deg1gprod - Mathbox for metakunt

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Theorem deg1gprod 42333

Description: Degree multiplication is a homomorphism. (Contributed by metakunt, 6-May-2025.)

**Hypotheses**
Ref	Expression
deg1gprod.1	⊢ (𝜑 → 𝑅 ∈ IDomn)
deg1gprod.2	⊢ (𝜑 → 𝑁 ∈ Fin)
deg1gprod.3	⊢ (𝜑 → ∀𝑥 ∈ 𝑁 (𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ 𝐶 ≠ (0_g‘(Poly₁‘𝑅))))

**Assertion**
Ref	Expression
deg1gprod	⊢ (𝜑 → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶))) = Σ𝑛 ∈ 𝑁 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶)))))

Distinct variable groups: 𝐶,𝑛 𝑛,𝑁,𝑥 𝑅,𝑛,𝑥 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥)

Proof of Theorem deg1gprod Dummy variables 𝑎 𝑏 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq1 5185	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ ∅ ↦ 𝐶))
2	1	oveq2d 7372	. . . . 5 ⊢ (𝑎 = ∅ → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)))
3	2	fveq2d 6836	. . . 4 ⊢ (𝑎 = ∅ → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
4		sumeq1 15610	. . . 4 ⊢ (𝑎 = ∅ → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
5	3, 4	eqeq12d 2750	. . 3 ⊢ (𝑎 = ∅ → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ↔ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛))))
6	3	breq2d 5108	. . 3 ⊢ (𝑎 = ∅ → (0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) ↔ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)))))
7	5, 6	anbi12d 632	. 2 ⊢ (𝑎 = ∅ → ((((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)))) ↔ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))))
8		mpteq1 5185	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑏 ↦ 𝐶))
9	8	oveq2d 7372	. . . . 5 ⊢ (𝑎 = 𝑏 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))
10	9	fveq2d 6836	. . . 4 ⊢ (𝑎 = 𝑏 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))
11		sumeq1 15610	. . . 4 ⊢ (𝑎 = 𝑏 → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
12	10, 11	eqeq12d 2750	. . 3 ⊢ (𝑎 = 𝑏 → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ↔ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛))))
13	10	breq2d 5108	. . 3 ⊢ (𝑎 = 𝑏 → (0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) ↔ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))))
14	12, 13	anbi12d 632	. 2 ⊢ (𝑎 = 𝑏 → ((((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)))) ↔ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))))
15		mpteq1 5185	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))
16	15	oveq2d 7372	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)))
17	16	fveq2d 6836	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))))
18		sumeq1 15610	. . . 4 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
19	17, 18	eqeq12d 2750	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ↔ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛))))
20	17	breq2d 5108	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) ↔ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)))))
21	19, 20	anbi12d 632	. 2 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)))) ↔ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))))))
22		mpteq1 5185	. . . . . 6 ⊢ (𝑎 = 𝑁 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶))
23	22	oveq2d 7372	. . . . 5 ⊢ (𝑎 = 𝑁 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶)))
24	23	fveq2d 6836	. . . 4 ⊢ (𝑎 = 𝑁 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶))))
25		sumeq1 15610	. . . 4 ⊢ (𝑎 = 𝑁 → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑁 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
26	24, 25	eqeq12d 2750	. . 3 ⊢ (𝑎 = 𝑁 → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ↔ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶))) = Σ𝑛 ∈ 𝑁 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛))))
27	24	breq2d 5108	. . 3 ⊢ (𝑎 = 𝑁 → (0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) ↔ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶)))))
28	26, 27	anbi12d 632	. 2 ⊢ (𝑎 = 𝑁 → ((((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)))) ↔ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶))) = Σ𝑛 ∈ 𝑁 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶))))))
29		mpt0 6632	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ 𝐶) = ∅
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ∅ ↦ 𝐶) = ∅)
31	30	oveq2d 7372	. . . . . . 7 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g ∅))
32		eqid 2734	. . . . . . . . 9 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
33	32	gsum0 18607	. . . . . . . 8 ⊢ ((mulGrp‘(Poly₁‘𝑅)) Σ_g ∅) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
34	33	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g ∅) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
35	31, 34	eqtrd 2769	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
36	35	fveq2d 6836	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = ((deg₁‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))))
37		deg1gprod.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ IDomn)
38	37	idomringd 20659	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
39		eqid 2734	. . . . . . . . . 10 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
40		eqid 2734	. . . . . . . . . 10 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
41		eqid 2734	. . . . . . . . . 10 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
42		eqid 2734	. . . . . . . . . . . 12 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
43		eqid 2734	. . . . . . . . . . . 12 ⊢ (1_r‘(Poly₁‘𝑅)) = (1_r‘(Poly₁‘𝑅))
44	42, 43	ringidval 20116	. . . . . . . . . . 11 ⊢ (1_r‘(Poly₁‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
45	44	eqcomi 2743	. . . . . . . . . 10 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (1_r‘(Poly₁‘𝑅))
46	39, 40, 41, 45	ply1scl1 22233	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
47	38, 46	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
48	47	eqcomd 2740	. . . . . . 7 ⊢ (𝜑 → (0_g‘(mulGrp‘(Poly₁‘𝑅))) = ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)))
49	48	fveq2d 6836	. . . . . 6 ⊢ (𝜑 → ((deg₁‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))))
50		eqid 2734	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
51	50, 41	ringidcl 20198	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
52	38, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Base‘𝑅))
53	37	idomdomd 20657	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Domn)
54		domnnzr 20637	. . . . . . . . 9 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
55	53, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ NzRing)
56		eqid 2734	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
57	41, 56	nzrnz 20446	. . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1_r‘𝑅) ≠ (0_g‘𝑅))
58	55, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ≠ (0_g‘𝑅))
59		eqid 2734	. . . . . . . 8 ⊢ (deg₁‘𝑅) = (deg₁‘𝑅)
60	59, 39, 50, 40, 56	deg1scl 26072	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1_r‘𝑅) ∈ (Base‘𝑅) ∧ (1_r‘𝑅) ≠ (0_g‘𝑅)) → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))) = 0)
61	38, 52, 58, 60	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))) = 0)
62	49, 61	eqtrd 2769	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = 0)
63	36, 62	eqtrd 2769	. . . 4 ⊢ (𝜑 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = 0)
64		sum0 15642	. . . . . 6 ⊢ Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = 0
65	64	eqcomi 2743	. . . . 5 ⊢ 0 = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛))
66	65	a1i 11	. . . 4 ⊢ (𝜑 → 0 = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
67	63, 66	eqtrd 2769	. . 3 ⊢ (𝜑 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
68		0red 11133	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
69	68	leidd 11701	. . . 4 ⊢ (𝜑 → 0 ≤ 0)
70	63	eqcomd 2740	. . . 4 ⊢ (𝜑 → 0 = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
71	69, 70	breqtrd 5122	. . 3 ⊢ (𝜑 → 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
72	67, 71	jca 511	. 2 ⊢ (𝜑 → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)))))
73		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐶
74		nfcsb1v 3871	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
75		csbeq1a 3861	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
76	73, 74, 75	cbvmpt 5198	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶) = (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶)
77	76	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶) = (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶))
78	77	oveq2d 7372	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶)))
79	78	fveq2d 6836	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶))))
80		eqid 2734	. . . . . . . 8 ⊢ (Base‘(mulGrp‘(Poly₁‘𝑅))) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
81		eqid 2734	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
82		isidom 20656	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
83	37, 82	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
84	83	simpld 494	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ CRing)
85	39	ply1crng 22137	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (Poly₁‘𝑅) ∈ CRing)
86	84, 85	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑅) ∈ CRing)
87	42	crngmgp 20174	. . . . . . . . . . 11 ⊢ ((Poly₁‘𝑅) ∈ CRing → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
89	88	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
90	89	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
91		deg1gprod.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ Fin)
92	91	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → 𝑁 ∈ Fin)
93		simplrl 776	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → 𝑏 ⊆ 𝑁)
94	92, 93	ssfid 9167	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → 𝑏 ∈ Fin)
95	93	sselda 3931	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) ∧ 𝑦 ∈ 𝑏) → 𝑦 ∈ 𝑁)
96		deg1gprod.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 (𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ 𝐶 ≠ (0_g‘(Poly₁‘𝑅))))
97		r19.26 3094	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝑁 (𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) ↔ (∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))))
98	97	biimpi 216	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ 𝑁 (𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → (∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))))
99	96, 98	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))))
100	99	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
101	100	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) ∧ 𝑦 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
102		rspcsbela 4388	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
103	95, 101, 102	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) ∧ 𝑦 ∈ 𝑏) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
104		eqid 2734	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
105	42, 104	mgpbas 20078	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
106	103, 105	eleqtrdi 2844	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) ∧ 𝑦 ∈ 𝑏) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(mulGrp‘(Poly₁‘𝑅))))
107		eldifi 4081	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝑁 ∖ 𝑏) → 𝑐 ∈ 𝑁)
108	107	adantl 481	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)) → 𝑐 ∈ 𝑁)
109	108	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑐 ∈ 𝑁)
110	109	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → 𝑐 ∈ 𝑁)
111		eldifn 4082	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝑁 ∖ 𝑏) → ¬ 𝑐 ∈ 𝑏)
112	111	adantl 481	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)) → ¬ 𝑐 ∈ 𝑏)
113	112	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ¬ 𝑐 ∈ 𝑏)
114	113	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ¬ 𝑐 ∈ 𝑏)
115	100	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
116		rspcsbela 4388	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
117	110, 115, 116	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
118	117, 105	eleqtrdi 2844	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(mulGrp‘(Poly₁‘𝑅))))
119		csbeq1 3850	. . . . . . . 8 ⊢ (𝑦 = 𝑐 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑐 / 𝑥⦌𝐶)
120	80, 81, 90, 94, 106, 110, 114, 118, 119	gsumunsn 19887	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶)) = (((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))(+_g‘(mulGrp‘(Poly₁‘𝑅)))⦋𝑐 / 𝑥⦌𝐶))
121	120	fveq2d 6836	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶))) = ((deg₁‘𝑅)‘(((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))(+_g‘(mulGrp‘(Poly₁‘𝑅)))⦋𝑐 / 𝑥⦌𝐶)))
122		eqid 2734	. . . . . . . . . 10 ⊢ (._r‘(Poly₁‘𝑅)) = (._r‘(Poly₁‘𝑅))
123	42, 122	mgpplusg 20077	. . . . . . . . 9 ⊢ (._r‘(Poly₁‘𝑅)) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
124	123	eqcomi 2743	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (._r‘(Poly₁‘𝑅))
125		eqid 2734	. . . . . . . 8 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
126	53	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑅 ∈ Domn)
127	126	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → 𝑅 ∈ Domn)
128	103	ralrimiva 3126	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ∀𝑦 ∈ 𝑏 ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
129	105, 90, 94, 128	gsummptcl 19894	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶)) ∈ (Base‘(Poly₁‘𝑅)))
130	39	ply1idom 26084	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ IDomn → (Poly₁‘𝑅) ∈ IDomn)
131	37, 130	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑅) ∈ IDomn)
132	131	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (Poly₁‘𝑅) ∈ IDomn)
133	132	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (Poly₁‘𝑅) ∈ IDomn)
134	99	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
135	134	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) ∧ 𝑦 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
136		rspcsbnea 42324	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑦 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
137	95, 135, 136	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) ∧ 𝑦 ∈ 𝑏) → ⦋𝑦 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
138	42, 133, 94, 103, 137	idomnnzgmulnz 42326	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶)) ≠ (0_g‘(Poly₁‘𝑅)))
139	134	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
140		rspcsbnea 42324	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
141	110, 139, 140	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
142	59, 39, 104, 124, 125, 127, 129, 138, 117, 141	deg1mul 26074	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘(((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))(+_g‘(mulGrp‘(Poly₁‘𝑅)))⦋𝑐 / 𝑥⦌𝐶)) = (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)))
143	73, 74, 75	cbvmpt 5198	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑏 ↦ 𝐶) = (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶)
144	143	eqcomi 2743	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑥 ∈ 𝑏 ↦ 𝐶)
145	144	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑥 ∈ 𝑏 ↦ 𝐶))
146	145	oveq2d 7372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))
147	146	fveq2d 6836	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))
148	147	oveq1d 7371	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)) = (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)))
149		simpl 482	. . . . . . . . . . 11 ⊢ ((((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
150	149	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
151	150	oveq1d 7371	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)) = (Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)))
152		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)))
153		nfcv 2896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐))
154	91	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑁 ∈ Fin)
155		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑏 ⊆ 𝑁)
156	154, 155	ssfid 9167	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑏 ∈ Fin)
157	73, 74, 75	cbvmpt 5198	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)
158	157	fveq1i 6833	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛) = ((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛)
159	158	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛) = ((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛))
160	159	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = ((deg₁‘𝑅)‘((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛)))
161		eqidd 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶))
162		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) ∧ 𝑦 = 𝑛) → 𝑦 = 𝑛)
163	162	csbeq1d 3851	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) ∧ 𝑦 = 𝑛) → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑛 / 𝑥⦌𝐶)
164	155	sselda 3931	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → 𝑛 ∈ 𝑁)
165	100	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
166	165	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
167		rspcsbela 4388	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑛 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
168	164, 166, 167	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ⦋𝑛 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
169	161, 163, 164, 168	fvmptd 6946	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛) = ⦋𝑛 / 𝑥⦌𝐶)
170	169	fveq2d 6836	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛)) = ((deg₁‘𝑅)‘⦋𝑛 / 𝑥⦌𝐶))
171	38	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑅 ∈ Ring)
172	171	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → 𝑅 ∈ Ring)
173	134	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
174		rspcsbnea 42324	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑛 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
175	164, 173, 174	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ⦋𝑛 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
176	59, 39, 125, 104	deg1nn0cl 26047	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ ⦋𝑛 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ ⦋𝑛 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ((deg₁‘𝑅)‘⦋𝑛 / 𝑥⦌𝐶) ∈ ℕ₀)
177	172, 168, 175, 176	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘⦋𝑛 / 𝑥⦌𝐶) ∈ ℕ₀)
178	170, 177	eqeltrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛)) ∈ ℕ₀)
179	160, 178	eqeltrd 2834	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℕ₀)
180	179	nn0cnd 12462	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℂ)
181		2fveq3 6837	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)))
182	109, 165, 116	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
183		eqid 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶)
184	183	fvmpts 6942	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑁 ∧ ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐) = ⦋𝑐 / 𝑥⦌𝐶)
185	109, 182, 184	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐) = ⦋𝑐 / 𝑥⦌𝐶)
186	185	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) = ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶))
187	108, 134, 140	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
188	59, 39, 125, 104	deg1nn0cl 26047	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶) ∈ ℕ₀)
189	171, 182, 187, 188	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶) ∈ ℕ₀)
190	186, 189	eqeltrd 2834	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℕ₀)
191	190	nn0cnd 12462	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℂ)
192	152, 153, 156, 109, 113, 180, 181, 191	fsumsplitsn 15665	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = (Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) + ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐))))
193	192	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = (Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) + ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐))))
194	185	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐) = ⦋𝑐 / 𝑥⦌𝐶)
195	194	fveq2d 6836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) = ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶))
196	195	oveq2d 7372	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) + ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐))) = (Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)))
197	193, 196	eqtrd 2769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = (Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)))
198	197	eqcomd 2740	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
199	151, 198	eqtrd 2769	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
200	148, 199	eqtrd 2769	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))) + ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶)) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
201	142, 200	eqtrd 2769	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘(((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶))(+_g‘(mulGrp‘(Poly₁‘𝑅)))⦋𝑐 / 𝑥⦌𝐶)) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
202	121, 201	eqtrd 2769	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶))) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
203	79, 202	eqtrd 2769	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
204	171	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → 𝑅 ∈ Ring)
205	110	snssd 4763	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → {𝑐} ⊆ 𝑁)
206	93, 205	unssd 4142	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (𝑏 ∪ {𝑐}) ⊆ 𝑁)
207	92, 206	ssfid 9167	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (𝑏 ∪ {𝑐}) ∈ Fin)
208	165	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
209		ssralv 4000	. . . . . . . . 9 ⊢ ((𝑏 ∪ {𝑐}) ⊆ 𝑁 → (∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)) → ∀𝑥 ∈ (𝑏 ∪ {𝑐})𝐶 ∈ (Base‘(Poly₁‘𝑅))))
210	206, 209	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)) → ∀𝑥 ∈ (𝑏 ∪ {𝑐})𝐶 ∈ (Base‘(Poly₁‘𝑅))))
211	208, 210	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ∀𝑥 ∈ (𝑏 ∪ {𝑐})𝐶 ∈ (Base‘(Poly₁‘𝑅)))
212	105, 90, 207, 211	gsummptcl 19894	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) ∈ (Base‘(Poly₁‘𝑅)))
213	76	oveq2i 7367	. . . . . . . . 9 ⊢ ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶))
214	213	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶)))
215	109	snssd 4763	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → {𝑐} ⊆ 𝑁)
216	155, 215	unssd 4142	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (𝑏 ∪ {𝑐}) ⊆ 𝑁)
217	154, 216	ssfid 9167	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (𝑏 ∪ {𝑐}) ∈ Fin)
218	216	sselda 3931	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → 𝑦 ∈ 𝑁)
219	165	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
220	218, 219, 102	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
221	134	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
222	218, 221, 136	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑦 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
223	42, 132, 217, 220, 222	idomnnzgmulnz 42326	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑦 ∈ (𝑏 ∪ {𝑐}) ↦ ⦋𝑦 / 𝑥⦌𝐶)) ≠ (0_g‘(Poly₁‘𝑅)))
224	214, 223	eqnetrd 2997	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) ≠ (0_g‘(Poly₁‘𝑅)))
225	224	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) ≠ (0_g‘(Poly₁‘𝑅)))
226	59, 39, 125, 104	deg1nn0cl 26047	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) ∈ (Base‘(Poly₁‘𝑅)) ∧ ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) ≠ (0_g‘(Poly₁‘𝑅))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) ∈ ℕ₀)
227	204, 212, 225, 226	syl3anc 1373	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) ∈ ℕ₀)
228	227	nn0ge0d 12463	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))))
229	203, 228	jca 511	. . 3 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))))) → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)))))
230	229	ex 412	. 2 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶))) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))) → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))) = Σ𝑛 ∈ (𝑏 ∪ {𝑐})((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))))))
231	7, 14, 21, 28, 72, 230, 91	findcard2d 9089	1 ⊢ (𝜑 → (((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶))) = Σ𝑛 ∈ 𝑁 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∧ 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∀wral 3049 ⦋csb 3847 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 ∅c0 4283 {csn 4578 class class class wbr 5096 ↦ cmpt 5177 ‘cfv 6490 (class class class)co 7356 Fincfn 8881 0cc0 11024 + caddc 11027 ≤ cle 11165 ℕ₀cn0 12399 Σcsu 15607 Basecbs 17134 +_gcplusg 17175 ._rcmulr 17176 0_gc0g 17357 Σ_g cgsu 17358 CMndccmn 19707 mulGrpcmgp 20073 1_rcur 20114 Ringcrg 20166 CRingccrg 20167 NzRingcnzr 20443 Domncdomn 20623 IDomncidom 20624 algSccascl 21805 Poly₁cpl1 22115 deg₁cdg1 26013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-sup 9343 df-oi 9413 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-rp 12904 df-fz 13422 df-fzo 13569 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-clim 15409 df-sum 15608 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-0g 17359 df-gsum 17360 df-prds 17365 df-pws 17367 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18706 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18996 df-subg 19051 df-ghm 19140 df-cntz 19244 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-nzr 20444 df-subrng 20477 df-subrg 20501 df-rlreg 20625 df-domn 20626 df-idom 20627 df-lmod 20811 df-lss 20881 df-cnfld 21308 df-ascl 21808 df-psr 21863 df-mvr 21864 df-mpl 21865 df-opsr 21867 df-psr1 22118 df-vr1 22119 df-ply1 22120 df-coe1 22121 df-mdeg 26014 df-deg1 26015

This theorem is referenced by: aks6d1c6lem1 42363

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