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Theorem deg1gprod 42123

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 5181	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ ∅ ↦ 𝐶))
2	1	oveq2d 7365	. . . . 5 ⊢ (𝑎 = ∅ → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)))
3	2	fveq2d 6826	. . . 4 ⊢ (𝑎 = ∅ → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
4		sumeq1 15596	. . . 4 ⊢ (𝑎 = ∅ → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
5	3, 4	eqeq12d 2745	. . 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 5104	. . 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 5181	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑏 ↦ 𝐶))
9	8	oveq2d 7365	. . . . 5 ⊢ (𝑎 = 𝑏 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))
10	9	fveq2d 6826	. . . 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 15596	. . . 4 ⊢ (𝑎 = 𝑏 → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑏 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
12	10, 11	eqeq12d 2745	. . 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 5104	. . 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 5181	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))
16	15	oveq2d 7365	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)))
17	16	fveq2d 6826	. . . 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 15596	. . . 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 2745	. . 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 5104	. . 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 5181	. . . . . 6 ⊢ (𝑎 = 𝑁 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶))
23	22	oveq2d 7365	. . . . 5 ⊢ (𝑎 = 𝑁 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶)))
24	23	fveq2d 6826	. . . 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 15596	. . . 4 ⊢ (𝑎 = 𝑁 → Σ𝑛 ∈ 𝑎 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑁 ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
26	24, 25	eqeq12d 2745	. . 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 5104	. . 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 6624	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ 𝐶) = ∅
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ∅ ↦ 𝐶) = ∅)
31	30	oveq2d 7365	. . . . . . 7 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g ∅))
32		eqid 2729	. . . . . . . . 9 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
33	32	gsum0 18558	. . . . . . . 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 2764	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
36	35	fveq2d 6826	. . . . 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 20613	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
39		eqid 2729	. . . . . . . . . 10 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
40		eqid 2729	. . . . . . . . . 10 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
41		eqid 2729	. . . . . . . . . 10 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
42		eqid 2729	. . . . . . . . . . . 12 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
43		eqid 2729	. . . . . . . . . . . 12 ⊢ (1_r‘(Poly₁‘𝑅)) = (1_r‘(Poly₁‘𝑅))
44	42, 43	ringidval 20068	. . . . . . . . . . 11 ⊢ (1_r‘(Poly₁‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
45	44	eqcomi 2738	. . . . . . . . . 10 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (1_r‘(Poly₁‘𝑅))
46	39, 40, 41, 45	ply1scl1 22177	. . . . . . . . 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 2735	. . . . . . 7 ⊢ (𝜑 → (0_g‘(mulGrp‘(Poly₁‘𝑅))) = ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)))
49	48	fveq2d 6826	. . . . . 6 ⊢ (𝜑 → ((deg₁‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = ((deg₁‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))))
50		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
51	50, 41	ringidcl 20150	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
52	38, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Base‘𝑅))
53	37	idomdomd 20611	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Domn)
54		domnnzr 20591	. . . . . . . . 9 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
55	53, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ NzRing)
56		eqid 2729	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
57	41, 56	nzrnz 20400	. . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1_r‘𝑅) ≠ (0_g‘𝑅))
58	55, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ≠ (0_g‘𝑅))
59		eqid 2729	. . . . . . . 8 ⊢ (deg₁‘𝑅) = (deg₁‘𝑅)
60	59, 39, 50, 40, 56	deg1scl 26016	. . . . . . 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 2764	. . . . 5 ⊢ (𝜑 → ((deg₁‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = 0)
63	36, 62	eqtrd 2764	. . . 4 ⊢ (𝜑 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = 0)
64		sum0 15628	. . . . . 6 ⊢ Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = 0
65	64	eqcomi 2738	. . . . 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 2764	. . 3 ⊢ (𝜑 → ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = Σ𝑛 ∈ ∅ ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
68		0red 11118	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
69	68	leidd 11686	. . . 4 ⊢ (𝜑 → 0 ≤ 0)
70	63	eqcomd 2735	. . . 4 ⊢ (𝜑 → 0 = ((deg₁‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
71	69, 70	breqtrd 5118	. . 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 2891	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐶
74		nfcsb1v 3875	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
75		csbeq1a 3865	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
76	73, 74, 75	cbvmpt 5194	. . . . . . . 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 7365	. . . . . 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 6826	. . . . 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 2729	. . . . . . . 8 ⊢ (Base‘(mulGrp‘(Poly₁‘𝑅))) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
81		eqid 2729	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
82		isidom 20610	. . . . . . . . . . . . . 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 22081	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (Poly₁‘𝑅) ∈ CRing)
86	84, 85	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑅) ∈ CRing)
87	42	crngmgp 20126	. . . . . . . . . . 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 9158	. . . . . . . 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 3935	. . . . . . . . . 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 3089	. . . . . . . . . . . . . 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 4389	. . . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
105	42, 104	mgpbas 20030	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
106	103, 105	eleqtrdi 2838	. . . . . . . 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 4082	. . . . . . . . . . 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 4083	. . . . . . . . . . 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 4389	. . . . . . . . . 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 2838	. . . . . . . 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 3854	. . . . . . . 8 ⊢ (𝑦 = 𝑐 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑐 / 𝑥⦌𝐶)
120	80, 81, 90, 94, 106, 110, 114, 118, 119	gsumunsn 19839	. . . . . . 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 6826	. . . . . 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 2729	. . . . . . . . . 10 ⊢ (._r‘(Poly₁‘𝑅)) = (._r‘(Poly₁‘𝑅))
123	42, 122	mgpplusg 20029	. . . . . . . . 9 ⊢ (._r‘(Poly₁‘𝑅)) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
124	123	eqcomi 2738	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (._r‘(Poly₁‘𝑅))
125		eqid 2729	. . . . . . . 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 3121	. . . . . . . . 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 19846	. . . . . . . 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 26028	. . . . . . . . . . . 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 42114	. . . . . . . . . 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 42116	. . . . . . . 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 42114	. . . . . . . . 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 26018	. . . . . . 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 5194	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑏 ↦ 𝐶) = (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶)
144	143	eqcomi 2738	. . . . . . . . . . . 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 7365	. . . . . . . . . 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 6826	. . . . . . . . 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 7364	. . . . . . . 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 7364	. . . . . . . . 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 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)))
153		nfcv 2891	. . . . . . . . . . . . 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 9158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑏 ∈ Fin)
157	73, 74, 75	cbvmpt 5194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)
158	157	fveq1i 6823	. . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 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 3855	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) ∧ 𝑦 = 𝑛) → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑛 / 𝑥⦌𝐶)
164	155	sselda 3935	. . . . . . . . . . . . . . . . . 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 4389	. . . . . . . . . . . . . . . . . . 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 6937	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛) = ⦋𝑛 / 𝑥⦌𝐶)
170	169	fveq2d 6826	. . . . . . . . . . . . . . . 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 42114	. . . . . . . . . . . . . . . . . 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 25991	. . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛)) ∈ ℕ₀)
179	160, 178	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℕ₀)
180	179	nn0cnd 12447	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℂ)
181		2fveq3 6827	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶)
184	183	fvmpts 6933	. . . . . . . . . . . . . . . . 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 6826	. . . . . . . . . . . . . . 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 25991	. . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℕ₀)
191	190	nn0cnd 12447	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((deg₁‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℂ)
192	152, 153, 156, 109, 113, 180, 181, 191	fsumsplitsn 15651	. . . . . . . . . . . 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 6826	. . . . . . . . . . . 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 7365	. . . . . . . . . . 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 2764	. . . . . . . . . 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 2735	. . . . . . . . 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 2764	. . . . . . . 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 2764	. . . . . . 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 2764	. . . . . 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 2764	. . . . 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 2764	. . . 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 4760	. . . . . . . . 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 4143	. . . . . . . 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 9158	. . . . . . 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 4004	. . . . . . . . 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 19846	. . . . . 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 7360	. . . . . . . . 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 4760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → {𝑐} ⊆ 𝑁)
216	155, 215	unssd 4143	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (𝑏 ∪ {𝑐}) ⊆ 𝑁)
217	154, 216	ssfid 9158	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (𝑏 ∪ {𝑐}) ∈ Fin)
218	216	sselda 3935	. . . . . . . . . 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 42116	. . . . . . . 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 2992	. . . . . . 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 25991	. . . . . 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 12448	. . . 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 9080	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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ⦋csb 3851 ∖ cdif 3900 ∪ cun 3901 ⊆ wss 3903 ∅c0 4284 {csn 4577 class class class wbr 5092 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 Fincfn 8872 0cc0 11009 + caddc 11012 ≤ cle 11150 ℕ₀cn0 12384 Σcsu 15593 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 0_gc0g 17343 Σ_g cgsu 17344 CMndccmn 19659 mulGrpcmgp 20025 1_rcur 20066 Ringcrg 20118 CRingccrg 20119 NzRingcnzr 20397 Domncdomn 20577 IDomncidom 20578 algSccascl 21759 Poly₁cpl1 22059 deg₁cdg1 25957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-sup 9332 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-mulg 18947 df-subg 19002 df-ghm 19092 df-cntz 19196 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-nzr 20398 df-subrng 20431 df-subrg 20455 df-rlreg 20579 df-domn 20580 df-idom 20581 df-lmod 20765 df-lss 20835 df-cnfld 21262 df-ascl 21762 df-psr 21816 df-mvr 21817 df-mpl 21818 df-opsr 21820 df-psr1 22062 df-vr1 22063 df-ply1 22064 df-coe1 22065 df-mdeg 25958 df-deg1 25959

This theorem is referenced by: aks6d1c6lem1 42153

W3C validator