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Theorem pm2mpmhmlem2 20624

Description: Lemma 2 for pm2mpmhm 20625. (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 790	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑁 ∈ Fin)
2		simplr 792	. . . . 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 20498	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
6	5	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ Ring)
7		simpl 473	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	adantl 482	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
9		simpr 477	. . . . . . 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 2622	. . . . . . 7 ⊢ (._r‘𝐶) = (._r‘𝐶)
13	11, 12	ringcl 18561	. . . . . 6 ⊢ ((𝐶 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
14	6, 8, 10, 13	syl3anc 1326	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(._r‘𝐶)𝑦) ∈ 𝐵)
15		eqid 2622	. . . . . 6 ⊢ ( ·_𝑠 ‘𝑄) = ( ·_𝑠 ‘𝑄)
16		eqid 2622	. . . . . 6 ⊢ (._g‘(mulGrp‘𝑄)) = (._g‘(mulGrp‘𝑄))
17		eqid 2622	. . . . . 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 20601	. . . . 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 1326	. . . 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		simpllr 799	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
24		simplr 792	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
25		simpr 477	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
26	3, 4, 11, 18	decpmatmul 20577	. . . . . . . 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))))))
27	23, 24, 25, 26	syl3anc 1326	. . . . . . 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))))))
28	27	oveq1d 6665	. . . . . 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))))
29	28	mpteq2dva 4744	. . . . 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)))))
30	29	oveq2d 6666	. . . 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))))))
31		eqid 2622	. . . . . . . 8 ⊢ (Base‘𝑄) = (Base‘𝑄)
32	18	matring 20249	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
33	32	ad2antrr 762	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝐴 ∈ Ring)
34		eqid 2622	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘𝐴)
35		eqid 2622	. . . . . . . 8 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
36		ringcmn 18581	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
37	32, 36	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ CMnd)
38	37	ad3antrrr 766	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
39		fzfid 12772	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
40	33	ad2antrr 762	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
41		simp-5r 809	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
42	8	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
43		elfznn0 12433	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → 𝑧 ∈ ℕ₀)
44	43	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ₀)
45	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ ℕ₀) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
46	41, 42, 44, 45	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
47	10	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
48		fznn0sub 12373	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (0...𝑘) → (𝑘 − 𝑧) ∈ ℕ₀)
49	48	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘 − 𝑧) ∈ ℕ₀)
50	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ (𝑘 − 𝑧) ∈ ℕ₀) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
51	41, 47, 49, 50	syl3anc 1326	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
52		eqid 2622	. . . . . . . . . . . . 13 ⊢ (._r‘𝐴) = (._r‘𝐴)
53	34, 52	ringcl 18561	. . . . . . . . . . . 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))
54	40, 46, 51, 53	syl3anc 1326	. . . . . . . . . . 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))
55	54	ralrimiva 2966	. . . . . . . . . 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))
56	34, 38, 39, 55	gsummptcl 18366	. . . . . . . . 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))
57	56	ralrimiva 2966	. . . . . . . 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))
58	3, 4, 11, 18, 52, 35	decpmatmulsumfsupp 20578	. . . . . . . . 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))
59	58	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))
60		simpr 477	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
61	19, 31, 17, 16, 33, 34, 15, 35, 57, 59, 60	gsummoncoe1 19674	. . . . . . 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))))))
62		csbov2g 6691	. . . . . . . . 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))))))
63		id 22	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℕ₀)
64		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (0...𝑘) = (0...𝑛))
65		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (𝑘 − 𝑧) = (𝑛 − 𝑧))
66	65	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → (𝑦 decompPMat (𝑘 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑧)))
67	66	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑘 − 𝑧))) = ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))))
68	64, 67	mpteq12dv 4733	. . . . . . . . . . . 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)))))
69	68	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)))))
70	63, 69	csbied 3560	. . . . . . . . . 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)))))
71	70	oveq2d 6666	. . . . . . . . 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))))))
72	62, 71	eqtrd 2656	. . . . . . . 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))))))
73	72	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))))))
74		eqidd 2623	. . . . . . . . 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)))))))
75		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑛 → (0...𝑟) = (0...𝑛))
76		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑛 → (𝑟 − 𝑙) = (𝑛 − 𝑙))
77	76	fveq2d 6195	. . . . . . . . . . . . 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)))
78	77	oveq2d 6666	. . . . . . . . . . . 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))))
79	75, 78	mpteq12dv 4733	. . . . . . . . . . 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)))))
80	79	oveq2d 6666	. . . . . . . . . 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))))))
81	80	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))))))
82		ovexd 6680	. . . . . . . . 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)
83	74, 81, 60, 82	fvmptd 6288	. . . . . . . 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))))))
84		eqid 2622	. . . . . . . . . 10 ⊢ (0_g‘𝑄) = (0_g‘𝑄)
85	19	ply1ring 19618	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
86	32, 85	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
87		ringcmn 18581	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ CMnd)
88	86, 87	syl 17	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ CMnd)
89	88	ad2antrr 762	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → 𝑄 ∈ CMnd)
90		nn0ex 11298	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
91	90	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → ℕ₀ ∈ V)
92	7	anim2i 593	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
93		df-3an 1039	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑥 ∈ 𝐵))
94	92, 93	sylibr 224	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
95	94	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
96	3, 4, 11, 15, 16, 17, 18, 19, 31	pm2mpghmlem1 20618	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
97	95, 96	sylan 488	. . . . . . . . . . 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))
98		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) = (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))
99	97, 98	fmptd 6385	. . . . . . . . . 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))
100	3, 4, 11, 15, 16, 17, 18, 19	pm2mpghmlem2 20617	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
101	95, 100	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))
102	31, 84, 89, 91, 99, 101	gsumcl 18316	. . . . . . . . 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))
103	9	anim2i 593	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
104		df-3an 1039	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑦 ∈ 𝐵))
105	103, 104	sylibr 224	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
106	105	adantr 481	. . . . . . . . . . . 12 ⊢ ((((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, 31	pm2mpghmlem1 20618	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) ∧ 𝑘 ∈ ℕ₀) → ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
108	106, 107	sylan 488	. . . . . . . . . . 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))
109		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) = (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))))
110	108, 109	fmptd 6385	. . . . . . . . . 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))
111	1, 2, 10	3jca 1242	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
112	111	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
113	3, 4, 11, 15, 16, 17, 18, 19	pm2mpghmlem2 20617	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)))) finSupp (0_g‘𝑄))
114	112, 113	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))
115	31, 84, 89, 91, 110, 114	gsumcl 18316	. . . . . . . . 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))
116		eqid 2622	. . . . . . . . . . 11 ⊢ (._r‘𝑄) = (._r‘𝑄)
117	19, 116, 52, 31	coe1mul 19640	. . . . . . . . . 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)))))))
118	117	fveq1d 6193	. . . . . . . . 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))
119	33, 102, 115, 118	syl3anc 1326	. . . . . . . 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))
120		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑥 decompPMat 𝑧) = (𝑥 decompPMat 𝑙))
121		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑙 → (𝑛 − 𝑧) = (𝑛 − 𝑙))
122	121	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑙 → (𝑦 decompPMat (𝑛 − 𝑧)) = (𝑦 decompPMat (𝑛 − 𝑙)))
123	120, 122	oveq12d 6668	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑙 → ((𝑥 decompPMat 𝑧)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑧))) = ((𝑥 decompPMat 𝑙)(._r‘𝐴)(𝑦 decompPMat (𝑛 − 𝑙))))
124	123	cbvmptv 4750	. . . . . . . . . 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))))
125	32	ad3antrrr 766	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → 𝐴 ∈ Ring)
126		simp-5r 809	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑅 ∈ Ring)
127	8	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
128		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑘 ∈ ℕ₀)
129	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
130	126, 127, 128, 129	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
131	130	ralrimiva 2966	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ₀ (𝑥 decompPMat 𝑘) ∈ (Base‘𝐴))
132	2, 8	jca 554	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
133	132	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵))
134	3, 4, 11, 18, 35	decpmatfsupp 20574	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑥 decompPMat 𝑘)) finSupp (0_g‘𝐴))
135	133, 134	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))
136		elfznn0 12433	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → 𝑙 ∈ ℕ₀)
137	136	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → 𝑙 ∈ ℕ₀)
138	19, 31, 17, 16, 125, 34, 15, 35, 131, 135, 137	gsummoncoe1 19674	. . . . . . . . . . . . 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))
139		csbov2g 6691	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘))
140		csbvarg 4003	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌𝑘 = 𝑙)
141	140	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑥 decompPMat ⦋𝑙 / 𝑘⦌𝑘) = (𝑥 decompPMat 𝑙))
142	139, 141	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (𝑙 ∈ (0...𝑛) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
143	142	adantl 482	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ⦋𝑙 / 𝑘⦌(𝑥 decompPMat 𝑘) = (𝑥 decompPMat 𝑙))
144	138, 143	eqtr2d 2657	. . . . . . . . . . . 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))
145	10	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → 𝑦 ∈ 𝐵)
146	3, 4, 11, 18, 34	decpmatcl 20572	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵 ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
147	126, 145, 128, 146	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) ∧ 𝑘 ∈ ℕ₀) → (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
148	147	ralrimiva 2966	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ∀𝑘 ∈ ℕ₀ (𝑦 decompPMat 𝑘) ∈ (Base‘𝐴))
149	2, 10	jca 554	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
150	149	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵))
151	3, 4, 11, 18, 35	decpmatfsupp 20574	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ (𝑦 decompPMat 𝑘)) finSupp (0_g‘𝐴))
152	150, 151	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))
153		fznn0sub 12373	. . . . . . . . . . . . . . 15 ⊢ (𝑙 ∈ (0...𝑛) → (𝑛 − 𝑙) ∈ ℕ₀)
154	153	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑛 − 𝑙) ∈ ℕ₀)
155	19, 31, 17, 16, 125, 34, 15, 35, 148, 152, 154	gsummoncoe1 19674	. . . . . . . . . . . . 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))
156		ovex 6678	. . . . . . . . . . . . . 14 ⊢ (𝑛 − 𝑙) ∈ V
157		csbov2g 6691	. . . . . . . . . . . . . 14 ⊢ ((𝑛 − 𝑙) ∈ V → ⦋(𝑛 − 𝑙) / 𝑘⦌(𝑦 decompPMat 𝑘) = (𝑦 decompPMat ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘))
158	156, 157	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))
159		csbvarg 4003	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 − 𝑙) ∈ V → ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘 = (𝑛 − 𝑙))
160	156, 159	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘 = (𝑛 − 𝑙))
161	160	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑛 ∈ ℕ₀) ∧ 𝑙 ∈ (0...𝑛)) → (𝑦 decompPMat ⦋(𝑛 − 𝑙) / 𝑘⦌𝑘) = (𝑦 decompPMat (𝑛 − 𝑙)))
162	155, 158, 161	3eqtrrd 2661	. . . . . . . . . . . 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)))
163	144, 162	oveq12d 6668	. . . . . . . . . . 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))))
164	163	mpteq2dva 4744	. . . . . . . . . 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)))))
165	124, 164	syl5eq 2668	. . . . . . . . 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)))))
166	165	oveq2d 6666	. . . . . . . 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))))))
167	83, 119, 166	3eqtr4rd 2667	. . . . . . 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))
168	61, 73, 167	3eqtrd 2660	. . . . . 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))
169	168	ralrimiva 2966	. . . . 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))
170	32	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐴 ∈ Ring)
171	88	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ CMnd)
172	90	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ℕ₀ ∈ V)
173	19	ply1lmod 19622	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
174	32, 173	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
175	174	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑄 ∈ LMod)
176	37	ad2antrr 762	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ CMnd)
177		fzfid 12772	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (0...𝑘) ∈ Fin)
178	32	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝐴 ∈ Ring)
179		simp-4r 807	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑅 ∈ Ring)
180		simplrl 800	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑥 ∈ 𝐵)
181	180	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑥 ∈ 𝐵)
182	43	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑧 ∈ ℕ₀)
183	179, 181, 182, 45	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑥 decompPMat 𝑧) ∈ (Base‘𝐴))
184		simplrr 801	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝑦 ∈ 𝐵)
185	184	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → 𝑦 ∈ 𝐵)
186	48	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑘 − 𝑧) ∈ ℕ₀)
187	179, 185, 186, 50	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) ∧ 𝑧 ∈ (0...𝑘)) → (𝑦 decompPMat (𝑘 − 𝑧)) ∈ (Base‘𝐴))
188	178, 183, 187, 53	syl3anc 1326	. . . . . . . . . . . 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))
189	188	ralrimiva 2966	. . . . . . . . . . 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))
190	34, 176, 177, 189	gsummptcl 18366	. . . . . . . . . 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))
191	32	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 ∈ Ring)
192	19	ply1sca 19623	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
193	191, 192	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → 𝐴 = (Scalar‘𝑄))
194	193	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Scalar‘𝑄) = 𝐴)
195	194	fveq2d 6195	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (Base‘(Scalar‘𝑄)) = (Base‘𝐴))
196	190, 195	eleqtrrd 2704	. . . . . . . . 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)))
197		eqid 2622	. . . . . . . . . . 11 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
198	19, 17, 197, 16, 31	ply1moncl 19641	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Ring ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
199	191, 198	sylancom 701	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → (𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴)) ∈ (Base‘𝑄))
200		eqid 2622	. . . . . . . . . 10 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
201		eqid 2622	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
202	31, 200, 15, 201	lmodvscl 18880	. . . . . . . . 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))
203	175, 196, 199, 202	syl3anc 1326	. . . . . . . 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))
204		eqid 2622	. . . . . . . 8 ⊢ (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)))) = (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))))
205	203, 204	fmptd 6385	. . . . . . 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))
206	3, 4, 11, 15, 16, 17, 18, 19, 31, 20	pm2mpmhmlem1 20623	. . . . . . 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))
207	31, 84, 171, 172, 205, 206	gsumcl 18316	. . . . . 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))
208	86	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑄 ∈ Ring)
209	94, 96	sylan 488	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → ((𝑥 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
210	209, 98	fmptd 6385	. . . . . . . 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))
211	94, 100	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))
212	31, 84, 171, 172, 210, 211	gsumcl 18316	. . . . . . 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))
213	105, 107	sylan 488	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑘 ∈ ℕ₀) → ((𝑦 decompPMat 𝑘)( ·_𝑠 ‘𝑄)(𝑘(._g‘(mulGrp‘𝑄))(var₁‘𝐴))) ∈ (Base‘𝑄))
214	213, 109	fmptd 6385	. . . . . . . 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))
215	1, 2, 10, 113	syl3anc 1326	. . . . . . . 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))
216	31, 84, 171, 172, 214, 215	gsumcl 18316	. . . . . . 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))
217	31, 116	ringcl 18561	. . . . . . 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))
218	208, 212, 216, 217	syl3anc 1326	. . . . . 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))
219		eqid 2622	. . . . . . 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))))))
220		eqid 2622	. . . . . . 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)))))))
221	19, 31, 219, 220	ply1coe1eq 19668	. . . . . 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))))))))
222	170, 207, 218, 221	syl3anc 1326	. . . . 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))))))))
223	169, 222	mpbid 222	. . . 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)))))))
224	22, 30, 223	3eqtrd 2660	. . 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)))))))
225	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 20601	. . . . 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))))))
226	1, 2, 8, 225	syl3anc 1326	. . . 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))))))
227	3, 4, 11, 15, 16, 17, 18, 19, 20	pm2mpfval 20601	. . . . 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))))))
228	1, 2, 10, 227	syl3anc 1326	. . . 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))))))
229	226, 228	oveq12d 6668	. . 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)))))))
230	224, 229	eqtr4d 2659	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)))
231	230	ralrimivva 2971	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⦋csb 3533 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 finSupp cfsupp 8275 0cc0 9936 − cmin 10266 ℕ₀cn0 11292 ...cfz 12326 Basecbs 15857 ._rcmulr 15942 Scalarcsca 15944 ·_𝑠 cvsca 15945 0_gc0g 16100 Σ_g cgsu 16101 ._gcmg 17540 CMndccmn 18193 mulGrpcmgp 18489 Ringcrg 18547 LModclmod 18863 var₁cv1 19546 Poly₁cpl1 19547 coe₁cco1 19548 Mat cmat 20213 decompPMat cdecpmat 20567 pMatToMatPoly cpm2mp 20597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-srg 18506 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-decpmat 20568 df-pm2mp 20598

This theorem is referenced by: pm2mpmhm 20625

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