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	adantl 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
9		simpr 489	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10	9	adantl 486	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
11		pm2mpmhm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
12		eqid 2769	. . . . . . 7 ⊢ (._r‘𝐶) = (._r‘𝐶)
13	11, 12	ringcl 20328	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
14	6, 8, 10, 13	syl3anc 1396	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
15		eqid 2769	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑄) = ( ·_𝑠 ‘𝑄)
16		eqid 2769	. . . . . 6 ⊢ (._g‘(mulGrp‘𝑄)) = (._g‘(mulGrp‘𝑄))
17		eqid 2769	. . . . . 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 22918	. . . . 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 1396	. . . 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 22894	. . . . . . . 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 1192	. . . . . . 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 7423	. . . . . 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 5205	. . . . 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 7424	. . . 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 2769	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘𝑄)
29	18	matring 22565	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
30	29	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝐴 ∈ Ring)
31		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘𝐴)
32		eqid 2769	. . . . . . . 8 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
33		ringcmn 20361	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
34	29, 33	syl 18	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
35	34	ad3antrrr 742	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
36		fzfid 14005	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
37	30	ad2antrr 738	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
38		simp-5r 797	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
39	8	ad3antrrr 742	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
40		elfznn0 13644	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ₀)
41	40	adantl 486	. . . . . . . . . . . . 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 22889	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ ℕ₀) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
43	38, 39, 41, 42	syl3anc 1396	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
44	10	ad3antrrr 742	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
45		fznn0sub 13580	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → (𝑘 − 𝑧) ∈ ℕ₀)
46	45	adantl 486	. . . . . . . . . . . . 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 22889	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ (𝑘 − 𝑧) ∈ ℕ₀) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
48	38, 44, 46, 47	syl3anc 1396	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
49		eqid 2769	. . . . . . . . . . . . 13 ⊢ (._r‘𝐴) = (._r‘𝐴)
50	31, 49	ringcl 20328	. . . . . . . . . . . 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 1396	. . . . . . . . . . 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 3163	. . . . . . . . . 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 20033	. . . . . . . . 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 3163	. . . . . . . 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 22895	. . . . . . . . 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 485	. . . . . . . 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 489	. . . . . . . 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 22433	. . . . . . 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 7456	. . . . . . . . 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 23	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
61		oveq2 7416	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
62		oveq1 7415	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑧) = (𝑛 − 𝑧))
63	62	oveq2d 7424	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑧)))
64	63	oveq2d 7424	. . . . . . . . . . . . 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 5199	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 3897	. . . . . . . . . 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 7424	. . . . . . . . 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 2804	. . . . . . . 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 486	. . . . . . 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 2770	. . . . . . . . 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 7416	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
73		fvoveq1 7431	. . . . . . . . . . . . 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 7424	. . . . . . . . . . . 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 5199	. . . . . . . . . . 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 7424	. . . . . . . . . 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 486	. . . . . . . . 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 7443	. . . . . . . . 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 6995	. . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
81	19	ply1ring 22372	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
82	29, 81	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
83		ringcmn 20361	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
84	82, 83	syl 18	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
85	84	ad2antrr 738	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝑄 ∈ CMnd)
86		nn0ex 12506	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
87	86	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ℕ₀ ∈ V)
88	7	anim2i 628	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
89		df-3an 1103	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
90	88, 89	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
91	90	adantr 485	. . . . . . . . . . . 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 22935	. . . . . . . . . . . 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 591	. . . . . . . . . . 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 7108	. . . . . . . . . 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 22934	. . . . . . . . . . 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 18	. . . . . . . . . 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 19981	. . . . . . . . 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 628	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
99		df-3an 1103	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
100	98, 99	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
101	100	adantr 485	. . . . . . . . . . . 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 22935	. . . . . . . . . . . 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 591	. . . . . . . . . . 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 7108	. . . . . . . . . 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 1144	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
106	105	adantr 485	. . . . . . . . . . 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 22934	. . . . . . . . . . 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 18	. . . . . . . . . 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 19981	. . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (._r‘𝑄) = (._r‘𝑄)
111	19, 110, 49, 28	coe1mul 22396	. . . . . . . . . 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 6881	. . . . . . . . 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 1396	. . . . . . . 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 7416	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
115		oveq2 7416	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑙 → (𝑛 − 𝑧) = (𝑛 − 𝑙))
116	115	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑙)))
117	114, 116	oveq12d 7426	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))) = ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙))))
118	117	cbvmptv 5216	. . . . . . . . . 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 742	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
120		simp-5r 797	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
121	8	ad3antrrr 742	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
122		simpr 489	. . . . . . . . . . . . . . . 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 22889	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
124	120, 121, 122, 123	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
125	124	ralrimiva 3163	. . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
127	126	ad2antrr 738	. . . . . . . . . . . . . . 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 22891	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑥 decompPMat 𝑘)) finSupp (0_g‘𝐴))
129	127, 128	syl 18	. . . . . . . . . . . . . 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 13644	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ₀)
131	130	adantl 486	. . . . . . . . . . . . . 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 22433	. . . . . . . . . . . . 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 7456	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘))
134		csbvarg 4397	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌𝑘 = 𝑙)
135	134	oveq2d 7424	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘) = (𝑥 decompPMat 𝑙))
136	133, 135	eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
137	136	adantl 486	. . . . . . . . . . . . 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 2805	. . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . 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 22889	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
141	120, 139, 122, 140	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
142	141	ralrimiva 3163	. . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
144	143	ad2antrr 738	. . . . . . . . . . . . . . 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 22891	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑦 decompPMat 𝑘)) finSupp (0_g‘𝐴))
146	144, 145	syl 18	. . . . . . . . . . . . . 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 13580	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ₀)
148	147	adantl 486	. . . . . . . . . . . . . 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 22433	. . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 𝑙) ∈ V
151		csbov2g 7456	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 𝑙) ∈ V → ⦋(𝑛 − 𝑙) / 𝑘⦌(𝑦 decompPMat 𝑘) = (𝑦 decompPMat ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘))
152	150, 151	mp1i 14	. . . . . . . . . . . . 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 4397	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 𝑙) ∈ V → ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘 = (𝑛 − 𝑙))
154	150, 153	mp1i 14	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘 = (𝑛 − 𝑙))
155	154	oveq2d 7424	. . . . . . . . . . . . 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 2809	. . . . . . . . . . . 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 7426	. . . . . . . . . . 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 5205	. . . . . . . . . 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 2816	. . . . . . . . 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 7424	. . . . . . . 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 2815	. . . . . . 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 2808	. . . . . 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 3163	. . . . 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 485	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring)
165	84	adantr 485	. . . . . . 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 22376	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
168	29, 167	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
169	168	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑄 ∈ LMod)
170	34	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
171		fzfid 14005	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
172	29	ad3antrrr 742	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
173		simp-4r 795	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
174		simplrl 788	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
175	174	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
176	40	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ₀)
177	173, 175, 176, 42	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
178		simplrr 789	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑦 ∈ 𝐵)
179	178	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
180	45	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘 − 𝑧) ∈ ℕ₀)
181	173, 179, 180, 47	syl3anc 1396	. . . . . . . . . . . . 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 1396	. . . . . . . . . . . 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 3163	. . . . . . . . . . 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 20033	. . . . . . . . . 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 738	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
186	19	ply1sca 22377	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
187	185, 186	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 = (Scalar‘𝑄))
188	187	eqcomd 2775	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Scalar‘𝑄) = 𝐴)
189	188	fveq2d 6883	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
190	184, 189	eleqtrrd 2872	. . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
192	19, 17, 191, 16, 28	ply1moncl 22397	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
193	185, 192	sylancom 599	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
194		eqid 2769	. . . . . . . . . 10 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
195		eqid 2769	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
196	28, 194, 15, 195	lmodvscl 20973	. . . . . . . . 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 1396	. . . . . . . 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 7108	. . . . . . 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 22940	. . . . . . 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 19981	. . . . . 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 485	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ Ring)
202	90, 92	sylan 591	. . . . . . . . 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 7108	. . . . . . . 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 18	. . . . . . . 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 19981	. . . . . . 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 591	. . . . . . . . 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 7108	. . . . . . . 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 1396	. . . . . . . 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 19981	. . . . . . 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 20328	. . . . . . 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 1396	. . . . . 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 2769	. . . . . . 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 2769	. . . . . . 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 22425	. . . . . 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 1396	. . . . 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 235	. . . 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 2808	. . 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 22918	. . . . 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 1396	. . . 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 22918	. . . . 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 1396	. . . 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 7426	. . 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 2807	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)))
224	223	ralrimivva 3214	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 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 Vcvv 3463 ⦋csb 3861 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6533 (class class class)co 7408 Fincfn 8939 finSupp cfsupp 9317 0cc0 11096 − cmin 11437 ℕ₀cn0 12500 ...cfz 13531 Basecbs 17265 ._rcmulr 17307 Scalarcsca 17309 ·_𝑠 cvsca 17310 0_gc0g 17488 Σ_g cgsu 17489 ._gcmg 19129 CMndccmn 19846 mulGrpcmgp 20212 Ringcrg 20311 LModclmod 20955 var₁cv1 22301 Poly₁cpl1 22302 coe₁cco1 22303 Mat cmat 22529 decompPMat cdecpmat 22884 pMatToMatPoly cpm2mp 22914

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-ofr 7673 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-oi 9468 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-fz 13532 df-fzo 13679 df-seq 14034 df-hash 14363 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-hom 17330 df-cco 17331 df-0g 17490 df-gsum 17491 df-prds 17496 df-pws 17498 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-mhm 18837 df-submnd 18838 df-grp 18999 df-minusg 19000 df-sbg 19001 df-mulg 19130 df-subg 19185 df-ghm 19280 df-cntz 19383 df-cmn 19848 df-abl 19849 df-mgp 20213 df-rng 20227 df-ur 20260 df-srg 20265 df-ring 20313 df-subrng 20627 df-subrg 20651 df-lmod 20957 df-lss 21027 df-sra 21268 df-rgmod 21269 df-dsmm 21847 df-frlm 21862 df-psr 22024 df-mvr 22025 df-mpl 22026 df-opsr 22028 df-psr1 22305 df-vr1 22306 df-ply1 22307 df-coe1 22308 df-mamu 22513 df-mat 22530 df-decpmat 22885 df-pm2mp 22915

This theorem is referenced by: pm2mpmhm 22942

W3C validator