rngunsnply - Mathbox for Stefan O'Rear

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Theorem rngunsnply 39780

Description: Adjoining one element to a ring results in a set of polynomial evaluations. (Contributed by Stefan O'Rear, 30-Nov-2014.)

**Hypotheses**
Ref	Expression
rngunsnply.b	⊢ (𝜑 → 𝐵 ∈ (SubRing‘ℂ_fld))
rngunsnply.x	⊢ (𝜑 → 𝑋 ∈ ℂ)
rngunsnply.s	⊢ (𝜑 → 𝑆 = ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))

**Assertion**
Ref	Expression
rngunsnply	⊢ (𝜑 → (𝑉 ∈ 𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋)))

Distinct variable groups: 𝜑,𝑝 𝐵,𝑝 𝑋,𝑝 𝑉,𝑝 Allowed substitution hint: 𝑆(𝑝)

Proof of Theorem rngunsnply Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngunsnply.s	. . 3 ⊢ (𝜑 → 𝑆 = ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
2	1	eleq2d 2900	. 2 ⊢ (𝜑 → (𝑉 ∈ 𝑆 ↔ 𝑉 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋}))))
3		cnring 20569	. . . . . . 7 ⊢ ℂ_fld ∈ Ring
4	3	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ_fld ∈ Ring)
5		cnfldbas 20551	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
6	5	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ = (Base‘ℂ_fld))
7		rngunsnply.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘ℂ_fld))
8	5	subrgss 19538	. . . . . . . 8 ⊢ (𝐵 ∈ (SubRing‘ℂ_fld) → 𝐵 ⊆ ℂ)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ ℂ)
10		rngunsnply.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℂ)
11	10	snssd 4744	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ ℂ)
12	9, 11	unssd 4164	. . . . . 6 ⊢ (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ℂ)
13		eqidd 2824	. . . . . 6 ⊢ (𝜑 → (RingSpan‘ℂ_fld) = (RingSpan‘ℂ_fld))
14		eqidd 2824	. . . . . 6 ⊢ (𝜑 → ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) = ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
15		eqidd 2824	. . . . . . 7 ⊢ (𝜑 → (ℂ_fld ↾_s {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}) = (ℂ_fld ↾_s {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}))
16		cnfld0 20571	. . . . . . . 8 ⊢ 0 = (0_g‘ℂ_fld)
17	16	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 = (0_g‘ℂ_fld))
18		cnfldadd 20552	. . . . . . . 8 ⊢ + = (+_g‘ℂ_fld)
19	18	a1i 11	. . . . . . 7 ⊢ (𝜑 → + = (+_g‘ℂ_fld))
20		plyf 24790	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (Poly‘𝐵) → 𝑝:ℂ⟶ℂ)
21		ffvelrn 6851	. . . . . . . . . . . 12 ⊢ ((𝑝:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑝‘𝑋) ∈ ℂ)
22	20, 10, 21	syl2anr 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → (𝑝‘𝑋) ∈ ℂ)
23		eleq1 2902	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑝‘𝑋) → (𝑎 ∈ ℂ ↔ (𝑝‘𝑋) ∈ ℂ))
24	22, 23	syl5ibrcom 249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → (𝑎 = (𝑝‘𝑋) → 𝑎 ∈ ℂ))
25	24	rexlimdva 3286	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) → 𝑎 ∈ ℂ))
26	25	ss2abdv 4046	. . . . . . . 8 ⊢ (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ⊆ {𝑎 ∣ 𝑎 ∈ ℂ})
27		abid2 2959	. . . . . . . . 9 ⊢ {𝑎 ∣ 𝑎 ∈ ℂ} = ℂ
28	27, 5	eqtri 2846	. . . . . . . 8 ⊢ {𝑎 ∣ 𝑎 ∈ ℂ} = (Base‘ℂ_fld)
29	26, 28	sseqtrdi 4019	. . . . . . 7 ⊢ (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ⊆ (Base‘ℂ_fld))
30		abid2 2959	. . . . . . . . 9 ⊢ {𝑎 ∣ 𝑎 ∈ 𝐵} = 𝐵
31		plyconst 24798	. . . . . . . . . . . . 13 ⊢ ((𝐵 ⊆ ℂ ∧ 𝑎 ∈ 𝐵) → (ℂ × {𝑎}) ∈ (Poly‘𝐵))
32	9, 31	sylan 582	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (ℂ × {𝑎}) ∈ (Poly‘𝐵))
33	10	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ ℂ)
34		vex 3499	. . . . . . . . . . . . . . 15 ⊢ 𝑎 ∈ V
35	34	fvconst2 6968	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ ℂ → ((ℂ × {𝑎})‘𝑋) = 𝑎)
36	33, 35	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((ℂ × {𝑎})‘𝑋) = 𝑎)
37	36	eqcomd 2829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 = ((ℂ × {𝑎})‘𝑋))
38		fveq1 6671	. . . . . . . . . . . . 13 ⊢ (𝑝 = (ℂ × {𝑎}) → (𝑝‘𝑋) = ((ℂ × {𝑎})‘𝑋))
39	38	rspceeqv 3640	. . . . . . . . . . . 12 ⊢ (((ℂ × {𝑎}) ∈ (Poly‘𝐵) ∧ 𝑎 = ((ℂ × {𝑎})‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋))
40	32, 37, 39	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋))
41	40	ex 415	. . . . . . . . . 10 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)))
42	41	ss2abdv 4046	. . . . . . . . 9 ⊢ (𝜑 → {𝑎 ∣ 𝑎 ∈ 𝐵} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
43	30, 42	eqsstrrid 4018	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
44		subrgsubg 19543	. . . . . . . . . 10 ⊢ (𝐵 ∈ (SubRing‘ℂ_fld) → 𝐵 ∈ (SubGrp‘ℂ_fld))
45	7, 44	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘ℂ_fld))
46	16	subg0cl 18289	. . . . . . . . 9 ⊢ (𝐵 ∈ (SubGrp‘ℂ_fld) → 0 ∈ 𝐵)
47	45, 46	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
48	43, 47	sseldd 3970	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
49		biid 263	. . . . . . . . 9 ⊢ (𝜑 ↔ 𝜑)
50		vex 3499	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
51		eqeq1 2827	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (𝑎 = (𝑝‘𝑋) ↔ 𝑏 = (𝑝‘𝑋)))
52	51	rexbidv 3299	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝‘𝑋)))
53		fveq1 6671	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑒 → (𝑝‘𝑋) = (𝑒‘𝑋))
54	53	eqeq2d 2834	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑒 → (𝑏 = (𝑝‘𝑋) ↔ 𝑏 = (𝑒‘𝑋)))
55	54	cbvrexvw 3452	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝‘𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋))
56	52, 55	syl6bb 289	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋)))
57	50, 56	elab 3669	. . . . . . . . 9 ⊢ (𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋))
58		vex 3499	. . . . . . . . . 10 ⊢ 𝑐 ∈ V
59		eqeq1 2827	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → (𝑎 = (𝑝‘𝑋) ↔ 𝑐 = (𝑝‘𝑋)))
60	59	rexbidv 3299	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝‘𝑋)))
61		fveq1 6671	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑑 → (𝑝‘𝑋) = (𝑑‘𝑋))
62	61	eqeq2d 2834	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑑 → (𝑐 = (𝑝‘𝑋) ↔ 𝑐 = (𝑑‘𝑋)))
63	62	cbvrexvw 3452	. . . . . . . . . . 11 ⊢ (∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝‘𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋))
64	60, 63	syl6bb 289	. . . . . . . . . 10 ⊢ (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋)))
65	58, 64	elab 3669	. . . . . . . . 9 ⊢ (𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋))
66		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 ∈ (Poly‘𝐵))
67		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 ∈ (Poly‘𝐵))
68	18	subrgacl 19548	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ (SubRing‘ℂ_fld) ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
69	68	3expb 1116	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ (SubRing‘ℂ_fld) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
70	7, 69	sylan 582	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
71	70	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
72	71	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
73	66, 67, 72	plyadd 24809	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒 ∘_f + 𝑑) ∈ (Poly‘𝐵))
74		plyf 24790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑒 ∈ (Poly‘𝐵) → 𝑒:ℂ⟶ℂ)
75	74	ffnd 6517	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑒 ∈ (Poly‘𝐵) → 𝑒 Fn ℂ)
76	75	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 Fn ℂ)
77		plyf 24790	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 ∈ (Poly‘𝐵) → 𝑑:ℂ⟶ℂ)
78	77	ffnd 6517	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 ∈ (Poly‘𝐵) → 𝑑 Fn ℂ)
79	78	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 Fn ℂ)
80		cnex 10620	. . . . . . . . . . . . . . . . . 18 ⊢ ℂ ∈ V
81	80	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ℂ ∈ V)
82	10	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑋 ∈ ℂ)
83		fnfvof 7425	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒 ∘_f + 𝑑)‘𝑋) = ((𝑒‘𝑋) + (𝑑‘𝑋)))
84	76, 79, 81, 82, 83	syl22anc 836	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒 ∘_f + 𝑑)‘𝑋) = ((𝑒‘𝑋) + (𝑑‘𝑋)))
85	84	eqcomd 2829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒‘𝑋) + (𝑑‘𝑋)) = ((𝑒 ∘_f + 𝑑)‘𝑋))
86		fveq1 6671	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑒 ∘_f + 𝑑) → (𝑝‘𝑋) = ((𝑒 ∘_f + 𝑑)‘𝑋))
87	86	rspceeqv 3640	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∘_f + 𝑑) ∈ (Poly‘𝐵) ∧ ((𝑒‘𝑋) + (𝑑‘𝑋)) = ((𝑒 ∘_f + 𝑑)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + (𝑑‘𝑋)) = (𝑝‘𝑋))
88	73, 85, 87	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + (𝑑‘𝑋)) = (𝑝‘𝑋))
89		oveq2 7166	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑑‘𝑋) → ((𝑒‘𝑋) + 𝑐) = ((𝑒‘𝑋) + (𝑑‘𝑋)))
90	89	eqeq1d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑑‘𝑋) → (((𝑒‘𝑋) + 𝑐) = (𝑝‘𝑋) ↔ ((𝑒‘𝑋) + (𝑑‘𝑋)) = (𝑝‘𝑋)))
91	90	rexbidv 3299	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑑‘𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + 𝑐) = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + (𝑑‘𝑋)) = (𝑝‘𝑋)))
92	88, 91	syl5ibrcom 249	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + 𝑐) = (𝑝‘𝑋)))
93	92	rexlimdva 3286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + 𝑐) = (𝑝‘𝑋)))
94		oveq1 7165	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑒‘𝑋) → (𝑏 + 𝑐) = ((𝑒‘𝑋) + 𝑐))
95	94	eqeq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑒‘𝑋) → ((𝑏 + 𝑐) = (𝑝‘𝑋) ↔ ((𝑒‘𝑋) + 𝑐) = (𝑝‘𝑋)))
96	95	rexbidv 3299	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑒‘𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + 𝑐) = (𝑝‘𝑋)))
97	96	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑒‘𝑋) → ((∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋)) ↔ (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) + 𝑐) = (𝑝‘𝑋))))
98	93, 97	syl5ibrcom 249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒‘𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋))))
99	98	rexlimdva 3286	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋))))
100	99	3imp 1107	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋))
101	49, 57, 65, 100	syl3anb 1157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋))
102		ovex 7191	. . . . . . . . 9 ⊢ (𝑏 + 𝑐) ∈ V
103		eqeq1 2827	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 + 𝑐) → (𝑎 = (𝑝‘𝑋) ↔ (𝑏 + 𝑐) = (𝑝‘𝑋)))
104	103	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 + 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋)))
105	102, 104	elab 3669	. . . . . . . 8 ⊢ ((𝑏 + 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝‘𝑋))
106	101, 105	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}) → (𝑏 + 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
107		ax-1cn 10597	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
108		cnfldneg 20573	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℂ → ((inv_g‘ℂ_fld)‘1) = -1)
109	107, 108	mp1i 13	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((inv_g‘ℂ_fld)‘1) = -1)
110		cnfld1 20572	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 = (1_r‘ℂ_fld)
111	110	subrg1cl 19545	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ (SubRing‘ℂ_fld) → 1 ∈ 𝐵)
112	7, 111	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 ∈ 𝐵)
113		eqid 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
114	113	subginvcl 18290	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ (SubGrp‘ℂ_fld) ∧ 1 ∈ 𝐵) → ((inv_g‘ℂ_fld)‘1) ∈ 𝐵)
115	45, 112, 114	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((inv_g‘ℂ_fld)‘1) ∈ 𝐵)
116	109, 115	eqeltrrd 2916	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → -1 ∈ 𝐵)
117		plyconst 24798	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ⊆ ℂ ∧ -1 ∈ 𝐵) → (ℂ × {-1}) ∈ (Poly‘𝐵))
118	9, 116, 117	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℂ × {-1}) ∈ (Poly‘𝐵))
119	118	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (ℂ × {-1}) ∈ (Poly‘𝐵))
120		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → 𝑒 ∈ (Poly‘𝐵))
121		cnfldmul 20553	. . . . . . . . . . . . . . . . . 18 ⊢ · = (._r‘ℂ_fld)
122	121	subrgmcl 19549	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ (SubRing‘ℂ_fld) ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 · 𝑏) ∈ 𝐵)
123	122	3expb 1116	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ (SubRing‘ℂ_fld) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
124	7, 123	sylan 582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
125	124	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
126	119, 120, 71, 125	plymul 24810	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1}) ∘_f · 𝑒) ∈ (Poly‘𝐵))
127		ffvelrn 6851	. . . . . . . . . . . . . . . 16 ⊢ ((𝑒:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑒‘𝑋) ∈ ℂ)
128	74, 10, 127	syl2anr 598	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (𝑒‘𝑋) ∈ ℂ)
129		cnfldneg 20573	. . . . . . . . . . . . . . 15 ⊢ ((𝑒‘𝑋) ∈ ℂ → ((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = -(𝑒‘𝑋))
130	128, 129	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → ((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = -(𝑒‘𝑋))
131		negex 10886	. . . . . . . . . . . . . . . . 17 ⊢ -1 ∈ V
132		fnconstg 6569	. . . . . . . . . . . . . . . . 17 ⊢ (-1 ∈ V → (ℂ × {-1}) Fn ℂ)
133	131, 132	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (ℂ × {-1}) Fn ℂ)
134	75	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → 𝑒 Fn ℂ)
135	80	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → ℂ ∈ V)
136	10	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → 𝑋 ∈ ℂ)
137		fnfvof 7425	. . . . . . . . . . . . . . . 16 ⊢ ((((ℂ × {-1}) Fn ℂ ∧ 𝑒 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → (((ℂ × {-1}) ∘_f · 𝑒)‘𝑋) = (((ℂ × {-1})‘𝑋) · (𝑒‘𝑋)))
138	133, 134, 135, 136, 137	syl22anc 836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘_f · 𝑒)‘𝑋) = (((ℂ × {-1})‘𝑋) · (𝑒‘𝑋)))
139	131	fvconst2 6968	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℂ → ((ℂ × {-1})‘𝑋) = -1)
140	136, 139	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1})‘𝑋) = -1)
141	140	oveq1d 7173	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1})‘𝑋) · (𝑒‘𝑋)) = (-1 · (𝑒‘𝑋)))
142	128	mulm1d 11094	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (-1 · (𝑒‘𝑋)) = -(𝑒‘𝑋))
143	138, 141, 142	3eqtrd 2862	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘_f · 𝑒)‘𝑋) = -(𝑒‘𝑋))
144	130, 143	eqtr4d 2861	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → ((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = (((ℂ × {-1}) ∘_f · 𝑒)‘𝑋))
145		fveq1 6671	. . . . . . . . . . . . . 14 ⊢ (𝑝 = ((ℂ × {-1}) ∘_f · 𝑒) → (𝑝‘𝑋) = (((ℂ × {-1}) ∘_f · 𝑒)‘𝑋))
146	145	rspceeqv 3640	. . . . . . . . . . . . 13 ⊢ ((((ℂ × {-1}) ∘_f · 𝑒) ∈ (Poly‘𝐵) ∧ ((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = (((ℂ × {-1}) ∘_f · 𝑒)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = (𝑝‘𝑋))
147	126, 144, 146	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = (𝑝‘𝑋))
148		fveqeq2 6681	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑒‘𝑋) → (((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋) ↔ ((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = (𝑝‘𝑋)))
149	148	rexbidv 3299	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑒‘𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘(𝑒‘𝑋)) = (𝑝‘𝑋)))
150	147, 149	syl5ibrcom 249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋)))
151	150	rexlimdva 3286	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋)))
152	151	imp 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋))
153	57, 152	sylan2b 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋))
154		fvex 6685	. . . . . . . . 9 ⊢ ((inv_g‘ℂ_fld)‘𝑏) ∈ V
155		eqeq1 2827	. . . . . . . . . 10 ⊢ (𝑎 = ((inv_g‘ℂ_fld)‘𝑏) → (𝑎 = (𝑝‘𝑋) ↔ ((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋)))
156	155	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑎 = ((inv_g‘ℂ_fld)‘𝑏) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋)))
157	154, 156	elab 3669	. . . . . . . 8 ⊢ (((inv_g‘ℂ_fld)‘𝑏) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)((inv_g‘ℂ_fld)‘𝑏) = (𝑝‘𝑋))
158	153, 157	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}) → ((inv_g‘ℂ_fld)‘𝑏) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
159	110	a1i 11	. . . . . . 7 ⊢ (𝜑 → 1 = (1_r‘ℂ_fld))
160	121	a1i 11	. . . . . . 7 ⊢ (𝜑 → · = (._r‘ℂ_fld))
161	43, 112	sseldd 3970	. . . . . . 7 ⊢ (𝜑 → 1 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
162	125	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
163	66, 67, 72, 162	plymul 24810	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒 ∘_f · 𝑑) ∈ (Poly‘𝐵))
164		fnfvof 7425	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒 ∘_f · 𝑑)‘𝑋) = ((𝑒‘𝑋) · (𝑑‘𝑋)))
165	76, 79, 81, 82, 164	syl22anc 836	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒 ∘_f · 𝑑)‘𝑋) = ((𝑒‘𝑋) · (𝑑‘𝑋)))
166	165	eqcomd 2829	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒‘𝑋) · (𝑑‘𝑋)) = ((𝑒 ∘_f · 𝑑)‘𝑋))
167		fveq1 6671	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑒 ∘_f · 𝑑) → (𝑝‘𝑋) = ((𝑒 ∘_f · 𝑑)‘𝑋))
168	167	rspceeqv 3640	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∘_f · 𝑑) ∈ (Poly‘𝐵) ∧ ((𝑒‘𝑋) · (𝑑‘𝑋)) = ((𝑒 ∘_f · 𝑑)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · (𝑑‘𝑋)) = (𝑝‘𝑋))
169	163, 166, 168	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · (𝑑‘𝑋)) = (𝑝‘𝑋))
170		oveq2 7166	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑑‘𝑋) → ((𝑒‘𝑋) · 𝑐) = ((𝑒‘𝑋) · (𝑑‘𝑋)))
171	170	eqeq1d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑑‘𝑋) → (((𝑒‘𝑋) · 𝑐) = (𝑝‘𝑋) ↔ ((𝑒‘𝑋) · (𝑑‘𝑋)) = (𝑝‘𝑋)))
172	171	rexbidv 3299	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑑‘𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · 𝑐) = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · (𝑑‘𝑋)) = (𝑝‘𝑋)))
173	169, 172	syl5ibrcom 249	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · 𝑐) = (𝑝‘𝑋)))
174	173	rexlimdva 3286	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · 𝑐) = (𝑝‘𝑋)))
175		oveq1 7165	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑒‘𝑋) → (𝑏 · 𝑐) = ((𝑒‘𝑋) · 𝑐))
176	175	eqeq1d 2825	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑒‘𝑋) → ((𝑏 · 𝑐) = (𝑝‘𝑋) ↔ ((𝑒‘𝑋) · 𝑐) = (𝑝‘𝑋)))
177	176	rexbidv 3299	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑒‘𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · 𝑐) = (𝑝‘𝑋)))
178	177	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑒‘𝑋) → ((∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋)) ↔ (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒‘𝑋) · 𝑐) = (𝑝‘𝑋))))
179	174, 178	syl5ibrcom 249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒‘𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋))))
180	179	rexlimdva 3286	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋))))
181	180	3imp 1107	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒‘𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋))
182	49, 57, 65, 181	syl3anb 1157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋))
183		ovex 7191	. . . . . . . . 9 ⊢ (𝑏 · 𝑐) ∈ V
184		eqeq1 2827	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 · 𝑐) → (𝑎 = (𝑝‘𝑋) ↔ (𝑏 · 𝑐) = (𝑝‘𝑋)))
185	184	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 · 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋)))
186	183, 185	elab 3669	. . . . . . . 8 ⊢ ((𝑏 · 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝‘𝑋))
187	182, 186	sylibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}) → (𝑏 · 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
188	15, 17, 19, 29, 48, 106, 158, 159, 160, 161, 187, 4	issubrngd2 19963	. . . . . 6 ⊢ (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ∈ (SubRing‘ℂ_fld))
189		plyid 24801	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ ℂ ∧ 1 ∈ 𝐵) → X_p ∈ (Poly‘𝐵))
190	9, 112, 189	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → X_p ∈ (Poly‘𝐵))
191		df-idp 24781	. . . . . . . . . . . 12 ⊢ X_p = ( I ↾ ℂ)
192	191	fveq1i 6673	. . . . . . . . . . 11 ⊢ (X_p‘𝑋) = (( I ↾ ℂ)‘𝑋)
193		fvresi 6937	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ ℂ → (( I ↾ ℂ)‘𝑋) = 𝑋)
194	10, 193	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (( I ↾ ℂ)‘𝑋) = 𝑋)
195	192, 194	syl5req 2871	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 = (X_p‘𝑋))
196		fveq1 6671	. . . . . . . . . . 11 ⊢ (𝑝 = X_p → (𝑝‘𝑋) = (X_p‘𝑋))
197	196	rspceeqv 3640	. . . . . . . . . 10 ⊢ ((X_p ∈ (Poly‘𝐵) ∧ 𝑋 = (X_p‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝‘𝑋))
198	190, 195, 197	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝‘𝑋))
199		eqeq1 2827	. . . . . . . . . 10 ⊢ (𝑎 = 𝑋 → (𝑎 = (𝑝‘𝑋) ↔ 𝑋 = (𝑝‘𝑋)))
200	199	rexbidv 3299	. . . . . . . . 9 ⊢ (𝑎 = 𝑋 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝‘𝑋)))
201	10, 198, 200	elabd 3671	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
202	201	snssd 4744	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
203	43, 202	unssd 4164	. . . . . 6 ⊢ (𝜑 → (𝐵 ∪ {𝑋}) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
204	4, 6, 12, 13, 14, 188, 203	rgspnmin 39778	. . . . 5 ⊢ (𝜑 → ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)})
205	204	sseld 3968	. . . 4 ⊢ (𝜑 → (𝑉 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) → 𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)}))
206		fvex 6685	. . . . . . 7 ⊢ (𝑝‘𝑋) ∈ V
207		eleq1 2902	. . . . . . 7 ⊢ (𝑉 = (𝑝‘𝑋) → (𝑉 ∈ V ↔ (𝑝‘𝑋) ∈ V))
208	206, 207	mpbiri 260	. . . . . 6 ⊢ (𝑉 = (𝑝‘𝑋) → 𝑉 ∈ V)
209	208	rexlimivw 3284	. . . . 5 ⊢ (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋) → 𝑉 ∈ V)
210		eqeq1 2827	. . . . . 6 ⊢ (𝑎 = 𝑉 → (𝑎 = (𝑝‘𝑋) ↔ 𝑉 = (𝑝‘𝑋)))
211	210	rexbidv 3299	. . . . 5 ⊢ (𝑎 = 𝑉 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋)))
212	209, 211	elab3 3676	. . . 4 ⊢ (𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝‘𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋))
213	205, 212	syl6ib 253	. . 3 ⊢ (𝜑 → (𝑉 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) → ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋)))
214	4, 6, 12, 13, 14	rgspncl 39776	. . . . . . 7 ⊢ (𝜑 → ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂ_fld))
215	214	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂ_fld))
216		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → 𝑝 ∈ (Poly‘𝐵))
217	4, 6, 12, 13, 14	rgspnssid 39777	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
218	217	unssbd 4166	. . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
219		snidg 4601	. . . . . . . . 9 ⊢ (𝑋 ∈ ℂ → 𝑋 ∈ {𝑋})
220	10, 219	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
221	218, 220	sseldd 3970	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
222	221	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → 𝑋 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
223	217	unssad 4165	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
224	223	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → 𝐵 ⊆ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
225	215, 216, 222, 224	cnsrplycl 39774	. . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → (𝑝‘𝑋) ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})))
226		eleq1 2902	. . . . 5 ⊢ (𝑉 = (𝑝‘𝑋) → (𝑉 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) ↔ (𝑝‘𝑋) ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋}))))
227	225, 226	syl5ibrcom 249	. . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ (Poly‘𝐵)) → (𝑉 = (𝑝‘𝑋) → 𝑉 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋}))))
228	227	rexlimdva 3286	. . 3 ⊢ (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋) → 𝑉 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋}))))
229	213, 228	impbid 214	. 2 ⊢ (𝜑 → (𝑉 ∈ ((RingSpan‘ℂ_fld)‘(𝐵 ∪ {𝑋})) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋)))
230	2, 229	bitrd 281	1 ⊢ (𝜑 → (𝑉 ∈ 𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 Vcvv 3496 ∪ cun 3936 ⊆ wss 3938 {csn 4569 I cid 5461 × cxp 5555 ↾ cres 5559 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∘_f cof 7409 ℂcc 10537 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 -cneg 10873 Basecbs 16485 ↾_s cress 16486 +_gcplusg 16567 ._rcmulr 16568 0_gc0g 16715 inv_gcminusg 18106 SubGrpcsubg 18275 1_rcur 19253 Ringcrg 19299 SubRingcsubrg 19533 RingSpancrgspn 19534 ℂ_fldccnfld 20547 Polycply 24776 X_pcidp 24777

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-cmn 18910 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-subrg 19535 df-rgspn 19536 df-cnfld 20548 df-0p 24273 df-ply 24780 df-idp 24781 df-coe 24782 df-dgr 24783

This theorem is referenced by: (None)

W3C validator