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Theorem deg1gprod 41644

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 5245	. . . . . 6 ⊢ (𝑎 = ∅ → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ ∅ ↦ 𝐶))
2	1	oveq2d 7442	. . . . 5 ⊢ (𝑎 = ∅ → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)))
3	2	fveq2d 6906	. . . 4 ⊢ (𝑎 = ∅ → (( deg₁ ‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶))) = (( deg₁ ‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
4		sumeq1 15675	. . . 4 ⊢ (𝑎 = ∅ → Σ𝑛 ∈ 𝑎 (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ ∅ (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
5	3, 4	eqeq12d 2744	. . 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 5164	. . 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 630	. 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 5245	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑏 ↦ 𝐶))
9	8	oveq2d 7442	. . . . 5 ⊢ (𝑎 = 𝑏 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑏 ↦ 𝐶)))
10	9	fveq2d 6906	. . . 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 15675	. . . 4 ⊢ (𝑎 = 𝑏 → Σ𝑛 ∈ 𝑎 (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑏 (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
12	10, 11	eqeq12d 2744	. . 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 5164	. . 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 630	. 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 5245	. . . . . 6 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶))
16	15	oveq2d 7442	. . . . 5 ⊢ (𝑎 = (𝑏 ∪ {𝑐}) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)))
17	16	fveq2d 6906	. . . 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 15675	. . . 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 2744	. . 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 5164	. . 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 630	. 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 5245	. . . . . 6 ⊢ (𝑎 = 𝑁 → (𝑥 ∈ 𝑎 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶))
23	22	oveq2d 7442	. . . . 5 ⊢ (𝑎 = 𝑁 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑎 ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ 𝑁 ↦ 𝐶)))
24	23	fveq2d 6906	. . . 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 15675	. . . 4 ⊢ (𝑎 = 𝑁 → Σ𝑛 ∈ 𝑎 (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = Σ𝑛 ∈ 𝑁 (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
26	24, 25	eqeq12d 2744	. . 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 5164	. . 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 630	. 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 6702	. . . . . . . . 9 ⊢ (𝑥 ∈ ∅ ↦ 𝐶) = ∅
30	29	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ∅ ↦ 𝐶) = ∅)
31	30	oveq2d 7442	. . . . . . 7 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = ((mulGrp‘(Poly₁‘𝑅)) Σ_g ∅))
32		eqid 2728	. . . . . . . . 9 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
33	32	gsum0 18651	. . . . . . . 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 2768	. . . . . 6 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶)) = (0_g‘(mulGrp‘(Poly₁‘𝑅))))
36	35	fveq2d 6906	. . . . 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 21264	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
39		eqid 2728	. . . . . . . . . 10 ⊢ (Poly₁‘𝑅) = (Poly₁‘𝑅)
40		eqid 2728	. . . . . . . . . 10 ⊢ (algSc‘(Poly₁‘𝑅)) = (algSc‘(Poly₁‘𝑅))
41		eqid 2728	. . . . . . . . . 10 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
42		eqid 2728	. . . . . . . . . . . 12 ⊢ (mulGrp‘(Poly₁‘𝑅)) = (mulGrp‘(Poly₁‘𝑅))
43		eqid 2728	. . . . . . . . . . . 12 ⊢ (1_r‘(Poly₁‘𝑅)) = (1_r‘(Poly₁‘𝑅))
44	42, 43	ringidval 20130	. . . . . . . . . . 11 ⊢ (1_r‘(Poly₁‘𝑅)) = (0_g‘(mulGrp‘(Poly₁‘𝑅)))
45	44	eqcomi 2737	. . . . . . . . . 10 ⊢ (0_g‘(mulGrp‘(Poly₁‘𝑅))) = (1_r‘(Poly₁‘𝑅))
46	39, 40, 41, 45	ply1scl1 22219	. . . . . . . . 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 2734	. . . . . . 7 ⊢ (𝜑 → (0_g‘(mulGrp‘(Poly₁‘𝑅))) = ((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅)))
49	48	fveq2d 6906	. . . . . 6 ⊢ (𝜑 → (( deg₁ ‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = (( deg₁ ‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))))
50		eqid 2728	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
51	50, 41	ringidcl 20209	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
52	38, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ∈ (Base‘𝑅))
53	37	idomdomd 21262	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Domn)
54		domnnzr 21249	. . . . . . . . 9 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
55	53, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ NzRing)
56		eqid 2728	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
57	41, 56	nzrnz 20461	. . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1_r‘𝑅) ≠ (0_g‘𝑅))
58	55, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → (1_r‘𝑅) ≠ (0_g‘𝑅))
59		eqid 2728	. . . . . . . 8 ⊢ ( deg₁ ‘𝑅) = ( deg₁ ‘𝑅)
60	59, 39, 50, 40, 56	deg1scl 26069	. . . . . . 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 1368	. . . . . 6 ⊢ (𝜑 → (( deg₁ ‘𝑅)‘((algSc‘(Poly₁‘𝑅))‘(1_r‘𝑅))) = 0)
62	49, 61	eqtrd 2768	. . . . 5 ⊢ (𝜑 → (( deg₁ ‘𝑅)‘(0_g‘(mulGrp‘(Poly₁‘𝑅)))) = 0)
63	36, 62	eqtrd 2768	. . . 4 ⊢ (𝜑 → (( deg₁ ‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = 0)
64		sum0 15707	. . . . . 6 ⊢ Σ𝑛 ∈ ∅ (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = 0
65	64	eqcomi 2737	. . . . 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 2768	. . 3 ⊢ (𝜑 → (( deg₁ ‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))) = Σ𝑛 ∈ ∅ (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)))
68		0red 11255	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
69	68	leidd 11818	. . . 4 ⊢ (𝜑 → 0 ≤ 0)
70	63	eqcomd 2734	. . . 4 ⊢ (𝜑 → 0 = (( deg₁ ‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
71	69, 70	breqtrd 5178	. . 3 ⊢ (𝜑 → 0 ≤ (( deg₁ ‘𝑅)‘((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ ∅ ↦ 𝐶))))
72	67, 71	jca 510	. 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 2899	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐶
74		nfcsb1v 3919	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
75		csbeq1a 3908	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
76	73, 74, 75	cbvmpt 5263	. . . . . . . 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 7442	. . . . . 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 6906	. . . . 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 2728	. . . . . . . 8 ⊢ (Base‘(mulGrp‘(Poly₁‘𝑅))) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
81		eqid 2728	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
82		isidom 21261	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
83	37, 82	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
84	83	simpld 493	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ CRing)
85	39	ply1crng 22124	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → (Poly₁‘𝑅) ∈ CRing)
86	84, 85	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑅) ∈ CRing)
87	42	crngmgp 20188	. . . . . . . . . . 11 ⊢ ((Poly₁‘𝑅) ∈ CRing → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
89	88	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (mulGrp‘(Poly₁‘𝑅)) ∈ CMnd)
90	89	adantr 479	. . . . . . . 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 724	. . . . . . . . 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 775	. . . . . . . . 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 9298	. . . . . . . 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 3982	. . . . . . . . . 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 3108	. . . . . . . . . . . . . 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 215	. . . . . . . . . . . . 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 493	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
101	100	ad3antrrr 728	. . . . . . . . . 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 4439	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
103	95, 101, 102	syl2anc 582	. . . . . . . . 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 2728	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(Poly₁‘𝑅))
105	42, 104	mgpbas 20087	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝑅)) = (Base‘(mulGrp‘(Poly₁‘𝑅)))
106	103, 105	eleqtrdi 2839	. . . . . . . 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 4127	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝑁 ∖ 𝑏) → 𝑐 ∈ 𝑁)
108	107	adantl 480	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)) → 𝑐 ∈ 𝑁)
109	108	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑐 ∈ 𝑁)
110	109	adantr 479	. . . . . . . 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 4128	. . . . . . . . . . 11 ⊢ (𝑐 ∈ (𝑁 ∖ 𝑏) → ¬ 𝑐 ∈ 𝑏)
112	111	adantl 480	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)) → ¬ 𝑐 ∈ 𝑏)
113	112	adantl 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ¬ 𝑐 ∈ 𝑏)
114	113	adantr 479	. . . . . . . 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 724	. . . . . . . . . 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 4439	. . . . . . . . . 10 ⊢ ((𝑐 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
117	110, 115, 116	syl2anc 582	. . . . . . . . 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 2839	. . . . . . . 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 3897	. . . . . . . 8 ⊢ (𝑦 = 𝑐 → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑐 / 𝑥⦌𝐶)
120	80, 81, 90, 94, 106, 110, 114, 118, 119	gsumunsn 19922	. . . . . . 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 6906	. . . . . 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 2728	. . . . . . . . . 10 ⊢ (._r‘(Poly₁‘𝑅)) = (._r‘(Poly₁‘𝑅))
123	42, 122	mgpplusg 20085	. . . . . . . . 9 ⊢ (._r‘(Poly₁‘𝑅)) = (+_g‘(mulGrp‘(Poly₁‘𝑅)))
124	123	eqcomi 2737	. . . . . . . 8 ⊢ (+_g‘(mulGrp‘(Poly₁‘𝑅))) = (._r‘(Poly₁‘𝑅))
125		eqid 2728	. . . . . . . 8 ⊢ (0_g‘(Poly₁‘𝑅)) = (0_g‘(Poly₁‘𝑅))
126	53	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑅 ∈ Domn)
127	126	adantr 479	. . . . . . . 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 3143	. . . . . . . . 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 19929	. . . . . . . 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 26080	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ IDomn → (Poly₁‘𝑅) ∈ IDomn)
131	37, 130	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝑅) ∈ IDomn)
132	131	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (Poly₁‘𝑅) ∈ IDomn)
133	132	adantr 479	. . . . . . . . 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 494	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
135	134	ad3antrrr 728	. . . . . . . . . 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 41634	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑦 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
137	95, 135, 136	syl2anc 582	. . . . . . . . 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 41636	. . . . . . . 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 724	. . . . . . . . 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 41634	. . . . . . . . 9 ⊢ ((𝑐 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
141	110, 139, 140	syl2anc 582	. . . . . . . 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 41643	. . . . . . 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 5263	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑏 ↦ 𝐶) = (𝑦 ∈ 𝑏 ↦ ⦋𝑦 / 𝑥⦌𝐶)
144	143	eqcomi 2737	. . . . . . . . . . . 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 7442	. . . . . . . . . 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 6906	. . . . . . . . 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 7441	. . . . . . . 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 481	. . . . . . . . . . 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 480	. . . . . . . . . 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 7441	. . . . . . . . 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 1909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏)))
153		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛(( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐))
154	91	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑁 ∈ Fin)
155		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑏 ⊆ 𝑁)
156	154, 155	ssfid 9298	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑏 ∈ Fin)
157	73, 74, 75	cbvmpt 5263	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)
158	157	fveq1i 6903	. . . . . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶))
162		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) ∧ 𝑦 = 𝑛) → 𝑦 = 𝑛)
163	162	csbeq1d 3898	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) ∧ 𝑦 = 𝑛) → ⦋𝑦 / 𝑥⦌𝐶 = ⦋𝑛 / 𝑥⦌𝐶)
164	155	sselda 3982	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → 𝑛 ∈ 𝑁)
165	100	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
166	165	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
167		rspcsbela 4439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ⦋𝑛 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
168	164, 166, 167	syl2anc 582	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ⦋𝑛 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
169	161, 163, 164, 168	fvmptd 7017	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛) = ⦋𝑛 / 𝑥⦌𝐶)
170	169	fveq2d 6906	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → 𝑅 ∈ Ring)
172	171	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → 𝑅 ∈ Ring)
173	134	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
174		rspcsbnea 41634	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ 𝑁 ∧ ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅))) → ⦋𝑛 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
175	164, 173, 174	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → ⦋𝑛 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
176	59, 39, 125, 104	deg1nn0cl 26044	. . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → (( deg₁ ‘𝑅)‘⦋𝑛 / 𝑥⦌𝐶) ∈ ℕ₀)
178	170, 177	eqeltrd 2829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → (( deg₁ ‘𝑅)‘((𝑦 ∈ 𝑁 ↦ ⦋𝑦 / 𝑥⦌𝐶)‘𝑛)) ∈ ℕ₀)
179	160, 178	eqeltrd 2829	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℕ₀)
180	179	nn0cnd 12572	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑛 ∈ 𝑏) → (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) ∈ ℂ)
181		2fveq3 6907	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑐 → (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑛)) = (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)))
182	109, 165, 116	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
183		eqid 2728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑁 ↦ 𝐶) = (𝑥 ∈ 𝑁 ↦ 𝐶)
184	183	fvmpts 7013	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ 𝑁 ∧ ⦋𝑐 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅))) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐) = ⦋𝑐 / 𝑥⦌𝐶)
185	109, 182, 184	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐) = ⦋𝑐 / 𝑥⦌𝐶)
186	185	fveq2d 6906	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) = (( deg₁ ‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶))
187	108, 134, 140	syl2anr 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ⦋𝑐 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
188	59, 39, 125, 104	deg1nn0cl 26044	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (( deg₁ ‘𝑅)‘⦋𝑐 / 𝑥⦌𝐶) ∈ ℕ₀)
190	186, 189	eqeltrd 2829	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℕ₀)
191	190	nn0cnd 12572	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (( deg₁ ‘𝑅)‘((𝑥 ∈ 𝑁 ↦ 𝐶)‘𝑐)) ∈ ℂ)
192	152, 153, 156, 109, 113, 180, 181, 191	fsumsplitsn 15730	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 479	. . . . . . . . . . . . 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 6906	. . . . . . . . . . . 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 7442	. . . . . . . . . . 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 2768	. . . . . . . . . 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 2734	. . . . . . . . 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 2768	. . . . . . . 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 2768	. . . . . . 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 2768	. . . . . 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 2768	. . . . 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 2768	. . . 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 479	. . . . . 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 4817	. . . . . . . . 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 4188	. . . . . . . 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 9298	. . . . . . 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 479	. . . . . . . 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 4050	. . . . . . . . 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 19929	. . . . . 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 7437	. . . . . . . . 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 4817	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → {𝑐} ⊆ 𝑁)
216	155, 215	unssd 4188	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (𝑏 ∪ {𝑐}) ⊆ 𝑁)
217	154, 216	ssfid 9298	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → (𝑏 ∪ {𝑐}) ∈ Fin)
218	216	sselda 3982	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → 𝑦 ∈ 𝑁)
219	165	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ∀𝑥 ∈ 𝑁 𝐶 ∈ (Base‘(Poly₁‘𝑅)))
220	218, 219, 102	syl2anc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑦 / 𝑥⦌𝐶 ∈ (Base‘(Poly₁‘𝑅)))
221	134	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ∀𝑥 ∈ 𝑁 𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
222	218, 221, 136	syl2anc 582	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) ∧ 𝑦 ∈ (𝑏 ∪ {𝑐})) → ⦋𝑦 / 𝑥⦌𝐶 ≠ (0_g‘(Poly₁‘𝑅)))
223	42, 132, 217, 220, 222	idomnnzgmulnz 41636	. . . . . . . 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 3005	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝑁 ∧ 𝑐 ∈ (𝑁 ∖ 𝑏))) → ((mulGrp‘(Poly₁‘𝑅)) Σ_g (𝑥 ∈ (𝑏 ∪ {𝑐}) ↦ 𝐶)) ≠ (0_g‘(Poly₁‘𝑅)))
225	224	adantr 479	. . . . . 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 26044	. . . . . 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 1368	. . . . 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 12573	. . . 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 510	. . 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 411	. 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 9197	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 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∀wral 3058 ⦋csb 3894 ∖ cdif 3946 ∪ cun 3947 ⊆ wss 3949 ∅c0 4326 {csn 4632 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 0cc0 11146 + caddc 11149 ≤ cle 11287 ℕ₀cn0 12510 Σcsu 15672 Basecbs 17187 +_gcplusg 17240 ._rcmulr 17241 0_gc0g 17428 Σ_g cgsu 17429 CMndccmn 19742 mulGrpcmgp 20081 1_rcur 20128 Ringcrg 20180 CRingccrg 20181 NzRingcnzr 20458 Domncdomn 21234 IDomncidom 21235 algSccascl 21793 Poly₁cpl1 22103 deg₁ cdg1 26007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-nzr 20459 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-rlreg 21237 df-domn 21238 df-idom 21239 df-cnfld 21287 df-ascl 21796 df-psr 21849 df-mvr 21850 df-mpl 21851 df-opsr 21853 df-psr1 22106 df-vr1 22107 df-ply1 22108 df-coe1 22109 df-mdeg 26008 df-deg1 26009

This theorem is referenced by: aks6d1c6lem1 41674

W3C validator