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Theorem rngunsnply 41486
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 (𝜑𝐵 ∈ (SubRing‘ℂfld))
rngunsnply.x (𝜑𝑋 ∈ ℂ)
rngunsnply.s (𝜑𝑆 = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
Assertion
Ref Expression
rngunsnply (𝜑 → (𝑉𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
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.
StepHypRef Expression
1 rngunsnply.s . . 3 (𝜑𝑆 = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
21eleq2d 2823 . 2 (𝜑 → (𝑉𝑆𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
3 cnring 20819 . . . . . . 7 fld ∈ Ring
43a1i 11 . . . . . 6 (𝜑 → ℂfld ∈ Ring)
5 cnfldbas 20800 . . . . . . 7 ℂ = (Base‘ℂfld)
65a1i 11 . . . . . 6 (𝜑 → ℂ = (Base‘ℂfld))
7 rngunsnply.b . . . . . . . 8 (𝜑𝐵 ∈ (SubRing‘ℂfld))
85subrgss 20223 . . . . . . . 8 (𝐵 ∈ (SubRing‘ℂfld) → 𝐵 ⊆ ℂ)
97, 8syl 17 . . . . . . 7 (𝜑𝐵 ⊆ ℂ)
10 rngunsnply.x . . . . . . . 8 (𝜑𝑋 ∈ ℂ)
1110snssd 4769 . . . . . . 7 (𝜑 → {𝑋} ⊆ ℂ)
129, 11unssd 4146 . . . . . 6 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ℂ)
13 eqidd 2737 . . . . . 6 (𝜑 → (RingSpan‘ℂfld) = (RingSpan‘ℂfld))
14 eqidd 2737 . . . . . 6 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
15 eqidd 2737 . . . . . . 7 (𝜑 → (ℂflds {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) = (ℂflds {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}))
16 cnfld0 20821 . . . . . . . 8 0 = (0g‘ℂfld)
1716a1i 11 . . . . . . 7 (𝜑 → 0 = (0g‘ℂfld))
18 cnfldadd 20801 . . . . . . . 8 + = (+g‘ℂfld)
1918a1i 11 . . . . . . 7 (𝜑 → + = (+g‘ℂfld))
20 plyf 25559 . . . . . . . . . . . 12 (𝑝 ∈ (Poly‘𝐵) → 𝑝:ℂ⟶ℂ)
21 ffvelcdm 7032 . . . . . . . . . . . 12 ((𝑝:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑝𝑋) ∈ ℂ)
2220, 10, 21syl2anr 597 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑝𝑋) ∈ ℂ)
23 eleq1 2825 . . . . . . . . . . 11 (𝑎 = (𝑝𝑋) → (𝑎 ∈ ℂ ↔ (𝑝𝑋) ∈ ℂ))
2422, 23syl5ibrcom 246 . . . . . . . . . 10 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑎 = (𝑝𝑋) → 𝑎 ∈ ℂ))
2524rexlimdva 3152 . . . . . . . . 9 (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) → 𝑎 ∈ ℂ))
2625ss2abdv 4020 . . . . . . . 8 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ⊆ {𝑎𝑎 ∈ ℂ})
27 abid2 2875 . . . . . . . . 9 {𝑎𝑎 ∈ ℂ} = ℂ
2827, 5eqtri 2764 . . . . . . . 8 {𝑎𝑎 ∈ ℂ} = (Base‘ℂfld)
2926, 28sseqtrdi 3994 . . . . . . 7 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ⊆ (Base‘ℂfld))
30 abid2 2875 . . . . . . . . 9 {𝑎𝑎𝐵} = 𝐵
31 plyconst 25567 . . . . . . . . . . . . 13 ((𝐵 ⊆ ℂ ∧ 𝑎𝐵) → (ℂ × {𝑎}) ∈ (Poly‘𝐵))
329, 31sylan 580 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (ℂ × {𝑎}) ∈ (Poly‘𝐵))
3310adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐵) → 𝑋 ∈ ℂ)
34 vex 3449 . . . . . . . . . . . . . . 15 𝑎 ∈ V
3534fvconst2 7153 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → ((ℂ × {𝑎})‘𝑋) = 𝑎)
3633, 35syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → ((ℂ × {𝑎})‘𝑋) = 𝑎)
3736eqcomd 2742 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → 𝑎 = ((ℂ × {𝑎})‘𝑋))
38 fveq1 6841 . . . . . . . . . . . . 13 (𝑝 = (ℂ × {𝑎}) → (𝑝𝑋) = ((ℂ × {𝑎})‘𝑋))
3938rspceeqv 3595 . . . . . . . . . . . 12 (((ℂ × {𝑎}) ∈ (Poly‘𝐵) ∧ 𝑎 = ((ℂ × {𝑎})‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋))
4032, 37, 39syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋))
4140ex 413 . . . . . . . . . 10 (𝜑 → (𝑎𝐵 → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)))
4241ss2abdv 4020 . . . . . . . . 9 (𝜑 → {𝑎𝑎𝐵} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
4330, 42eqsstrrid 3993 . . . . . . . 8 (𝜑𝐵 ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
44 subrgsubg 20228 . . . . . . . . . 10 (𝐵 ∈ (SubRing‘ℂfld) → 𝐵 ∈ (SubGrp‘ℂfld))
457, 44syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ (SubGrp‘ℂfld))
4616subg0cl 18936 . . . . . . . . 9 (𝐵 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝐵)
4745, 46syl 17 . . . . . . . 8 (𝜑 → 0 ∈ 𝐵)
4843, 47sseldd 3945 . . . . . . 7 (𝜑 → 0 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
49 biid 260 . . . . . . . . 9 (𝜑𝜑)
50 vex 3449 . . . . . . . . . 10 𝑏 ∈ V
51 eqeq1 2740 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑎 = (𝑝𝑋) ↔ 𝑏 = (𝑝𝑋)))
5251rexbidv 3175 . . . . . . . . . . 11 (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝𝑋)))
53 fveq1 6841 . . . . . . . . . . . . 13 (𝑝 = 𝑒 → (𝑝𝑋) = (𝑒𝑋))
5453eqeq2d 2747 . . . . . . . . . . . 12 (𝑝 = 𝑒 → (𝑏 = (𝑝𝑋) ↔ 𝑏 = (𝑒𝑋)))
5554cbvrexvw 3226 . . . . . . . . . . 11 (∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋))
5652, 55bitrdi 286 . . . . . . . . . 10 (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋)))
5750, 56elab 3630 . . . . . . . . 9 (𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋))
58 vex 3449 . . . . . . . . . 10 𝑐 ∈ V
59 eqeq1 2740 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎 = (𝑝𝑋) ↔ 𝑐 = (𝑝𝑋)))
6059rexbidv 3175 . . . . . . . . . . 11 (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝𝑋)))
61 fveq1 6841 . . . . . . . . . . . . 13 (𝑝 = 𝑑 → (𝑝𝑋) = (𝑑𝑋))
6261eqeq2d 2747 . . . . . . . . . . . 12 (𝑝 = 𝑑 → (𝑐 = (𝑝𝑋) ↔ 𝑐 = (𝑑𝑋)))
6362cbvrexvw 3226 . . . . . . . . . . 11 (∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋))
6460, 63bitrdi 286 . . . . . . . . . 10 (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)))
6558, 64elab 3630 . . . . . . . . 9 (𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋))
66 simplr 767 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 ∈ (Poly‘𝐵))
67 simpr 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 ∈ (Poly‘𝐵))
6818subrgacl 20233 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
69683expb 1120 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ (SubRing‘ℂfld) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
707, 69sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7170adantlr 713 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7271adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7366, 67, 72plyadd 25578 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒f + 𝑑) ∈ (Poly‘𝐵))
74 plyf 25559 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ (Poly‘𝐵) → 𝑒:ℂ⟶ℂ)
7574ffnd 6669 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (Poly‘𝐵) → 𝑒 Fn ℂ)
7675ad2antlr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 Fn ℂ)
77 plyf 25559 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ (Poly‘𝐵) → 𝑑:ℂ⟶ℂ)
7877ffnd 6669 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (Poly‘𝐵) → 𝑑 Fn ℂ)
7978adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 Fn ℂ)
80 cnex 11132 . . . . . . . . . . . . . . . . . 18 ℂ ∈ V
8180a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ℂ ∈ V)
8210ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑋 ∈ ℂ)
83 fnfvof 7634 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒f + 𝑑)‘𝑋) = ((𝑒𝑋) + (𝑑𝑋)))
8476, 79, 81, 82, 83syl22anc 837 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒f + 𝑑)‘𝑋) = ((𝑒𝑋) + (𝑑𝑋)))
8584eqcomd 2742 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑋) + (𝑑𝑋)) = ((𝑒f + 𝑑)‘𝑋))
86 fveq1 6841 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒f + 𝑑) → (𝑝𝑋) = ((𝑒f + 𝑑)‘𝑋))
8786rspceeqv 3595 . . . . . . . . . . . . . . 15 (((𝑒f + 𝑑) ∈ (Poly‘𝐵) ∧ ((𝑒𝑋) + (𝑑𝑋)) = ((𝑒f + 𝑑)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋))
8873, 85, 87syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋))
89 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑑𝑋) → ((𝑒𝑋) + 𝑐) = ((𝑒𝑋) + (𝑑𝑋)))
9089eqeq1d 2738 . . . . . . . . . . . . . . 15 (𝑐 = (𝑑𝑋) → (((𝑒𝑋) + 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋)))
9190rexbidv 3175 . . . . . . . . . . . . . 14 (𝑐 = (𝑑𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋)))
9288, 91syl5ibrcom 246 . . . . . . . . . . . . 13 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9392rexlimdva 3152 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
94 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒𝑋) → (𝑏 + 𝑐) = ((𝑒𝑋) + 𝑐))
9594eqeq1d 2738 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝑋) → ((𝑏 + 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9695rexbidv 3175 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9796imbi2d 340 . . . . . . . . . . . 12 (𝑏 = (𝑒𝑋) → ((∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋)) ↔ (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋))))
9893, 97syl5ibrcom 246 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))))
9998rexlimdva 3152 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))))
100993imp 1111 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
10149, 57, 65, 100syl3anb 1161 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
102 ovex 7390 . . . . . . . . 9 (𝑏 + 𝑐) ∈ V
103 eqeq1 2740 . . . . . . . . . 10 (𝑎 = (𝑏 + 𝑐) → (𝑎 = (𝑝𝑋) ↔ (𝑏 + 𝑐) = (𝑝𝑋)))
104103rexbidv 3175 . . . . . . . . 9 (𝑎 = (𝑏 + 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋)))
105102, 104elab 3630 . . . . . . . 8 ((𝑏 + 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
106101, 105sylibr 233 . . . . . . 7 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → (𝑏 + 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
107 ax-1cn 11109 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
108 cnfldneg 20823 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℂ → ((invg‘ℂfld)‘1) = -1)
109107, 108mp1i 13 . . . . . . . . . . . . . . . . 17 (𝜑 → ((invg‘ℂfld)‘1) = -1)
110 cnfld1 20822 . . . . . . . . . . . . . . . . . . . 20 1 = (1r‘ℂfld)
111110subrg1cl 20230 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ (SubRing‘ℂfld) → 1 ∈ 𝐵)
1127, 111syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → 1 ∈ 𝐵)
113 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (invg‘ℂfld) = (invg‘ℂfld)
114113subginvcl 18937 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ (SubGrp‘ℂfld) ∧ 1 ∈ 𝐵) → ((invg‘ℂfld)‘1) ∈ 𝐵)
11545, 112, 114syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → ((invg‘ℂfld)‘1) ∈ 𝐵)
116109, 115eqeltrrd 2839 . . . . . . . . . . . . . . . 16 (𝜑 → -1 ∈ 𝐵)
117 plyconst 25567 . . . . . . . . . . . . . . . 16 ((𝐵 ⊆ ℂ ∧ -1 ∈ 𝐵) → (ℂ × {-1}) ∈ (Poly‘𝐵))
1189, 116, 117syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (ℂ × {-1}) ∈ (Poly‘𝐵))
119118adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (ℂ × {-1}) ∈ (Poly‘𝐵))
120 simpr 485 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → 𝑒 ∈ (Poly‘𝐵))
121 cnfldmul 20802 . . . . . . . . . . . . . . . . . 18 · = (.r‘ℂfld)
122121subrgmcl 20234 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 · 𝑏) ∈ 𝐵)
1231223expb 1120 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ (SubRing‘ℂfld) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
1247, 123sylan 580 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
125124adantlr 713 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
126119, 120, 71, 125plymul 25579 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1}) ∘f · 𝑒) ∈ (Poly‘𝐵))
127 ffvelcdm 7032 . . . . . . . . . . . . . . . 16 ((𝑒:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑒𝑋) ∈ ℂ)
12874, 10, 127syl2anr 597 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑒𝑋) ∈ ℂ)
129 cnfldneg 20823 . . . . . . . . . . . . . . 15 ((𝑒𝑋) ∈ ℂ → ((invg‘ℂfld)‘(𝑒𝑋)) = -(𝑒𝑋))
130128, 129syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((invg‘ℂfld)‘(𝑒𝑋)) = -(𝑒𝑋))
131 negex 11399 . . . . . . . . . . . . . . . . 17 -1 ∈ V
132 fnconstg 6730 . . . . . . . . . . . . . . . . 17 (-1 ∈ V → (ℂ × {-1}) Fn ℂ)
133131, 132mp1i 13 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (ℂ × {-1}) Fn ℂ)
13475adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → 𝑒 Fn ℂ)
13580a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ℂ ∈ V)
13610adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → 𝑋 ∈ ℂ)
137 fnfvof 7634 . . . . . . . . . . . . . . . 16 ((((ℂ × {-1}) Fn ℂ ∧ 𝑒 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → (((ℂ × {-1}) ∘f · 𝑒)‘𝑋) = (((ℂ × {-1})‘𝑋) · (𝑒𝑋)))
138133, 134, 135, 136, 137syl22anc 837 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘f · 𝑒)‘𝑋) = (((ℂ × {-1})‘𝑋) · (𝑒𝑋)))
139131fvconst2 7153 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ℂ → ((ℂ × {-1})‘𝑋) = -1)
140136, 139syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1})‘𝑋) = -1)
141140oveq1d 7372 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1})‘𝑋) · (𝑒𝑋)) = (-1 · (𝑒𝑋)))
142128mulm1d 11607 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (-1 · (𝑒𝑋)) = -(𝑒𝑋))
143138, 141, 1423eqtrd 2780 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘f · 𝑒)‘𝑋) = -(𝑒𝑋))
144130, 143eqtr4d 2779 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((invg‘ℂfld)‘(𝑒𝑋)) = (((ℂ × {-1}) ∘f · 𝑒)‘𝑋))
145 fveq1 6841 . . . . . . . . . . . . . 14 (𝑝 = ((ℂ × {-1}) ∘f · 𝑒) → (𝑝𝑋) = (((ℂ × {-1}) ∘f · 𝑒)‘𝑋))
146145rspceeqv 3595 . . . . . . . . . . . . 13 ((((ℂ × {-1}) ∘f · 𝑒) ∈ (Poly‘𝐵) ∧ ((invg‘ℂfld)‘(𝑒𝑋)) = (((ℂ × {-1}) ∘f · 𝑒)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋))
147126, 144, 146syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋))
148 fveqeq2 6851 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (((invg‘ℂfld)‘𝑏) = (𝑝𝑋) ↔ ((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋)))
149148rexbidv 3175 . . . . . . . . . . . 12 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋)))
150147, 149syl5ibrcom 246 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
151150rexlimdva 3152 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
152151imp 407 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
15357, 152sylan2b 594 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
154 fvex 6855 . . . . . . . . 9 ((invg‘ℂfld)‘𝑏) ∈ V
155 eqeq1 2740 . . . . . . . . . 10 (𝑎 = ((invg‘ℂfld)‘𝑏) → (𝑎 = (𝑝𝑋) ↔ ((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
156155rexbidv 3175 . . . . . . . . 9 (𝑎 = ((invg‘ℂfld)‘𝑏) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
157154, 156elab 3630 . . . . . . . 8 (((invg‘ℂfld)‘𝑏) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
158153, 157sylibr 233 . . . . . . 7 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ((invg‘ℂfld)‘𝑏) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
159110a1i 11 . . . . . . 7 (𝜑 → 1 = (1r‘ℂfld))
160121a1i 11 . . . . . . 7 (𝜑 → · = (.r‘ℂfld))
16143, 112sseldd 3945 . . . . . . 7 (𝜑 → 1 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
162125adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
16366, 67, 72, 162plymul 25579 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒f · 𝑑) ∈ (Poly‘𝐵))
164 fnfvof 7634 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒f · 𝑑)‘𝑋) = ((𝑒𝑋) · (𝑑𝑋)))
16576, 79, 81, 82, 164syl22anc 837 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒f · 𝑑)‘𝑋) = ((𝑒𝑋) · (𝑑𝑋)))
166165eqcomd 2742 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑋) · (𝑑𝑋)) = ((𝑒f · 𝑑)‘𝑋))
167 fveq1 6841 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒f · 𝑑) → (𝑝𝑋) = ((𝑒f · 𝑑)‘𝑋))
168167rspceeqv 3595 . . . . . . . . . . . . . . 15 (((𝑒f · 𝑑) ∈ (Poly‘𝐵) ∧ ((𝑒𝑋) · (𝑑𝑋)) = ((𝑒f · 𝑑)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋))
169163, 166, 168syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋))
170 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑑𝑋) → ((𝑒𝑋) · 𝑐) = ((𝑒𝑋) · (𝑑𝑋)))
171170eqeq1d 2738 . . . . . . . . . . . . . . 15 (𝑐 = (𝑑𝑋) → (((𝑒𝑋) · 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋)))
172171rexbidv 3175 . . . . . . . . . . . . . 14 (𝑐 = (𝑑𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋)))
173169, 172syl5ibrcom 246 . . . . . . . . . . . . 13 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
174173rexlimdva 3152 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
175 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒𝑋) → (𝑏 · 𝑐) = ((𝑒𝑋) · 𝑐))
176175eqeq1d 2738 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝑋) → ((𝑏 · 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
177176rexbidv 3175 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
178177imbi2d 340 . . . . . . . . . . . 12 (𝑏 = (𝑒𝑋) → ((∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋)) ↔ (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋))))
179174, 178syl5ibrcom 246 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))))
180179rexlimdva 3152 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))))
1811803imp 1111 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
18249, 57, 65, 181syl3anb 1161 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
183 ovex 7390 . . . . . . . . 9 (𝑏 · 𝑐) ∈ V
184 eqeq1 2740 . . . . . . . . . 10 (𝑎 = (𝑏 · 𝑐) → (𝑎 = (𝑝𝑋) ↔ (𝑏 · 𝑐) = (𝑝𝑋)))
185184rexbidv 3175 . . . . . . . . 9 (𝑎 = (𝑏 · 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋)))
186183, 185elab 3630 . . . . . . . 8 ((𝑏 · 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
187182, 186sylibr 233 . . . . . . 7 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → (𝑏 · 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
18815, 17, 19, 29, 48, 106, 158, 159, 160, 161, 187, 4issubrngd2 20658 . . . . . 6 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∈ (SubRing‘ℂfld))
189 plyid 25570 . . . . . . . . . . 11 ((𝐵 ⊆ ℂ ∧ 1 ∈ 𝐵) → Xp ∈ (Poly‘𝐵))
1909, 112, 189syl2anc 584 . . . . . . . . . 10 (𝜑Xp ∈ (Poly‘𝐵))
191 df-idp 25550 . . . . . . . . . . . 12 Xp = ( I ↾ ℂ)
192191fveq1i 6843 . . . . . . . . . . 11 (Xp𝑋) = (( I ↾ ℂ)‘𝑋)
193 fvresi 7119 . . . . . . . . . . . 12 (𝑋 ∈ ℂ → (( I ↾ ℂ)‘𝑋) = 𝑋)
19410, 193syl 17 . . . . . . . . . . 11 (𝜑 → (( I ↾ ℂ)‘𝑋) = 𝑋)
195192, 194eqtr2id 2789 . . . . . . . . . 10 (𝜑𝑋 = (Xp𝑋))
196 fveq1 6841 . . . . . . . . . . 11 (𝑝 = Xp → (𝑝𝑋) = (Xp𝑋))
197196rspceeqv 3595 . . . . . . . . . 10 ((Xp ∈ (Poly‘𝐵) ∧ 𝑋 = (Xp𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋))
198190, 195, 197syl2anc 584 . . . . . . . . 9 (𝜑 → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋))
199 eqeq1 2740 . . . . . . . . . 10 (𝑎 = 𝑋 → (𝑎 = (𝑝𝑋) ↔ 𝑋 = (𝑝𝑋)))
200199rexbidv 3175 . . . . . . . . 9 (𝑎 = 𝑋 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋)))
20110, 198, 200elabd 3633 . . . . . . . 8 (𝜑𝑋 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
202201snssd 4769 . . . . . . 7 (𝜑 → {𝑋} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
20343, 202unssd 4146 . . . . . 6 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
2044, 6, 12, 13, 14, 188, 203rgspnmin 41484 . . . . 5 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
205204sseld 3943 . . . 4 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) → 𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}))
206 fvex 6855 . . . . . . 7 (𝑝𝑋) ∈ V
207 eleq1 2825 . . . . . . 7 (𝑉 = (𝑝𝑋) → (𝑉 ∈ V ↔ (𝑝𝑋) ∈ V))
208206, 207mpbiri 257 . . . . . 6 (𝑉 = (𝑝𝑋) → 𝑉 ∈ V)
209208rexlimivw 3148 . . . . 5 (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋) → 𝑉 ∈ V)
210 eqeq1 2740 . . . . . 6 (𝑎 = 𝑉 → (𝑎 = (𝑝𝑋) ↔ 𝑉 = (𝑝𝑋)))
211210rexbidv 3175 . . . . 5 (𝑎 = 𝑉 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
212209, 211elab3 3638 . . . 4 (𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋))
213205, 212syl6ib 250 . . 3 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) → ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
2144, 6, 12, 13, 14rgspncl 41482 . . . . . . 7 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂfld))
215214adantr 481 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂfld))
216 simpr 485 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝑝 ∈ (Poly‘𝐵))
2174, 6, 12, 13, 14rgspnssid 41483 . . . . . . . . 9 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
218217unssbd 4148 . . . . . . . 8 (𝜑 → {𝑋} ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
219 snidg 4620 . . . . . . . . 9 (𝑋 ∈ ℂ → 𝑋 ∈ {𝑋})
22010, 219syl 17 . . . . . . . 8 (𝜑𝑋 ∈ {𝑋})
221218, 220sseldd 3945 . . . . . . 7 (𝜑𝑋 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
222221adantr 481 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝑋 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
223217unssad 4147 . . . . . . 7 (𝜑𝐵 ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
224223adantr 481 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝐵 ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
225215, 216, 222, 224cnsrplycl 41480 . . . . 5 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑝𝑋) ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
226 eleq1 2825 . . . . 5 (𝑉 = (𝑝𝑋) → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ↔ (𝑝𝑋) ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
227225, 226syl5ibrcom 246 . . . 4 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑉 = (𝑝𝑋) → 𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
228227rexlimdva 3152 . . 3 (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋) → 𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
229213, 228impbid 211 . 2 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
2302, 229bitrd 278 1 (𝜑 → (𝑉𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wrex 3073  Vcvv 3445  cun 3908  wss 3910  {csn 4586   I cid 5530   × cxp 5631  cres 5635   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  -cneg 11386  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  0gc0g 17321  invgcminusg 18749  SubGrpcsubg 18922  1rcur 19913  Ringcrg 19964  SubRingcsubrg 20218  RingSpancrgspn 20219  fldccnfld 20796  Polycply 25545  Xpcidp 25546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-cmn 19564  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-rgspn 20221  df-cnfld 20797  df-0p 25034  df-ply 25549  df-idp 25550  df-coe 25551  df-dgr 25552
This theorem is referenced by: (None)
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