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Theorem rngunsnply 38244
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 2871 . 2 (𝜑 → (𝑉𝑆𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
3 cnring 19976 . . . . . . 7 fld ∈ Ring
43a1i 11 . . . . . 6 (𝜑 → ℂfld ∈ Ring)
5 cnfldbas 19958 . . . . . . 7 ℂ = (Base‘ℂfld)
65a1i 11 . . . . . 6 (𝜑 → ℂ = (Base‘ℂfld))
7 rngunsnply.b . . . . . . . 8 (𝜑𝐵 ∈ (SubRing‘ℂfld))
85subrgss 18985 . . . . . . . 8 (𝐵 ∈ (SubRing‘ℂfld) → 𝐵 ⊆ ℂ)
97, 8syl 17 . . . . . . 7 (𝜑𝐵 ⊆ ℂ)
10 rngunsnply.x . . . . . . . 8 (𝜑𝑋 ∈ ℂ)
1110snssd 4530 . . . . . . 7 (𝜑 → {𝑋} ⊆ ℂ)
129, 11unssd 3988 . . . . . 6 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ℂ)
13 eqidd 2807 . . . . . 6 (𝜑 → (RingSpan‘ℂfld) = (RingSpan‘ℂfld))
14 eqidd 2807 . . . . . 6 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
15 eqidd 2807 . . . . . . 7 (𝜑 → (ℂflds {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) = (ℂflds {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}))
16 cnfld0 19978 . . . . . . . 8 0 = (0g‘ℂfld)
1716a1i 11 . . . . . . 7 (𝜑 → 0 = (0g‘ℂfld))
18 cnfldadd 19959 . . . . . . . 8 + = (+g‘ℂfld)
1918a1i 11 . . . . . . 7 (𝜑 → + = (+g‘ℂfld))
20 plyf 24168 . . . . . . . . . . . 12 (𝑝 ∈ (Poly‘𝐵) → 𝑝:ℂ⟶ℂ)
21 ffvelrn 6579 . . . . . . . . . . . 12 ((𝑝:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑝𝑋) ∈ ℂ)
2220, 10, 21syl2anr 586 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑝𝑋) ∈ ℂ)
23 eleq1 2873 . . . . . . . . . . 11 (𝑎 = (𝑝𝑋) → (𝑎 ∈ ℂ ↔ (𝑝𝑋) ∈ ℂ))
2422, 23syl5ibrcom 238 . . . . . . . . . 10 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑎 = (𝑝𝑋) → 𝑎 ∈ ℂ))
2524rexlimdva 3219 . . . . . . . . 9 (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) → 𝑎 ∈ ℂ))
2625ss2abdv 3872 . . . . . . . 8 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ⊆ {𝑎𝑎 ∈ ℂ})
27 abid2 2929 . . . . . . . . 9 {𝑎𝑎 ∈ ℂ} = ℂ
2827, 5eqtri 2828 . . . . . . . 8 {𝑎𝑎 ∈ ℂ} = (Base‘ℂfld)
2926, 28syl6sseq 3848 . . . . . . 7 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ⊆ (Base‘ℂfld))
30 abid2 2929 . . . . . . . . 9 {𝑎𝑎𝐵} = 𝐵
31 plyconst 24176 . . . . . . . . . . . . 13 ((𝐵 ⊆ ℂ ∧ 𝑎𝐵) → (ℂ × {𝑎}) ∈ (Poly‘𝐵))
329, 31sylan 571 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (ℂ × {𝑎}) ∈ (Poly‘𝐵))
3310adantr 468 . . . . . . . . . . . . . 14 ((𝜑𝑎𝐵) → 𝑋 ∈ ℂ)
34 vex 3394 . . . . . . . . . . . . . . 15 𝑎 ∈ V
3534fvconst2 6694 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → ((ℂ × {𝑎})‘𝑋) = 𝑎)
3633, 35syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → ((ℂ × {𝑎})‘𝑋) = 𝑎)
3736eqcomd 2812 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → 𝑎 = ((ℂ × {𝑎})‘𝑋))
38 fveq1 6407 . . . . . . . . . . . . 13 (𝑝 = (ℂ × {𝑎}) → (𝑝𝑋) = ((ℂ × {𝑎})‘𝑋))
3938rspceeqv 3520 . . . . . . . . . . . 12 (((ℂ × {𝑎}) ∈ (Poly‘𝐵) ∧ 𝑎 = ((ℂ × {𝑎})‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋))
4032, 37, 39syl2anc 575 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋))
4140ex 399 . . . . . . . . . 10 (𝜑 → (𝑎𝐵 → ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)))
4241ss2abdv 3872 . . . . . . . . 9 (𝜑 → {𝑎𝑎𝐵} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
4330, 42syl5eqssr 3847 . . . . . . . 8 (𝜑𝐵 ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
44 subrgsubg 18990 . . . . . . . . . 10 (𝐵 ∈ (SubRing‘ℂfld) → 𝐵 ∈ (SubGrp‘ℂfld))
457, 44syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ (SubGrp‘ℂfld))
4616subg0cl 17804 . . . . . . . . 9 (𝐵 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝐵)
4745, 46syl 17 . . . . . . . 8 (𝜑 → 0 ∈ 𝐵)
4843, 47sseldd 3799 . . . . . . 7 (𝜑 → 0 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
49 biid 252 . . . . . . . . 9 (𝜑𝜑)
50 vex 3394 . . . . . . . . . 10 𝑏 ∈ V
51 eqeq1 2810 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑎 = (𝑝𝑋) ↔ 𝑏 = (𝑝𝑋)))
5251rexbidv 3240 . . . . . . . . . . 11 (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝𝑋)))
53 fveq1 6407 . . . . . . . . . . . . 13 (𝑝 = 𝑒 → (𝑝𝑋) = (𝑒𝑋))
5453eqeq2d 2816 . . . . . . . . . . . 12 (𝑝 = 𝑒 → (𝑏 = (𝑝𝑋) ↔ 𝑏 = (𝑒𝑋)))
5554cbvrexv 3361 . . . . . . . . . . 11 (∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋))
5652, 55syl6bb 278 . . . . . . . . . 10 (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋)))
5750, 56elab 3545 . . . . . . . . 9 (𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋))
58 vex 3394 . . . . . . . . . 10 𝑐 ∈ V
59 eqeq1 2810 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎 = (𝑝𝑋) ↔ 𝑐 = (𝑝𝑋)))
6059rexbidv 3240 . . . . . . . . . . 11 (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝𝑋)))
61 fveq1 6407 . . . . . . . . . . . . 13 (𝑝 = 𝑑 → (𝑝𝑋) = (𝑑𝑋))
6261eqeq2d 2816 . . . . . . . . . . . 12 (𝑝 = 𝑑 → (𝑐 = (𝑝𝑋) ↔ 𝑐 = (𝑑𝑋)))
6362cbvrexv 3361 . . . . . . . . . . 11 (∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋))
6460, 63syl6bb 278 . . . . . . . . . 10 (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)))
6558, 64elab 3545 . . . . . . . . 9 (𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋))
66 simplr 776 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 ∈ (Poly‘𝐵))
67 simpr 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 ∈ (Poly‘𝐵))
6818subrgacl 18995 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
69683expb 1142 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ (SubRing‘ℂfld) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
707, 69sylan 571 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7170adantlr 697 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7271adantlr 697 . . . . . . . . . . . . . . . 16 ((((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7366, 67, 72plyadd 24187 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒𝑓 + 𝑑) ∈ (Poly‘𝐵))
74 plyf 24168 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ (Poly‘𝐵) → 𝑒:ℂ⟶ℂ)
7574ffnd 6257 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (Poly‘𝐵) → 𝑒 Fn ℂ)
7675ad2antlr 709 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 Fn ℂ)
77 plyf 24168 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ (Poly‘𝐵) → 𝑑:ℂ⟶ℂ)
7877ffnd 6257 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (Poly‘𝐵) → 𝑑 Fn ℂ)
7978adantl 469 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 Fn ℂ)
80 cnex 10302 . . . . . . . . . . . . . . . . . 18 ℂ ∈ V
8180a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ℂ ∈ V)
8210ad2antrr 708 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑋 ∈ ℂ)
83 fnfvof 7141 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒𝑓 + 𝑑)‘𝑋) = ((𝑒𝑋) + (𝑑𝑋)))
8476, 79, 81, 82, 83syl22anc 858 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑓 + 𝑑)‘𝑋) = ((𝑒𝑋) + (𝑑𝑋)))
8584eqcomd 2812 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑋) + (𝑑𝑋)) = ((𝑒𝑓 + 𝑑)‘𝑋))
86 fveq1 6407 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒𝑓 + 𝑑) → (𝑝𝑋) = ((𝑒𝑓 + 𝑑)‘𝑋))
8786rspceeqv 3520 . . . . . . . . . . . . . . 15 (((𝑒𝑓 + 𝑑) ∈ (Poly‘𝐵) ∧ ((𝑒𝑋) + (𝑑𝑋)) = ((𝑒𝑓 + 𝑑)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋))
8873, 85, 87syl2anc 575 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋))
89 oveq2 6882 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑑𝑋) → ((𝑒𝑋) + 𝑐) = ((𝑒𝑋) + (𝑑𝑋)))
9089eqeq1d 2808 . . . . . . . . . . . . . . 15 (𝑐 = (𝑑𝑋) → (((𝑒𝑋) + 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋)))
9190rexbidv 3240 . . . . . . . . . . . . . 14 (𝑐 = (𝑑𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋)))
9288, 91syl5ibrcom 238 . . . . . . . . . . . . 13 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9392rexlimdva 3219 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
94 oveq1 6881 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒𝑋) → (𝑏 + 𝑐) = ((𝑒𝑋) + 𝑐))
9594eqeq1d 2808 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝑋) → ((𝑏 + 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9695rexbidv 3240 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9796imbi2d 331 . . . . . . . . . . . 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 238 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))))
9998rexlimdva 3219 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))))
100993imp 1130 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
10149, 57, 65, 100syl3anb 1193 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
102 ovex 6906 . . . . . . . . 9 (𝑏 + 𝑐) ∈ V
103 eqeq1 2810 . . . . . . . . . 10 (𝑎 = (𝑏 + 𝑐) → (𝑎 = (𝑝𝑋) ↔ (𝑏 + 𝑐) = (𝑝𝑋)))
104103rexbidv 3240 . . . . . . . . 9 (𝑎 = (𝑏 + 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋)))
105102, 104elab 3545 . . . . . . . 8 ((𝑏 + 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
106101, 105sylibr 225 . . . . . . 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 10279 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
108 cnfldneg 19980 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℂ → ((invg‘ℂfld)‘1) = -1)
109107, 108mp1i 13 . . . . . . . . . . . . . . . . 17 (𝜑 → ((invg‘ℂfld)‘1) = -1)
110 cnfld1 19979 . . . . . . . . . . . . . . . . . . . 20 1 = (1r‘ℂfld)
111110subrg1cl 18992 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ (SubRing‘ℂfld) → 1 ∈ 𝐵)
1127, 111syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → 1 ∈ 𝐵)
113 eqid 2806 . . . . . . . . . . . . . . . . . . 19 (invg‘ℂfld) = (invg‘ℂfld)
114113subginvcl 17805 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ (SubGrp‘ℂfld) ∧ 1 ∈ 𝐵) → ((invg‘ℂfld)‘1) ∈ 𝐵)
11545, 112, 114syl2anc 575 . . . . . . . . . . . . . . . . 17 (𝜑 → ((invg‘ℂfld)‘1) ∈ 𝐵)
116109, 115eqeltrrd 2886 . . . . . . . . . . . . . . . 16 (𝜑 → -1 ∈ 𝐵)
117 plyconst 24176 . . . . . . . . . . . . . . . 16 ((𝐵 ⊆ ℂ ∧ -1 ∈ 𝐵) → (ℂ × {-1}) ∈ (Poly‘𝐵))
1189, 116, 117syl2anc 575 . . . . . . . . . . . . . . 15 (𝜑 → (ℂ × {-1}) ∈ (Poly‘𝐵))
119118adantr 468 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (ℂ × {-1}) ∈ (Poly‘𝐵))
120 simpr 473 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → 𝑒 ∈ (Poly‘𝐵))
121 cnfldmul 19960 . . . . . . . . . . . . . . . . . 18 · = (.r‘ℂfld)
122121subrgmcl 18996 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 · 𝑏) ∈ 𝐵)
1231223expb 1142 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ (SubRing‘ℂfld) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
1247, 123sylan 571 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
125124adantlr 697 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
126119, 120, 71, 125plymul 24188 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1}) ∘𝑓 · 𝑒) ∈ (Poly‘𝐵))
127 ffvelrn 6579 . . . . . . . . . . . . . . . 16 ((𝑒:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑒𝑋) ∈ ℂ)
12874, 10, 127syl2anr 586 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑒𝑋) ∈ ℂ)
129 cnfldneg 19980 . . . . . . . . . . . . . . 15 ((𝑒𝑋) ∈ ℂ → ((invg‘ℂfld)‘(𝑒𝑋)) = -(𝑒𝑋))
130128, 129syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((invg‘ℂfld)‘(𝑒𝑋)) = -(𝑒𝑋))
131 negex 10564 . . . . . . . . . . . . . . . . 17 -1 ∈ V
132 fnconstg 6308 . . . . . . . . . . . . . . . . 17 (-1 ∈ V → (ℂ × {-1}) Fn ℂ)
133131, 132mp1i 13 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (ℂ × {-1}) Fn ℂ)
13475adantl 469 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → 𝑒 Fn ℂ)
13580a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ℂ ∈ V)
13610adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → 𝑋 ∈ ℂ)
137 fnfvof 7141 . . . . . . . . . . . . . . . 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 858 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋) = (((ℂ × {-1})‘𝑋) · (𝑒𝑋)))
139131fvconst2 6694 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ℂ → ((ℂ × {-1})‘𝑋) = -1)
140136, 139syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1})‘𝑋) = -1)
141140oveq1d 6889 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1})‘𝑋) · (𝑒𝑋)) = (-1 · (𝑒𝑋)))
142128mulm1d 10767 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (-1 · (𝑒𝑋)) = -(𝑒𝑋))
143138, 141, 1423eqtrd 2844 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋) = -(𝑒𝑋))
144130, 143eqtr4d 2843 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((invg‘ℂfld)‘(𝑒𝑋)) = (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋))
145 fveq1 6407 . . . . . . . . . . . . . 14 (𝑝 = ((ℂ × {-1}) ∘𝑓 · 𝑒) → (𝑝𝑋) = (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋))
146145rspceeqv 3520 . . . . . . . . . . . . 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 575 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋))
148 fveqeq2 6417 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (((invg‘ℂfld)‘𝑏) = (𝑝𝑋) ↔ ((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋)))
149148rexbidv 3240 . . . . . . . . . . . 12 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋)))
150147, 149syl5ibrcom 238 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
151150rexlimdva 3219 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
152151imp 395 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
15357, 152sylan2b 583 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
154 fvex 6421 . . . . . . . . 9 ((invg‘ℂfld)‘𝑏) ∈ V
155 eqeq1 2810 . . . . . . . . . 10 (𝑎 = ((invg‘ℂfld)‘𝑏) → (𝑎 = (𝑝𝑋) ↔ ((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
156155rexbidv 3240 . . . . . . . . 9 (𝑎 = ((invg‘ℂfld)‘𝑏) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
157154, 156elab 3545 . . . . . . . 8 (((invg‘ℂfld)‘𝑏) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
158153, 157sylibr 225 . . . . . . 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 3799 . . . . . . 7 (𝜑 → 1 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
162125adantlr 697 . . . . . . . . . . . . . . . 16 ((((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
16366, 67, 72, 162plymul 24188 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒𝑓 · 𝑑) ∈ (Poly‘𝐵))
164 fnfvof 7141 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒𝑓 · 𝑑)‘𝑋) = ((𝑒𝑋) · (𝑑𝑋)))
16576, 79, 81, 82, 164syl22anc 858 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑓 · 𝑑)‘𝑋) = ((𝑒𝑋) · (𝑑𝑋)))
166165eqcomd 2812 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑋) · (𝑑𝑋)) = ((𝑒𝑓 · 𝑑)‘𝑋))
167 fveq1 6407 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒𝑓 · 𝑑) → (𝑝𝑋) = ((𝑒𝑓 · 𝑑)‘𝑋))
168167rspceeqv 3520 . . . . . . . . . . . . . . 15 (((𝑒𝑓 · 𝑑) ∈ (Poly‘𝐵) ∧ ((𝑒𝑋) · (𝑑𝑋)) = ((𝑒𝑓 · 𝑑)‘𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋))
169163, 166, 168syl2anc 575 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋))
170 oveq2 6882 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑑𝑋) → ((𝑒𝑋) · 𝑐) = ((𝑒𝑋) · (𝑑𝑋)))
171170eqeq1d 2808 . . . . . . . . . . . . . . 15 (𝑐 = (𝑑𝑋) → (((𝑒𝑋) · 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋)))
172171rexbidv 3240 . . . . . . . . . . . . . 14 (𝑐 = (𝑑𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋)))
173169, 172syl5ibrcom 238 . . . . . . . . . . . . 13 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
174173rexlimdva 3219 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
175 oveq1 6881 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒𝑋) → (𝑏 · 𝑐) = ((𝑒𝑋) · 𝑐))
176175eqeq1d 2808 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝑋) → ((𝑏 · 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
177176rexbidv 3240 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
178177imbi2d 331 . . . . . . . . . . . 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 238 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))))
180179rexlimdva 3219 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))))
1811803imp 1130 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
18249, 57, 65, 181syl3anb 1193 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
183 ovex 6906 . . . . . . . . 9 (𝑏 · 𝑐) ∈ V
184 eqeq1 2810 . . . . . . . . . 10 (𝑎 = (𝑏 · 𝑐) → (𝑎 = (𝑝𝑋) ↔ (𝑏 · 𝑐) = (𝑝𝑋)))
185184rexbidv 3240 . . . . . . . . 9 (𝑎 = (𝑏 · 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋)))
186183, 185elab 3545 . . . . . . . 8 ((𝑏 · 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
187182, 186sylibr 225 . . . . . . 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 19398 . . . . . 6 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∈ (SubRing‘ℂfld))
189 plyid 24179 . . . . . . . . . . 11 ((𝐵 ⊆ ℂ ∧ 1 ∈ 𝐵) → Xp ∈ (Poly‘𝐵))
1909, 112, 189syl2anc 575 . . . . . . . . . 10 (𝜑Xp ∈ (Poly‘𝐵))
191 df-idp 24159 . . . . . . . . . . . 12 Xp = ( I ↾ ℂ)
192191fveq1i 6409 . . . . . . . . . . 11 (Xp𝑋) = (( I ↾ ℂ)‘𝑋)
193 fvresi 6664 . . . . . . . . . . . 12 (𝑋 ∈ ℂ → (( I ↾ ℂ)‘𝑋) = 𝑋)
19410, 193syl 17 . . . . . . . . . . 11 (𝜑 → (( I ↾ ℂ)‘𝑋) = 𝑋)
195192, 194syl5req 2853 . . . . . . . . . 10 (𝜑𝑋 = (Xp𝑋))
196 fveq1 6407 . . . . . . . . . . 11 (𝑝 = Xp → (𝑝𝑋) = (Xp𝑋))
197196rspceeqv 3520 . . . . . . . . . 10 ((Xp ∈ (Poly‘𝐵) ∧ 𝑋 = (Xp𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋))
198190, 195, 197syl2anc 575 . . . . . . . . 9 (𝜑 → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋))
199 eqeq1 2810 . . . . . . . . . . . 12 (𝑎 = 𝑋 → (𝑎 = (𝑝𝑋) ↔ 𝑋 = (𝑝𝑋)))
200199rexbidv 3240 . . . . . . . . . . 11 (𝑎 = 𝑋 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋)))
201200elabg 3546 . . . . . . . . . 10 (𝑋 ∈ ℂ → (𝑋 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋)))
20210, 201syl 17 . . . . . . . . 9 (𝜑 → (𝑋 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋)))
203198, 202mpbird 248 . . . . . . . 8 (𝜑𝑋 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
204203snssd 4530 . . . . . . 7 (𝜑 → {𝑋} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
20543, 204unssd 3988 . . . . . 6 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
2064, 6, 12, 13, 14, 188, 205rgspnmin 38242 . . . . 5 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
207206sseld 3797 . . . 4 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) → 𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}))
208 fvex 6421 . . . . . . 7 (𝑝𝑋) ∈ V
209 eleq1 2873 . . . . . . 7 (𝑉 = (𝑝𝑋) → (𝑉 ∈ V ↔ (𝑝𝑋) ∈ V))
210208, 209mpbiri 249 . . . . . 6 (𝑉 = (𝑝𝑋) → 𝑉 ∈ V)
211210rexlimivw 3217 . . . . 5 (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋) → 𝑉 ∈ V)
212 eqeq1 2810 . . . . . 6 (𝑎 = 𝑉 → (𝑎 = (𝑝𝑋) ↔ 𝑉 = (𝑝𝑋)))
213212rexbidv 3240 . . . . 5 (𝑎 = 𝑉 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
214211, 213elab3 3553 . . . 4 (𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋))
215207, 214syl6ib 242 . . 3 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) → ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
2164, 6, 12, 13, 14rgspncl 38240 . . . . . . 7 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂfld))
217216adantr 468 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂfld))
218 simpr 473 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝑝 ∈ (Poly‘𝐵))
2194, 6, 12, 13, 14rgspnssid 38241 . . . . . . . . 9 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
220219unssbd 3990 . . . . . . . 8 (𝜑 → {𝑋} ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
221 snidg 4400 . . . . . . . . 9 (𝑋 ∈ ℂ → 𝑋 ∈ {𝑋})
22210, 221syl 17 . . . . . . . 8 (𝜑𝑋 ∈ {𝑋})
223220, 222sseldd 3799 . . . . . . 7 (𝜑𝑋 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
224223adantr 468 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝑋 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
225219unssad 3989 . . . . . . 7 (𝜑𝐵 ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
226225adantr 468 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝐵 ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
227217, 218, 224, 226cnsrplycl 38238 . . . . 5 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑝𝑋) ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
228 eleq1 2873 . . . . 5 (𝑉 = (𝑝𝑋) → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ↔ (𝑝𝑋) ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
229227, 228syl5ibrcom 238 . . . 4 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑉 = (𝑝𝑋) → 𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
230229rexlimdva 3219 . . 3 (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋) → 𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
231215, 230impbid 203 . 2 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
2322, 231bitrd 270 1 (𝜑 → (𝑉𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  wrex 3097  Vcvv 3391  cun 3767  wss 3769  {csn 4370   I cid 5218   × cxp 5309  cres 5313   Fn wfn 6096  wf 6097  cfv 6101  (class class class)co 6874  𝑓 cof 7125  cc 10219  0cc0 10221  1c1 10222   + caddc 10224   · cmul 10226  -cneg 10552  Basecbs 16068  s cress 16069  +gcplusg 16153  .rcmulr 16154  0gc0g 16305  invgcminusg 17628  SubGrpcsubg 17790  1rcur 18703  Ringcrg 18749  SubRingcsubrg 18980  RingSpancrgspn 18981  fldccnfld 19954  Polycply 24154  Xpcidp 24155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299  ax-addf 10300  ax-mulf 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-sup 8587  df-inf 8588  df-oi 8654  df-card 9048  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-4 11366  df-5 11367  df-6 11368  df-7 11369  df-8 11370  df-9 11371  df-n0 11560  df-z 11644  df-dec 11760  df-uz 11905  df-rp 12047  df-fz 12550  df-fzo 12690  df-fl 12817  df-seq 13025  df-exp 13084  df-hash 13338  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-clim 14442  df-rlim 14443  df-sum 14640  df-struct 16070  df-ndx 16071  df-slot 16072  df-base 16074  df-sets 16075  df-ress 16076  df-plusg 16166  df-mulr 16167  df-starv 16168  df-tset 16172  df-ple 16173  df-ds 16175  df-unif 16176  df-0g 16307  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-grp 17630  df-minusg 17631  df-subg 17793  df-cmn 18396  df-mgp 18692  df-ur 18704  df-ring 18751  df-cring 18752  df-subrg 18982  df-rgspn 18983  df-cnfld 19955  df-0p 23651  df-ply 24158  df-idp 24159  df-coe 24160  df-dgr 24161
This theorem is referenced by: (None)
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