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Theorem deg1gprod 43009

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 5194	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ ∅ ↦ 𝐶))
2	1	oveq2d 7430	. . . . 5 ⊢ (𝑎 = ∅ → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)))
3	2	fveq2d 6883	. . . 4 ⊢ (𝑎 = ∅ → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
4		sumeq1 15779	. . . 4 ⊢ (𝑎 = ∅ → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
5	3, 4	eqeq12d 2776	. . 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 5115	. . 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 644	. 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 5194	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑏 ↦ 𝐶))
9	8	oveq2d 7430	. . . . 5 ⊢ (𝑎 = 𝑏 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))
10	9	fveq2d 6883	. . . 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 15779	. . . 4 ⊢ (𝑎 = 𝑏 → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
12	10, 11	eqeq12d 2776	. . 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 5115	. . 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 644	. 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 5194	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))
16	15	oveq2d 7430	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)))
17	16	fveq2d 6883	. . . 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 15779	. . . 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 2776	. . 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 5115	. . 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 644	. 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 5194	. . . . . 6 ⊢ (𝑎 = 𝑁 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶))
23	22	oveq2d 7430	. . . . 5 ⊢ (𝑎 = 𝑁 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶)))
24	23	fveq2d 6883	. . . 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 15779	. . . 4 ⊢ (𝑎 = 𝑁 → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑁 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
26	24, 25	eqeq12d 2776	. . 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 5115	. . 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 644	. 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 6675	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ 𝐶) = ∅
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ∅ ↦ 𝐶) = ∅)
31	30	oveq2d 7430	. . . . . . 7 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g ∅))
32		eqid 2760	. . . . . . . . 9 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
33	32	gsum0 18789	. . . . . . . 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 2795	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
36	35	fveq2d 6883	. . . . 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 20892	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
39		eqid 2760	. . . . . . . . . 10 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
40		eqid 2760	. . . . . . . . . 10 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
41		eqid 2760	. . . . . . . . . 10 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
42		eqid 2760	. . . . . . . . . . . 12 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
43		eqid 2760	. . . . . . . . . . . 12 ⊢ (1_r‘(Poly₁‘𝑅)) = (1_r‘(Poly₁‘𝑅))
44	42, 43	ringidval 20325	. . . . . . . . . . 11 ⊢ (1_r‘(Poly₁‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
45	44	eqcomi 2769	. . . . . . . . . 10 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (1_r‘(Poly₁‘𝑅))
46	39, 40, 41, 45	ply1scl1 22521	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
47	38, 46	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
48	47	eqcomd 2766	. . . . . . 7 ⊢ (𝜑 → (0_g‘(mulGrp‘(Poly₁‘𝑅))) = ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)))
49	48	fveq2d 6883	. . . . . 6 ⊢ (𝜑 → ((deg₁‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))))
50		eqid 2760	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
51	50, 41	ringidcl 20409	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
52	38, 51	syl 18	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Base‘𝑅))
53	37	idomdomd 20890	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Domn)
54		domnnzr 20871	. . . . . . . . 9 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
55	53, 54	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ NzRing)
56		eqid 2760	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
57	41, 56	nzrnz 20678	. . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1_r‘𝑅) ≠ (0_g‘𝑅))
58	55, 57	syl 18	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ≠ (0_g‘𝑅))
59		eqid 2760	. . . . . . . 8 ⊢ (deg₁‘𝑅) = (deg₁‘𝑅)
60	59, 39, 50, 40, 56	deg1scl 26341	. . . . . . 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 1398	. . . . . 6 ⊢ (𝜑 → ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))) = 0)
62	49, 61	eqtrd 2795	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = 0)
63	36, 62	eqtrd 2795	. . . 4 ⊢ (𝜑 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = 0)
64		sum0 15810	. . . . . 6 ⊢ Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = 0
65	64	eqcomi 2769	. . . . 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 2795	. . 3 ⊢ (𝜑 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
68		0red 11238	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
69	68	leidd 11807	. . . 4 ⊢ (𝜑 → 0 ≤ 0)
70	63	eqcomd 2766	. . . 4 ⊢ (𝜑 → 0 = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
71	69, 70	breqtrd 5131	. . 3 ⊢ (𝜑 → 0 ≤ ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
72	67, 71	jca 521	. 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 2922	. . . . . . . . 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 5207	. . . . . . . 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 7430	. . . . . 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 6883	. . . . 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 2760	. . . . . . . 8 ⊢ (Base‘(mulGrp‘(Poly₁‘𝑅))) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
81		eqid 2760	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
82		isidom 20889	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
83	37, 82	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
84	83	simpld 500	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ CRing)
85	39	ply1crng 22426	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (Poly₁‘𝑅) ∈ CRing)
86	84, 85	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑅) ∈ CRing)
87	42	crngmgp 20383	. . . . . . . . . . 11 ⊢ ((Poly₁‘𝑅) ∈ CRing → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
88	86, 87	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
89	88	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
90	89	adantr 486	. . . . . . . 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 739	. . . . . . . . 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 789	. . . . . . . . 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 9242	. . . . . . . 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 3122	. . . . . . . . . . . . . 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 219	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)) ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))))
100	99	simpld 500	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
101	100	ad3antrrr 743	. . . . . . . . . 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 4396	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
103	95, 101, 102	syl2anc 596	. . . . . . . . 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 2760	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
105	42, 104	mgpbas 20281	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
106	103, 105	eleqtrdi 2870	. . . . . . . 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 4078	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝑁 ∖ 𝑏) → 𝑐 ∈ 𝑁)
108	107	adantl 487	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)) → 𝑐 ∈ 𝑁)
109	108	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑐 ∈ 𝑁)
110	109	adantr 486	. . . . . . . 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 4079	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝑁 ∖ 𝑏) → ¬ 𝑐 ∈ 𝑏)
112	111	adantl 487	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)) → ¬ 𝑐 ∈ 𝑏)
113	112	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ¬ 𝑐 ∈ 𝑏)
114	113	adantr 486	. . . . . . . 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 739	. . . . . . . . . 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 4396	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
117	110, 115, 116	syl2anc 596	. . . . . . . . 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 2870	. . . . . . . 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 20090	. . . . . . 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 6883	. . . . . 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 2760	. . . . . . . . . 10 ⊢ (._r‘(Poly₁‘𝑅)) = (._r‘(Poly₁‘𝑅))
123	42, 122	mgpplusg 20280	. . . . . . . . 9 ⊢ (._r‘(Poly₁‘𝑅)) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
124	123	eqcomi 2769	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (._r‘(Poly₁‘𝑅))
125		eqid 2760	. . . . . . . 8 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
126	53	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑅 ∈ Domn)
127	126	adantr 486	. . . . . . . 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 3154	. . . . . . . . 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 20097	. . . . . . . 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 26353	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ IDomn → (Poly₁‘𝑅) ∈ IDomn)
131	37, 130	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑅) ∈ IDomn)
132	131	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (Poly₁‘𝑅) ∈ IDomn)
133	132	adantr 486	. . . . . . . . 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 501	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
135	134	ad3antrrr 743	. . . . . . . . . 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 43000	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑦 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
137	95, 135, 136	syl2anc 596	. . . . . . . . 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 43002	. . . . . . . 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 739	. . . . . . . . 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 43000	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
141	110, 139, 140	syl2anc 596	. . . . . . . 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 26343	. . . . . . 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 5207	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑏 ↦ 𝐶) = (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶)
144	143	eqcomi 2769	. . . . . . . . . . . 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 7430	. . . . . . . . . 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 6883	. . . . . . . . 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 7429	. . . . . . . 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 488	. . . . . . . . . . 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 487	. . . . . . . . . 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 7429	. . . . . . . . 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 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)))
153		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐))
154	91	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑁 ∈ Fin)
155		simprl 783	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑏 ⊆ 𝑁)
156	154, 155	ssfid 9242	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑏 ∈ Fin)
157	73, 74, 75	cbvmpt 5207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)
158	157	fveq1i 6880	. . . . . . . . . . . . . . . . 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 6883	. . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶))
162		simpr 490	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
166	165	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
167		rspcsbela 4396	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑛 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
168	164, 166, 167	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ⦋𝑛 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
169	161, 163, 164, 168	fvmptd 6995	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛) = ⦋𝑛 / 𝑥⦌𝐶)
170	169	fveq2d 6883	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑅 ∈ Ring)
172	171	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → 𝑅 ∈ Ring)
173	134	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
174		rspcsbnea 43000	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑛 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
175	164, 173, 174	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ⦋𝑛 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
176	59, 39, 125, 104	deg1nn0cl 26316	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘⦋𝑛 / 𝑥⦌𝐶) ∈ ℕ₀)
178	170, 177	eqeltrd 2860	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛)) ∈ ℕ₀)
179	160, 178	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℕ₀)
180	179	nn0cnd 12594	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℂ)
181		2fveq3 6884	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)))
182	109, 165, 116	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
183		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶)
184	183	fvmpts 6991	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑁 ∧ ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐) = ⦋𝑐 / 𝑥⦌𝐶)
185	109, 182, 184	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐) = ⦋𝑐 / 𝑥⦌𝐶)
186	185	fveq2d 6883	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) = ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶))
187	108, 134, 140	syl2anr 609	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
188	59, 39, 125, 104	deg1nn0cl 26316	. . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶) ∈ ℕ₀)
190	186, 189	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℕ₀)
191	190	nn0cnd 12594	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℂ)
192	152, 153, 156, 109, 113, 180, 181, 191	fsumsplitsn 15833	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 486	. . . . . . . . . . . . 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 6883	. . . . . . . . . . . 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 7430	. . . . . . . . . . 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 2795	. . . . . . . . . 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 2766	. . . . . . . . 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 2795	. . . . . . . 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 2795	. . . . . . 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 2795	. . . . . 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 2795	. . . . 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 2795	. . . 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 486	. . . . . 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 4747	. . . . . . . . 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 4138	. . . . . . . 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 9242	. . . . . . 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 486	. . . . . . . 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 18	. . . . . . . 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 16	. . . . . . 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 20097	. . . . . 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 7425	. . . . . . . . 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 4747	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → {𝑐} ⊆ 𝑁)
216	155, 215	unssd 4138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (𝑏 ∪ {𝑐}) ⊆ 𝑁)
217	154, 216	ssfid 9242	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
220	218, 219, 102	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
221	134	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
222	218, 221, 136	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑦 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
223	42, 132, 217, 220, 222	idomnnzgmulnz 43002	. . . . . . . 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 3022	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) ≠ (0_g‘(Poly₁‘𝑅)))
225	224	adantr 486	. . . . . 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 26316	. . . . . 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 1398	. . . . 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 12595	. . . 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 521	. . 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 418	. 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 9164	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ⦋csb 3847 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 ∅c0 4279 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7414 Fincfn 8955 0cc0 11127 + caddc 11130 ≤ cle 11271 ℕ₀cn0 12531 Σcsu 15776 Basecbs 17304 +_gcplusg 17345 ._rcmulr 17346 0_gc0g 17527 Σ_g cgsu 17528 CMndccmn 19910 mulGrpcmgp 20276 1_rcur 20323 Ringcrg 20375 CRingccrg 20376 NzRingcnzr 20675 Domncdomn 20857 IDomncidom 20858 algSccascl 22070 Poly₁cpl1 22405 deg₁cdg1 26282

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 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 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-rp 13046 df-fz 13565 df-fzo 13713 df-seq 14069 df-exp 14129 df-hash 14398 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-clim 15578 df-sum 15777 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ds 17367 df-unif 17368 df-hom 17369 df-cco 17370 df-0g 17529 df-gsum 17530 df-prds 17535 df-pws 17537 df-mre 17673 df-mrc 17674 df-acs 17676 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-mhm 18894 df-submnd 18895 df-grp 19063 df-minusg 19064 df-sbg 19065 df-mulg 19194 df-subg 19249 df-ghm 19344 df-cntz 19447 df-cmn 19912 df-abl 19913 df-mgp 20277 df-rng 20291 df-ur 20324 df-ring 20377 df-cring 20378 df-nzr 20676 df-subrng 20711 df-subrg 20735 df-rlreg 20859 df-domn 20860 df-idom 20861 df-lmod 21049 df-lss 21119 df-cnfld 21589 df-ascl 22073 df-psr 22127 df-mvr 22128 df-mpl 22129 df-opsr 22131 df-psr1 22408 df-vr1 22409 df-ply1 22410 df-coe1 22411 df-mdeg 26283 df-deg1 26284

This theorem is used by: aks6d1c6lem1 43039

W3C validator