Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngunsnply Structured version   Visualization version   GIF version

Theorem rngunsnply 39279
 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 2870 . 2 (𝜑 → (𝑉𝑆𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
3 cnring 20253 . . . . . . 7 fld ∈ Ring
43a1i 11 . . . . . 6 (𝜑 → ℂfld ∈ Ring)
5 cnfldbas 20235 . . . . . . 7 ℂ = (Base‘ℂfld)
65a1i 11 . . . . . 6 (𝜑 → ℂ = (Base‘ℂfld))
7 rngunsnply.b . . . . . . . 8 (𝜑𝐵 ∈ (SubRing‘ℂfld))
85subrgss 19230 . . . . . . . 8 (𝐵 ∈ (SubRing‘ℂfld) → 𝐵 ⊆ ℂ)
97, 8syl 17 . . . . . . 7 (𝜑𝐵 ⊆ ℂ)
10 rngunsnply.x . . . . . . . 8 (𝜑𝑋 ∈ ℂ)
1110snssd 4655 . . . . . . 7 (𝜑 → {𝑋} ⊆ ℂ)
129, 11unssd 4089 . . . . . 6 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ℂ)
13 eqidd 2798 . . . . . 6 (𝜑 → (RingSpan‘ℂfld) = (RingSpan‘ℂfld))
14 eqidd 2798 . . . . . 6 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) = ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
15 eqidd 2798 . . . . . . 7 (𝜑 → (ℂflds {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) = (ℂflds {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}))
16 cnfld0 20255 . . . . . . . 8 0 = (0g‘ℂfld)
1716a1i 11 . . . . . . 7 (𝜑 → 0 = (0g‘ℂfld))
18 cnfldadd 20236 . . . . . . . 8 + = (+g‘ℂfld)
1918a1i 11 . . . . . . 7 (𝜑 → + = (+g‘ℂfld))
20 plyf 24475 . . . . . . . . . . . 12 (𝑝 ∈ (Poly‘𝐵) → 𝑝:ℂ⟶ℂ)
21 ffvelrn 6721 . . . . . . . . . . . 12 ((𝑝:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑝𝑋) ∈ ℂ)
2220, 10, 21syl2anr 596 . . . . . . . . . . 11 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑝𝑋) ∈ ℂ)
23 eleq1 2872 . . . . . . . . . . 11 (𝑎 = (𝑝𝑋) → (𝑎 ∈ ℂ ↔ (𝑝𝑋) ∈ ℂ))
2422, 23syl5ibrcom 248 . . . . . . . . . 10 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑎 = (𝑝𝑋) → 𝑎 ∈ ℂ))
2524rexlimdva 3249 . . . . . . . . 9 (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) → 𝑎 ∈ ℂ))
2625ss2abdv 3971 . . . . . . . 8 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ⊆ {𝑎𝑎 ∈ ℂ})
27 abid2 2928 . . . . . . . . 9 {𝑎𝑎 ∈ ℂ} = ℂ
2827, 5eqtri 2821 . . . . . . . 8 {𝑎𝑎 ∈ ℂ} = (Base‘ℂfld)
2926, 28syl6sseq 3944 . . . . . . 7 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ⊆ (Base‘ℂfld))
30 abid2 2928 . . . . . . . . 9 {𝑎𝑎𝐵} = 𝐵
31 plyconst 24483 . . . . . . . . . . . . 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 3443 . . . . . . . . . . . . . . 15 𝑎 ∈ V
3534fvconst2 6840 . . . . . . . . . . . . . 14 (𝑋 ∈ ℂ → ((ℂ × {𝑎})‘𝑋) = 𝑎)
3633, 35syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → ((ℂ × {𝑎})‘𝑋) = 𝑎)
3736eqcomd 2803 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → 𝑎 = ((ℂ × {𝑎})‘𝑋))
38 fveq1 6544 . . . . . . . . . . . . 13 (𝑝 = (ℂ × {𝑎}) → (𝑝𝑋) = ((ℂ × {𝑎})‘𝑋))
3938rspceeqv 3579 . . . . . . . . . . . 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 3971 . . . . . . . . 9 (𝜑 → {𝑎𝑎𝐵} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
4330, 42eqsstrrid 3943 . . . . . . . 8 (𝜑𝐵 ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
44 subrgsubg 19235 . . . . . . . . . 10 (𝐵 ∈ (SubRing‘ℂfld) → 𝐵 ∈ (SubGrp‘ℂfld))
457, 44syl 17 . . . . . . . . 9 (𝜑𝐵 ∈ (SubGrp‘ℂfld))
4616subg0cl 18045 . . . . . . . . 9 (𝐵 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝐵)
4745, 46syl 17 . . . . . . . 8 (𝜑 → 0 ∈ 𝐵)
4843, 47sseldd 3896 . . . . . . 7 (𝜑 → 0 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
49 biid 262 . . . . . . . . 9 (𝜑𝜑)
50 vex 3443 . . . . . . . . . 10 𝑏 ∈ V
51 eqeq1 2801 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (𝑎 = (𝑝𝑋) ↔ 𝑏 = (𝑝𝑋)))
5251rexbidv 3262 . . . . . . . . . . 11 (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝𝑋)))
53 fveq1 6544 . . . . . . . . . . . . 13 (𝑝 = 𝑒 → (𝑝𝑋) = (𝑒𝑋))
5453eqeq2d 2807 . . . . . . . . . . . 12 (𝑝 = 𝑒 → (𝑏 = (𝑝𝑋) ↔ 𝑏 = (𝑒𝑋)))
5554cbvrexv 3406 . . . . . . . . . . 11 (∃𝑝 ∈ (Poly‘𝐵)𝑏 = (𝑝𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋))
5652, 55syl6bb 288 . . . . . . . . . 10 (𝑎 = 𝑏 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋)))
5750, 56elab 3608 . . . . . . . . 9 (𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋))
58 vex 3443 . . . . . . . . . 10 𝑐 ∈ V
59 eqeq1 2801 . . . . . . . . . . . 12 (𝑎 = 𝑐 → (𝑎 = (𝑝𝑋) ↔ 𝑐 = (𝑝𝑋)))
6059rexbidv 3262 . . . . . . . . . . 11 (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝𝑋)))
61 fveq1 6544 . . . . . . . . . . . . 13 (𝑝 = 𝑑 → (𝑝𝑋) = (𝑑𝑋))
6261eqeq2d 2807 . . . . . . . . . . . 12 (𝑝 = 𝑑 → (𝑐 = (𝑝𝑋) ↔ 𝑐 = (𝑑𝑋)))
6362cbvrexv 3406 . . . . . . . . . . 11 (∃𝑝 ∈ (Poly‘𝐵)𝑐 = (𝑝𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋))
6460, 63syl6bb 288 . . . . . . . . . 10 (𝑎 = 𝑐 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)))
6558, 64elab 3608 . . . . . . . . 9 (𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋))
66 simplr 765 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 ∈ (Poly‘𝐵))
67 simpr 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 ∈ (Poly‘𝐵))
6818subrgacl 19240 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
69683expb 1113 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∈ (SubRing‘ℂfld) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
707, 69sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7170adantlr 711 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7271adantlr 711 . . . . . . . . . . . . . . . 16 ((((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 + 𝑏) ∈ 𝐵)
7366, 67, 72plyadd 24494 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒𝑓 + 𝑑) ∈ (Poly‘𝐵))
74 plyf 24475 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ (Poly‘𝐵) → 𝑒:ℂ⟶ℂ)
7574ffnd 6390 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (Poly‘𝐵) → 𝑒 Fn ℂ)
7675ad2antlr 723 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑒 Fn ℂ)
77 plyf 24475 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ (Poly‘𝐵) → 𝑑:ℂ⟶ℂ)
7877ffnd 6390 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (Poly‘𝐵) → 𝑑 Fn ℂ)
7978adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑑 Fn ℂ)
80 cnex 10471 . . . . . . . . . . . . . . . . . 18 ℂ ∈ V
8180a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ℂ ∈ V)
8210ad2antrr 722 . . . . . . . . . . . . . . . . 17 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → 𝑋 ∈ ℂ)
83 fnfvof 7288 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒𝑓 + 𝑑)‘𝑋) = ((𝑒𝑋) + (𝑑𝑋)))
8476, 79, 81, 82, 83syl22anc 835 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑓 + 𝑑)‘𝑋) = ((𝑒𝑋) + (𝑑𝑋)))
8584eqcomd 2803 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑋) + (𝑑𝑋)) = ((𝑒𝑓 + 𝑑)‘𝑋))
86 fveq1 6544 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒𝑓 + 𝑑) → (𝑝𝑋) = ((𝑒𝑓 + 𝑑)‘𝑋))
8786rspceeqv 3579 . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑑𝑋) → ((𝑒𝑋) + 𝑐) = ((𝑒𝑋) + (𝑑𝑋)))
9089eqeq1d 2799 . . . . . . . . . . . . . . 15 (𝑐 = (𝑑𝑋) → (((𝑒𝑋) + 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋)))
9190rexbidv 3262 . . . . . . . . . . . . . 14 (𝑐 = (𝑑𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + (𝑑𝑋)) = (𝑝𝑋)))
9288, 91syl5ibrcom 248 . . . . . . . . . . . . 13 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9392rexlimdva 3249 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
94 oveq1 7030 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒𝑋) → (𝑏 + 𝑐) = ((𝑒𝑋) + 𝑐))
9594eqeq1d 2799 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝑋) → ((𝑏 + 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9695rexbidv 3262 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) + 𝑐) = (𝑝𝑋)))
9796imbi2d 342 . . . . . . . . . . . 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 248 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))))
9998rexlimdva 3249 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))))
100993imp 1104 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
10149, 57, 65, 100syl3anb 1154 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
102 ovex 7055 . . . . . . . . 9 (𝑏 + 𝑐) ∈ V
103 eqeq1 2801 . . . . . . . . . 10 (𝑎 = (𝑏 + 𝑐) → (𝑎 = (𝑝𝑋) ↔ (𝑏 + 𝑐) = (𝑝𝑋)))
104103rexbidv 3262 . . . . . . . . 9 (𝑎 = (𝑏 + 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋)))
105102, 104elab 3608 . . . . . . . 8 ((𝑏 + 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 + 𝑐) = (𝑝𝑋))
106101, 105sylibr 235 . . . . . . 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 10448 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
108 cnfldneg 20257 . . . . . . . . . . . . . . . . . 18 (1 ∈ ℂ → ((invg‘ℂfld)‘1) = -1)
109107, 108mp1i 13 . . . . . . . . . . . . . . . . 17 (𝜑 → ((invg‘ℂfld)‘1) = -1)
110 cnfld1 20256 . . . . . . . . . . . . . . . . . . . 20 1 = (1r‘ℂfld)
111110subrg1cl 19237 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ (SubRing‘ℂfld) → 1 ∈ 𝐵)
1127, 111syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → 1 ∈ 𝐵)
113 eqid 2797 . . . . . . . . . . . . . . . . . . 19 (invg‘ℂfld) = (invg‘ℂfld)
114113subginvcl 18046 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ (SubGrp‘ℂfld) ∧ 1 ∈ 𝐵) → ((invg‘ℂfld)‘1) ∈ 𝐵)
11545, 112, 114syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → ((invg‘ℂfld)‘1) ∈ 𝐵)
116109, 115eqeltrrd 2886 . . . . . . . . . . . . . . . 16 (𝜑 → -1 ∈ 𝐵)
117 plyconst 24483 . . . . . . . . . . . . . . . 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 20237 . . . . . . . . . . . . . . . . . 18 · = (.r‘ℂfld)
122121subrgmcl 19241 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ (SubRing‘ℂfld) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 · 𝑏) ∈ 𝐵)
1231223expb 1113 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ (SubRing‘ℂfld) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
1247, 123sylan 580 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
125124adantlr 711 . . . . . . . . . . . . . 14 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
126119, 120, 71, 125plymul 24495 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1}) ∘𝑓 · 𝑒) ∈ (Poly‘𝐵))
127 ffvelrn 6721 . . . . . . . . . . . . . . . 16 ((𝑒:ℂ⟶ℂ ∧ 𝑋 ∈ ℂ) → (𝑒𝑋) ∈ ℂ)
12874, 10, 127syl2anr 596 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑒𝑋) ∈ ℂ)
129 cnfldneg 20257 . . . . . . . . . . . . . . 15 ((𝑒𝑋) ∈ ℂ → ((invg‘ℂfld)‘(𝑒𝑋)) = -(𝑒𝑋))
130128, 129syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((invg‘ℂfld)‘(𝑒𝑋)) = -(𝑒𝑋))
131 negex 10737 . . . . . . . . . . . . . . . . 17 -1 ∈ V
132 fnconstg 6442 . . . . . . . . . . . . . . . . 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 7288 . . . . . . . . . . . . . . . 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 835 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋) = (((ℂ × {-1})‘𝑋) · (𝑒𝑋)))
139131fvconst2 6840 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ℂ → ((ℂ × {-1})‘𝑋) = -1)
140136, 139syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((ℂ × {-1})‘𝑋) = -1)
141140oveq1d 7038 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1})‘𝑋) · (𝑒𝑋)) = (-1 · (𝑒𝑋)))
142128mulm1d 10946 . . . . . . . . . . . . . . 15 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (-1 · (𝑒𝑋)) = -(𝑒𝑋))
143138, 141, 1423eqtrd 2837 . . . . . . . . . . . . . 14 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋) = -(𝑒𝑋))
144130, 143eqtr4d 2836 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ (Poly‘𝐵)) → ((invg‘ℂfld)‘(𝑒𝑋)) = (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋))
145 fveq1 6544 . . . . . . . . . . . . . 14 (𝑝 = ((ℂ × {-1}) ∘𝑓 · 𝑒) → (𝑝𝑋) = (((ℂ × {-1}) ∘𝑓 · 𝑒)‘𝑋))
146145rspceeqv 3579 . . . . . . . . . . . . 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 6554 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (((invg‘ℂfld)‘𝑏) = (𝑝𝑋) ↔ ((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋)))
149148rexbidv 3262 . . . . . . . . . . . 12 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘(𝑒𝑋)) = (𝑝𝑋)))
150147, 149syl5ibrcom 248 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
151150rexlimdva 3249 . . . . . . . . . 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 593 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
154 fvex 6558 . . . . . . . . 9 ((invg‘ℂfld)‘𝑏) ∈ V
155 eqeq1 2801 . . . . . . . . . 10 (𝑎 = ((invg‘ℂfld)‘𝑏) → (𝑎 = (𝑝𝑋) ↔ ((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
156155rexbidv 3262 . . . . . . . . 9 (𝑎 = ((invg‘ℂfld)‘𝑏) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋)))
157154, 156elab 3608 . . . . . . . 8 (((invg‘ℂfld)‘𝑏) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)((invg‘ℂfld)‘𝑏) = (𝑝𝑋))
158153, 157sylibr 235 . . . . . . 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 3896 . . . . . . 7 (𝜑 → 1 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
162125adantlr 711 . . . . . . . . . . . . . . . 16 ((((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) ∧ (𝑎𝐵𝑏𝐵)) → (𝑎 · 𝑏) ∈ 𝐵)
16366, 67, 72, 162plymul 24495 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑒𝑓 · 𝑑) ∈ (Poly‘𝐵))
164 fnfvof 7288 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn ℂ ∧ 𝑑 Fn ℂ) ∧ (ℂ ∈ V ∧ 𝑋 ∈ ℂ)) → ((𝑒𝑓 · 𝑑)‘𝑋) = ((𝑒𝑋) · (𝑑𝑋)))
16576, 79, 81, 82, 164syl22anc 835 . . . . . . . . . . . . . . . 16 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑓 · 𝑑)‘𝑋) = ((𝑒𝑋) · (𝑑𝑋)))
166165eqcomd 2803 . . . . . . . . . . . . . . 15 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → ((𝑒𝑋) · (𝑑𝑋)) = ((𝑒𝑓 · 𝑑)‘𝑋))
167 fveq1 6544 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒𝑓 · 𝑑) → (𝑝𝑋) = ((𝑒𝑓 · 𝑑)‘𝑋))
168167rspceeqv 3579 . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑑𝑋) → ((𝑒𝑋) · 𝑐) = ((𝑒𝑋) · (𝑑𝑋)))
171170eqeq1d 2799 . . . . . . . . . . . . . . 15 (𝑐 = (𝑑𝑋) → (((𝑒𝑋) · 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋)))
172171rexbidv 3262 . . . . . . . . . . . . . 14 (𝑐 = (𝑑𝑋) → (∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · (𝑑𝑋)) = (𝑝𝑋)))
173169, 172syl5ibrcom 248 . . . . . . . . . . . . 13 (((𝜑𝑒 ∈ (Poly‘𝐵)) ∧ 𝑑 ∈ (Poly‘𝐵)) → (𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
174173rexlimdva 3249 . . . . . . . . . . . 12 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
175 oveq1 7030 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒𝑋) → (𝑏 · 𝑐) = ((𝑒𝑋) · 𝑐))
176175eqeq1d 2799 . . . . . . . . . . . . . 14 (𝑏 = (𝑒𝑋) → ((𝑏 · 𝑐) = (𝑝𝑋) ↔ ((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
177176rexbidv 3262 . . . . . . . . . . . . 13 (𝑏 = (𝑒𝑋) → (∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)((𝑒𝑋) · 𝑐) = (𝑝𝑋)))
178177imbi2d 342 . . . . . . . . . . . 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 248 . . . . . . . . . . 11 ((𝜑𝑒 ∈ (Poly‘𝐵)) → (𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))))
180179rexlimdva 3249 . . . . . . . . . 10 (𝜑 → (∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) → (∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))))
1811803imp 1104 . . . . . . . . 9 ((𝜑 ∧ ∃𝑒 ∈ (Poly‘𝐵)𝑏 = (𝑒𝑋) ∧ ∃𝑑 ∈ (Poly‘𝐵)𝑐 = (𝑑𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
18249, 57, 65, 181syl3anb 1154 . . . . . . . 8 ((𝜑𝑏 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∧ 𝑐 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}) → ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
183 ovex 7055 . . . . . . . . 9 (𝑏 · 𝑐) ∈ V
184 eqeq1 2801 . . . . . . . . . 10 (𝑎 = (𝑏 · 𝑐) → (𝑎 = (𝑝𝑋) ↔ (𝑏 · 𝑐) = (𝑝𝑋)))
185184rexbidv 3262 . . . . . . . . 9 (𝑎 = (𝑏 · 𝑐) → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋)))
186183, 185elab 3608 . . . . . . . 8 ((𝑏 · 𝑐) ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)(𝑏 · 𝑐) = (𝑝𝑋))
187182, 186sylibr 235 . . . . . . 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 19655 . . . . . 6 (𝜑 → {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ∈ (SubRing‘ℂfld))
189 plyid 24486 . . . . . . . . . . 11 ((𝐵 ⊆ ℂ ∧ 1 ∈ 𝐵) → Xp ∈ (Poly‘𝐵))
1909, 112, 189syl2anc 584 . . . . . . . . . 10 (𝜑Xp ∈ (Poly‘𝐵))
191 df-idp 24466 . . . . . . . . . . . 12 Xp = ( I ↾ ℂ)
192191fveq1i 6546 . . . . . . . . . . 11 (Xp𝑋) = (( I ↾ ℂ)‘𝑋)
193 fvresi 6805 . . . . . . . . . . . 12 (𝑋 ∈ ℂ → (( I ↾ ℂ)‘𝑋) = 𝑋)
19410, 193syl 17 . . . . . . . . . . 11 (𝜑 → (( I ↾ ℂ)‘𝑋) = 𝑋)
195192, 194syl5req 2846 . . . . . . . . . 10 (𝜑𝑋 = (Xp𝑋))
196 fveq1 6544 . . . . . . . . . . 11 (𝑝 = Xp → (𝑝𝑋) = (Xp𝑋))
197196rspceeqv 3579 . . . . . . . . . 10 ((Xp ∈ (Poly‘𝐵) ∧ 𝑋 = (Xp𝑋)) → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋))
198190, 195, 197syl2anc 584 . . . . . . . . 9 (𝜑 → ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋))
199 eqeq1 2801 . . . . . . . . . . . 12 (𝑎 = 𝑋 → (𝑎 = (𝑝𝑋) ↔ 𝑋 = (𝑝𝑋)))
200199rexbidv 3262 . . . . . . . . . . 11 (𝑎 = 𝑋 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑋 = (𝑝𝑋)))
201200elabg 3607 . . . . . . . . . 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 258 . . . . . . . 8 (𝜑𝑋 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
204203snssd 4655 . . . . . . 7 (𝜑 → {𝑋} ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
20543, 204unssd 4089 . . . . . 6 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
2064, 6, 12, 13, 14, 188, 205rgspnmin 39277 . . . . 5 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ⊆ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)})
207206sseld 3894 . . . 4 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) → 𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)}))
208 fvex 6558 . . . . . . 7 (𝑝𝑋) ∈ V
209 eleq1 2872 . . . . . . 7 (𝑉 = (𝑝𝑋) → (𝑉 ∈ V ↔ (𝑝𝑋) ∈ V))
210208, 209mpbiri 259 . . . . . 6 (𝑉 = (𝑝𝑋) → 𝑉 ∈ V)
211210rexlimivw 3247 . . . . 5 (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋) → 𝑉 ∈ V)
212 eqeq1 2801 . . . . . 6 (𝑎 = 𝑉 → (𝑎 = (𝑝𝑋) ↔ 𝑉 = (𝑝𝑋)))
213212rexbidv 3262 . . . . 5 (𝑎 = 𝑉 → (∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
214211, 213elab3 3615 . . . 4 (𝑉 ∈ {𝑎 ∣ ∃𝑝 ∈ (Poly‘𝐵)𝑎 = (𝑝𝑋)} ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋))
215207, 214syl6ib 252 . . 3 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) → ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
2164, 6, 12, 13, 14rgspncl 39275 . . . . . . 7 (𝜑 → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂfld))
217216adantr 481 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ∈ (SubRing‘ℂfld))
218 simpr 485 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝑝 ∈ (Poly‘𝐵))
2194, 6, 12, 13, 14rgspnssid 39276 . . . . . . . . 9 (𝜑 → (𝐵 ∪ {𝑋}) ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
220219unssbd 4091 . . . . . . . 8 (𝜑 → {𝑋} ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
221 snidg 4510 . . . . . . . . 9 (𝑋 ∈ ℂ → 𝑋 ∈ {𝑋})
22210, 221syl 17 . . . . . . . 8 (𝜑𝑋 ∈ {𝑋})
223220, 222sseldd 3896 . . . . . . 7 (𝜑𝑋 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
224223adantr 481 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝑋 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
225219unssad 4090 . . . . . . 7 (𝜑𝐵 ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
226225adantr 481 . . . . . 6 ((𝜑𝑝 ∈ (Poly‘𝐵)) → 𝐵 ⊆ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
227217, 218, 224, 226cnsrplycl 39273 . . . . 5 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑝𝑋) ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})))
228 eleq1 2872 . . . . 5 (𝑉 = (𝑝𝑋) → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ↔ (𝑝𝑋) ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
229227, 228syl5ibrcom 248 . . . 4 ((𝜑𝑝 ∈ (Poly‘𝐵)) → (𝑉 = (𝑝𝑋) → 𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
230229rexlimdva 3249 . . 3 (𝜑 → (∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋) → 𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋}))))
231215, 230impbid 213 . 2 (𝜑 → (𝑉 ∈ ((RingSpan‘ℂfld)‘(𝐵 ∪ {𝑋})) ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
2322, 231bitrd 280 1 (𝜑 → (𝑉𝑆 ↔ ∃𝑝 ∈ (Poly‘𝐵)𝑉 = (𝑝𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083  {cab 2777  ∃wrex 3108  Vcvv 3440   ∪ cun 3863   ⊆ wss 3865  {csn 4478   I cid 5354   × cxp 5448   ↾ cres 5452   Fn wfn 6227  ⟶wf 6228  ‘cfv 6232  (class class class)co 7023   ∘𝑓 cof 7272  ℂcc 10388  0cc0 10390  1c1 10391   + caddc 10393   · cmul 10395  -cneg 10724  Basecbs 16316   ↾s cress 16317  +gcplusg 16398  .rcmulr 16399  0gc0g 16546  invgcminusg 17866  SubGrpcsubg 18031  1rcur 18945  Ringcrg 18991  SubRingcsubrg 19225  RingSpancrgspn 19226  ℂfldccnfld 20231  Polycply 24461  Xpcidp 24462 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468  ax-addf 10469  ax-mulf 10470 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-map 8265  df-pm 8266  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-oi 8827  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-z 11836  df-dec 11953  df-uz 12098  df-rp 12244  df-fz 12747  df-fzo 12888  df-fl 13016  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-rlim 14684  df-sum 14881  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-mulr 16412  df-starv 16413  df-tset 16417  df-ple 16418  df-ds 16420  df-unif 16421  df-0g 16548  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-grp 17868  df-minusg 17869  df-subg 18034  df-cmn 18639  df-mgp 18934  df-ur 18946  df-ring 18993  df-cring 18994  df-subrg 19227  df-rgspn 19228  df-cnfld 20232  df-0p 23958  df-ply 24465  df-idp 24466  df-coe 24467  df-dgr 24468 This theorem is referenced by: (None)
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