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Theorem rngunsnply 41543
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 (πœ‘ β†’ 𝐡 ∈ (SubRingβ€˜β„‚fld))
rngunsnply.x (πœ‘ β†’ 𝑋 ∈ β„‚)
rngunsnply.s (πœ‘ β†’ 𝑆 = ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
Assertion
Ref Expression
rngunsnply (πœ‘ β†’ (𝑉 ∈ 𝑆 ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹)))
Distinct variable groups:   πœ‘,𝑝   𝐡,𝑝   𝑋,𝑝   𝑉,𝑝
Allowed substitution hint:   𝑆(𝑝)

Proof of Theorem rngunsnply
Dummy variables π‘Ž 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngunsnply.s . . 3 (πœ‘ β†’ 𝑆 = ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
21eleq2d 2820 . 2 (πœ‘ β†’ (𝑉 ∈ 𝑆 ↔ 𝑉 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋}))))
3 cnring 20835 . . . . . . 7 β„‚fld ∈ Ring
43a1i 11 . . . . . 6 (πœ‘ β†’ β„‚fld ∈ Ring)
5 cnfldbas 20816 . . . . . . 7 β„‚ = (Baseβ€˜β„‚fld)
65a1i 11 . . . . . 6 (πœ‘ β†’ β„‚ = (Baseβ€˜β„‚fld))
7 rngunsnply.b . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ (SubRingβ€˜β„‚fld))
85subrgss 20237 . . . . . . . 8 (𝐡 ∈ (SubRingβ€˜β„‚fld) β†’ 𝐡 βŠ† β„‚)
97, 8syl 17 . . . . . . 7 (πœ‘ β†’ 𝐡 βŠ† β„‚)
10 rngunsnply.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ β„‚)
1110snssd 4770 . . . . . . 7 (πœ‘ β†’ {𝑋} βŠ† β„‚)
129, 11unssd 4147 . . . . . 6 (πœ‘ β†’ (𝐡 βˆͺ {𝑋}) βŠ† β„‚)
13 eqidd 2734 . . . . . 6 (πœ‘ β†’ (RingSpanβ€˜β„‚fld) = (RingSpanβ€˜β„‚fld))
14 eqidd 2734 . . . . . 6 (πœ‘ β†’ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) = ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
15 eqidd 2734 . . . . . . 7 (πœ‘ β†’ (β„‚fld β†Ύs {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}) = (β„‚fld β†Ύs {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}))
16 cnfld0 20837 . . . . . . . 8 0 = (0gβ€˜β„‚fld)
1716a1i 11 . . . . . . 7 (πœ‘ β†’ 0 = (0gβ€˜β„‚fld))
18 cnfldadd 20817 . . . . . . . 8 + = (+gβ€˜β„‚fld)
1918a1i 11 . . . . . . 7 (πœ‘ β†’ + = (+gβ€˜β„‚fld))
20 plyf 25575 . . . . . . . . . . . 12 (𝑝 ∈ (Polyβ€˜π΅) β†’ 𝑝:β„‚βŸΆβ„‚)
21 ffvelcdm 7033 . . . . . . . . . . . 12 ((𝑝:β„‚βŸΆβ„‚ ∧ 𝑋 ∈ β„‚) β†’ (π‘β€˜π‘‹) ∈ β„‚)
2220, 10, 21syl2anr 598 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ (π‘β€˜π‘‹) ∈ β„‚)
23 eleq1 2822 . . . . . . . . . . 11 (π‘Ž = (π‘β€˜π‘‹) β†’ (π‘Ž ∈ β„‚ ↔ (π‘β€˜π‘‹) ∈ β„‚))
2422, 23syl5ibrcom 247 . . . . . . . . . 10 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ (π‘Ž = (π‘β€˜π‘‹) β†’ π‘Ž ∈ β„‚))
2524rexlimdva 3149 . . . . . . . . 9 (πœ‘ β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) β†’ π‘Ž ∈ β„‚))
2625ss2abdv 4021 . . . . . . . 8 (πœ‘ β†’ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} βŠ† {π‘Ž ∣ π‘Ž ∈ β„‚})
27 abid2 2872 . . . . . . . . 9 {π‘Ž ∣ π‘Ž ∈ β„‚} = β„‚
2827, 5eqtri 2761 . . . . . . . 8 {π‘Ž ∣ π‘Ž ∈ β„‚} = (Baseβ€˜β„‚fld)
2926, 28sseqtrdi 3995 . . . . . . 7 (πœ‘ β†’ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} βŠ† (Baseβ€˜β„‚fld))
30 abid2 2872 . . . . . . . . 9 {π‘Ž ∣ π‘Ž ∈ 𝐡} = 𝐡
31 plyconst 25583 . . . . . . . . . . . . 13 ((𝐡 βŠ† β„‚ ∧ π‘Ž ∈ 𝐡) β†’ (β„‚ Γ— {π‘Ž}) ∈ (Polyβ€˜π΅))
329, 31sylan 581 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (β„‚ Γ— {π‘Ž}) ∈ (Polyβ€˜π΅))
3310adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝑋 ∈ β„‚)
34 vex 3448 . . . . . . . . . . . . . . 15 π‘Ž ∈ V
3534fvconst2 7154 . . . . . . . . . . . . . 14 (𝑋 ∈ β„‚ β†’ ((β„‚ Γ— {π‘Ž})β€˜π‘‹) = π‘Ž)
3633, 35syl 17 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ((β„‚ Γ— {π‘Ž})β€˜π‘‹) = π‘Ž)
3736eqcomd 2739 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ π‘Ž = ((β„‚ Γ— {π‘Ž})β€˜π‘‹))
38 fveq1 6842 . . . . . . . . . . . . 13 (𝑝 = (β„‚ Γ— {π‘Ž}) β†’ (π‘β€˜π‘‹) = ((β„‚ Γ— {π‘Ž})β€˜π‘‹))
3938rspceeqv 3596 . . . . . . . . . . . 12 (((β„‚ Γ— {π‘Ž}) ∈ (Polyβ€˜π΅) ∧ π‘Ž = ((β„‚ Γ— {π‘Ž})β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹))
4032, 37, 39syl2anc 585 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹))
4140ex 414 . . . . . . . . . 10 (πœ‘ β†’ (π‘Ž ∈ 𝐡 β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)))
4241ss2abdv 4021 . . . . . . . . 9 (πœ‘ β†’ {π‘Ž ∣ π‘Ž ∈ 𝐡} βŠ† {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
4330, 42eqsstrrid 3994 . . . . . . . 8 (πœ‘ β†’ 𝐡 βŠ† {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
44 subrgsubg 20242 . . . . . . . . . 10 (𝐡 ∈ (SubRingβ€˜β„‚fld) β†’ 𝐡 ∈ (SubGrpβ€˜β„‚fld))
457, 44syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐡 ∈ (SubGrpβ€˜β„‚fld))
4616subg0cl 18941 . . . . . . . . 9 (𝐡 ∈ (SubGrpβ€˜β„‚fld) β†’ 0 ∈ 𝐡)
4745, 46syl 17 . . . . . . . 8 (πœ‘ β†’ 0 ∈ 𝐡)
4843, 47sseldd 3946 . . . . . . 7 (πœ‘ β†’ 0 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
49 biid 261 . . . . . . . . 9 (πœ‘ ↔ πœ‘)
50 vex 3448 . . . . . . . . . 10 𝑏 ∈ V
51 eqeq1 2737 . . . . . . . . . . . 12 (π‘Ž = 𝑏 β†’ (π‘Ž = (π‘β€˜π‘‹) ↔ 𝑏 = (π‘β€˜π‘‹)))
5251rexbidv 3172 . . . . . . . . . . 11 (π‘Ž = 𝑏 β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑏 = (π‘β€˜π‘‹)))
53 fveq1 6842 . . . . . . . . . . . . 13 (𝑝 = 𝑒 β†’ (π‘β€˜π‘‹) = (π‘’β€˜π‘‹))
5453eqeq2d 2744 . . . . . . . . . . . 12 (𝑝 = 𝑒 β†’ (𝑏 = (π‘β€˜π‘‹) ↔ 𝑏 = (π‘’β€˜π‘‹)))
5554cbvrexvw 3225 . . . . . . . . . . 11 (βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑏 = (π‘β€˜π‘‹) ↔ βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹))
5652, 55bitrdi 287 . . . . . . . . . 10 (π‘Ž = 𝑏 β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹)))
5750, 56elab 3631 . . . . . . . . 9 (𝑏 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ↔ βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹))
58 vex 3448 . . . . . . . . . 10 𝑐 ∈ V
59 eqeq1 2737 . . . . . . . . . . . 12 (π‘Ž = 𝑐 β†’ (π‘Ž = (π‘β€˜π‘‹) ↔ 𝑐 = (π‘β€˜π‘‹)))
6059rexbidv 3172 . . . . . . . . . . 11 (π‘Ž = 𝑐 β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘β€˜π‘‹)))
61 fveq1 6842 . . . . . . . . . . . . 13 (𝑝 = 𝑑 β†’ (π‘β€˜π‘‹) = (π‘‘β€˜π‘‹))
6261eqeq2d 2744 . . . . . . . . . . . 12 (𝑝 = 𝑑 β†’ (𝑐 = (π‘β€˜π‘‹) ↔ 𝑐 = (π‘‘β€˜π‘‹)))
6362cbvrexvw 3225 . . . . . . . . . . 11 (βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘β€˜π‘‹) ↔ βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹))
6460, 63bitrdi 287 . . . . . . . . . 10 (π‘Ž = 𝑐 β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹)))
6558, 64elab 3631 . . . . . . . . 9 (𝑐 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ↔ βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹))
66 simplr 768 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ 𝑒 ∈ (Polyβ€˜π΅))
67 simpr 486 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ 𝑑 ∈ (Polyβ€˜π΅))
6818subrgacl 20247 . . . . . . . . . . . . . . . . . . . 20 ((𝐡 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (π‘Ž + 𝑏) ∈ 𝐡)
69683expb 1121 . . . . . . . . . . . . . . . . . . 19 ((𝐡 ∈ (SubRingβ€˜β„‚fld) ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž + 𝑏) ∈ 𝐡)
707, 69sylan 581 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž + 𝑏) ∈ 𝐡)
7170adantlr 714 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž + 𝑏) ∈ 𝐡)
7271adantlr 714 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž + 𝑏) ∈ 𝐡)
7366, 67, 72plyadd 25594 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ (𝑒 ∘f + 𝑑) ∈ (Polyβ€˜π΅))
74 plyf 25575 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ (Polyβ€˜π΅) β†’ 𝑒:β„‚βŸΆβ„‚)
7574ffnd 6670 . . . . . . . . . . . . . . . . . 18 (𝑒 ∈ (Polyβ€˜π΅) β†’ 𝑒 Fn β„‚)
7675ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ 𝑒 Fn β„‚)
77 plyf 25575 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∈ (Polyβ€˜π΅) β†’ 𝑑:β„‚βŸΆβ„‚)
7877ffnd 6670 . . . . . . . . . . . . . . . . . 18 (𝑑 ∈ (Polyβ€˜π΅) β†’ 𝑑 Fn β„‚)
7978adantl 483 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ 𝑑 Fn β„‚)
80 cnex 11137 . . . . . . . . . . . . . . . . . 18 β„‚ ∈ V
8180a1i 11 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ β„‚ ∈ V)
8210ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ 𝑋 ∈ β„‚)
83 fnfvof 7635 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn β„‚ ∧ 𝑑 Fn β„‚) ∧ (β„‚ ∈ V ∧ 𝑋 ∈ β„‚)) β†’ ((𝑒 ∘f + 𝑑)β€˜π‘‹) = ((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)))
8476, 79, 81, 82, 83syl22anc 838 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ ((𝑒 ∘f + 𝑑)β€˜π‘‹) = ((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)))
8584eqcomd 2739 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ ((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)) = ((𝑒 ∘f + 𝑑)β€˜π‘‹))
86 fveq1 6842 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒 ∘f + 𝑑) β†’ (π‘β€˜π‘‹) = ((𝑒 ∘f + 𝑑)β€˜π‘‹))
8786rspceeqv 3596 . . . . . . . . . . . . . . 15 (((𝑒 ∘f + 𝑑) ∈ (Polyβ€˜π΅) ∧ ((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)) = ((𝑒 ∘f + 𝑑)β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹))
8873, 85, 87syl2anc 585 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹))
89 oveq2 7366 . . . . . . . . . . . . . . . 16 (𝑐 = (π‘‘β€˜π‘‹) β†’ ((π‘’β€˜π‘‹) + 𝑐) = ((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)))
9089eqeq1d 2735 . . . . . . . . . . . . . . 15 (𝑐 = (π‘‘β€˜π‘‹) β†’ (((π‘’β€˜π‘‹) + 𝑐) = (π‘β€˜π‘‹) ↔ ((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹)))
9190rexbidv 3172 . . . . . . . . . . . . . 14 (𝑐 = (π‘‘β€˜π‘‹) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + 𝑐) = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹)))
9288, 91syl5ibrcom 247 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ (𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + 𝑐) = (π‘β€˜π‘‹)))
9392rexlimdva 3149 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + 𝑐) = (π‘β€˜π‘‹)))
94 oveq1 7365 . . . . . . . . . . . . . . 15 (𝑏 = (π‘’β€˜π‘‹) β†’ (𝑏 + 𝑐) = ((π‘’β€˜π‘‹) + 𝑐))
9594eqeq1d 2735 . . . . . . . . . . . . . 14 (𝑏 = (π‘’β€˜π‘‹) β†’ ((𝑏 + 𝑐) = (π‘β€˜π‘‹) ↔ ((π‘’β€˜π‘‹) + 𝑐) = (π‘β€˜π‘‹)))
9695rexbidv 3172 . . . . . . . . . . . . 13 (𝑏 = (π‘’β€˜π‘‹) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + 𝑐) = (π‘β€˜π‘‹)))
9796imbi2d 341 . . . . . . . . . . . 12 (𝑏 = (π‘’β€˜π‘‹) β†’ ((βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹)) ↔ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) + 𝑐) = (π‘β€˜π‘‹))))
9893, 97syl5ibrcom 247 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (𝑏 = (π‘’β€˜π‘‹) β†’ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹))))
9998rexlimdva 3149 . . . . . . . . . 10 (πœ‘ β†’ (βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹) β†’ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹))))
100993imp 1112 . . . . . . . . 9 ((πœ‘ ∧ βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹) ∧ βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹))
10149, 57, 65, 100syl3anb 1162 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ∧ 𝑐 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹))
102 ovex 7391 . . . . . . . . 9 (𝑏 + 𝑐) ∈ V
103 eqeq1 2737 . . . . . . . . . 10 (π‘Ž = (𝑏 + 𝑐) β†’ (π‘Ž = (π‘β€˜π‘‹) ↔ (𝑏 + 𝑐) = (π‘β€˜π‘‹)))
104103rexbidv 3172 . . . . . . . . 9 (π‘Ž = (𝑏 + 𝑐) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹)))
105102, 104elab 3631 . . . . . . . 8 ((𝑏 + 𝑐) ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 + 𝑐) = (π‘β€˜π‘‹))
106101, 105sylibr 233 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ∧ 𝑐 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}) β†’ (𝑏 + 𝑐) ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
107 ax-1cn 11114 . . . . . . . . . . . . . . . . . 18 1 ∈ β„‚
108 cnfldneg 20839 . . . . . . . . . . . . . . . . . 18 (1 ∈ β„‚ β†’ ((invgβ€˜β„‚fld)β€˜1) = -1)
109107, 108mp1i 13 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ((invgβ€˜β„‚fld)β€˜1) = -1)
110 cnfld1 20838 . . . . . . . . . . . . . . . . . . . 20 1 = (1rβ€˜β„‚fld)
111110subrg1cl 20244 . . . . . . . . . . . . . . . . . . 19 (𝐡 ∈ (SubRingβ€˜β„‚fld) β†’ 1 ∈ 𝐡)
1127, 111syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 1 ∈ 𝐡)
113 eqid 2733 . . . . . . . . . . . . . . . . . . 19 (invgβ€˜β„‚fld) = (invgβ€˜β„‚fld)
114113subginvcl 18942 . . . . . . . . . . . . . . . . . 18 ((𝐡 ∈ (SubGrpβ€˜β„‚fld) ∧ 1 ∈ 𝐡) β†’ ((invgβ€˜β„‚fld)β€˜1) ∈ 𝐡)
11545, 112, 114syl2anc 585 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ((invgβ€˜β„‚fld)β€˜1) ∈ 𝐡)
116109, 115eqeltrrd 2835 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ -1 ∈ 𝐡)
117 plyconst 25583 . . . . . . . . . . . . . . . 16 ((𝐡 βŠ† β„‚ ∧ -1 ∈ 𝐡) β†’ (β„‚ Γ— {-1}) ∈ (Polyβ€˜π΅))
1189, 116, 117syl2anc 585 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (β„‚ Γ— {-1}) ∈ (Polyβ€˜π΅))
119118adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (β„‚ Γ— {-1}) ∈ (Polyβ€˜π΅))
120 simpr 486 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ 𝑒 ∈ (Polyβ€˜π΅))
121 cnfldmul 20818 . . . . . . . . . . . . . . . . . 18 Β· = (.rβ€˜β„‚fld)
122121subrgmcl 20248 . . . . . . . . . . . . . . . . 17 ((𝐡 ∈ (SubRingβ€˜β„‚fld) ∧ π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (π‘Ž Β· 𝑏) ∈ 𝐡)
1231223expb 1121 . . . . . . . . . . . . . . . 16 ((𝐡 ∈ (SubRingβ€˜β„‚fld) ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž Β· 𝑏) ∈ 𝐡)
1247, 123sylan 581 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž Β· 𝑏) ∈ 𝐡)
125124adantlr 714 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž Β· 𝑏) ∈ 𝐡)
126119, 120, 71, 125plymul 25595 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ ((β„‚ Γ— {-1}) ∘f Β· 𝑒) ∈ (Polyβ€˜π΅))
127 ffvelcdm 7033 . . . . . . . . . . . . . . . 16 ((𝑒:β„‚βŸΆβ„‚ ∧ 𝑋 ∈ β„‚) β†’ (π‘’β€˜π‘‹) ∈ β„‚)
12874, 10, 127syl2anr 598 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (π‘’β€˜π‘‹) ∈ β„‚)
129 cnfldneg 20839 . . . . . . . . . . . . . . 15 ((π‘’β€˜π‘‹) ∈ β„‚ β†’ ((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = -(π‘’β€˜π‘‹))
130128, 129syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ ((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = -(π‘’β€˜π‘‹))
131 negex 11404 . . . . . . . . . . . . . . . . 17 -1 ∈ V
132 fnconstg 6731 . . . . . . . . . . . . . . . . 17 (-1 ∈ V β†’ (β„‚ Γ— {-1}) Fn β„‚)
133131, 132mp1i 13 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (β„‚ Γ— {-1}) Fn β„‚)
13475adantl 483 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ 𝑒 Fn β„‚)
13580a1i 11 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ β„‚ ∈ V)
13610adantr 482 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ 𝑋 ∈ β„‚)
137 fnfvof 7635 . . . . . . . . . . . . . . . 16 ((((β„‚ Γ— {-1}) Fn β„‚ ∧ 𝑒 Fn β„‚) ∧ (β„‚ ∈ V ∧ 𝑋 ∈ β„‚)) β†’ (((β„‚ Γ— {-1}) ∘f Β· 𝑒)β€˜π‘‹) = (((β„‚ Γ— {-1})β€˜π‘‹) Β· (π‘’β€˜π‘‹)))
138133, 134, 135, 136, 137syl22anc 838 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (((β„‚ Γ— {-1}) ∘f Β· 𝑒)β€˜π‘‹) = (((β„‚ Γ— {-1})β€˜π‘‹) Β· (π‘’β€˜π‘‹)))
139131fvconst2 7154 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ β„‚ β†’ ((β„‚ Γ— {-1})β€˜π‘‹) = -1)
140136, 139syl 17 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ ((β„‚ Γ— {-1})β€˜π‘‹) = -1)
141140oveq1d 7373 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (((β„‚ Γ— {-1})β€˜π‘‹) Β· (π‘’β€˜π‘‹)) = (-1 Β· (π‘’β€˜π‘‹)))
142128mulm1d 11612 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (-1 Β· (π‘’β€˜π‘‹)) = -(π‘’β€˜π‘‹))
143138, 141, 1423eqtrd 2777 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (((β„‚ Γ— {-1}) ∘f Β· 𝑒)β€˜π‘‹) = -(π‘’β€˜π‘‹))
144130, 143eqtr4d 2776 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ ((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = (((β„‚ Γ— {-1}) ∘f Β· 𝑒)β€˜π‘‹))
145 fveq1 6842 . . . . . . . . . . . . . 14 (𝑝 = ((β„‚ Γ— {-1}) ∘f Β· 𝑒) β†’ (π‘β€˜π‘‹) = (((β„‚ Γ— {-1}) ∘f Β· 𝑒)β€˜π‘‹))
146145rspceeqv 3596 . . . . . . . . . . . . 13 ((((β„‚ Γ— {-1}) ∘f Β· 𝑒) ∈ (Polyβ€˜π΅) ∧ ((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = (((β„‚ Γ— {-1}) ∘f Β· 𝑒)β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = (π‘β€˜π‘‹))
147126, 144, 146syl2anc 585 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = (π‘β€˜π‘‹))
148 fveqeq2 6852 . . . . . . . . . . . . 13 (𝑏 = (π‘’β€˜π‘‹) β†’ (((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹) ↔ ((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = (π‘β€˜π‘‹)))
149148rexbidv 3172 . . . . . . . . . . . 12 (𝑏 = (π‘’β€˜π‘‹) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜(π‘’β€˜π‘‹)) = (π‘β€˜π‘‹)))
150147, 149syl5ibrcom 247 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (𝑏 = (π‘’β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹)))
151150rexlimdva 3149 . . . . . . . . . 10 (πœ‘ β†’ (βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹)))
152151imp 408 . . . . . . . . 9 ((πœ‘ ∧ βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹))
15357, 152sylan2b 595 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹))
154 fvex 6856 . . . . . . . . 9 ((invgβ€˜β„‚fld)β€˜π‘) ∈ V
155 eqeq1 2737 . . . . . . . . . 10 (π‘Ž = ((invgβ€˜β„‚fld)β€˜π‘) β†’ (π‘Ž = (π‘β€˜π‘‹) ↔ ((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹)))
156155rexbidv 3172 . . . . . . . . 9 (π‘Ž = ((invgβ€˜β„‚fld)β€˜π‘) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹)))
157154, 156elab 3631 . . . . . . . 8 (((invgβ€˜β„‚fld)β€˜π‘) ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)((invgβ€˜β„‚fld)β€˜π‘) = (π‘β€˜π‘‹))
158153, 157sylibr 233 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}) β†’ ((invgβ€˜β„‚fld)β€˜π‘) ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
159110a1i 11 . . . . . . 7 (πœ‘ β†’ 1 = (1rβ€˜β„‚fld))
160121a1i 11 . . . . . . 7 (πœ‘ β†’ Β· = (.rβ€˜β„‚fld))
16143, 112sseldd 3946 . . . . . . 7 (πœ‘ β†’ 1 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
162125adantlr 714 . . . . . . . . . . . . . . . 16 ((((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž Β· 𝑏) ∈ 𝐡)
16366, 67, 72, 162plymul 25595 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ (𝑒 ∘f Β· 𝑑) ∈ (Polyβ€˜π΅))
164 fnfvof 7635 . . . . . . . . . . . . . . . . 17 (((𝑒 Fn β„‚ ∧ 𝑑 Fn β„‚) ∧ (β„‚ ∈ V ∧ 𝑋 ∈ β„‚)) β†’ ((𝑒 ∘f Β· 𝑑)β€˜π‘‹) = ((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)))
16576, 79, 81, 82, 164syl22anc 838 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ ((𝑒 ∘f Β· 𝑑)β€˜π‘‹) = ((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)))
166165eqcomd 2739 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ ((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)) = ((𝑒 ∘f Β· 𝑑)β€˜π‘‹))
167 fveq1 6842 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑒 ∘f Β· 𝑑) β†’ (π‘β€˜π‘‹) = ((𝑒 ∘f Β· 𝑑)β€˜π‘‹))
168167rspceeqv 3596 . . . . . . . . . . . . . . 15 (((𝑒 ∘f Β· 𝑑) ∈ (Polyβ€˜π΅) ∧ ((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)) = ((𝑒 ∘f Β· 𝑑)β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹))
169163, 166, 168syl2anc 585 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹))
170 oveq2 7366 . . . . . . . . . . . . . . . 16 (𝑐 = (π‘‘β€˜π‘‹) β†’ ((π‘’β€˜π‘‹) Β· 𝑐) = ((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)))
171170eqeq1d 2735 . . . . . . . . . . . . . . 15 (𝑐 = (π‘‘β€˜π‘‹) β†’ (((π‘’β€˜π‘‹) Β· 𝑐) = (π‘β€˜π‘‹) ↔ ((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹)))
172171rexbidv 3172 . . . . . . . . . . . . . 14 (𝑐 = (π‘‘β€˜π‘‹) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· 𝑐) = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· (π‘‘β€˜π‘‹)) = (π‘β€˜π‘‹)))
173169, 172syl5ibrcom 247 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) ∧ 𝑑 ∈ (Polyβ€˜π΅)) β†’ (𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· 𝑐) = (π‘β€˜π‘‹)))
174173rexlimdva 3149 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· 𝑐) = (π‘β€˜π‘‹)))
175 oveq1 7365 . . . . . . . . . . . . . . 15 (𝑏 = (π‘’β€˜π‘‹) β†’ (𝑏 Β· 𝑐) = ((π‘’β€˜π‘‹) Β· 𝑐))
176175eqeq1d 2735 . . . . . . . . . . . . . 14 (𝑏 = (π‘’β€˜π‘‹) β†’ ((𝑏 Β· 𝑐) = (π‘β€˜π‘‹) ↔ ((π‘’β€˜π‘‹) Β· 𝑐) = (π‘β€˜π‘‹)))
177176rexbidv 3172 . . . . . . . . . . . . 13 (𝑏 = (π‘’β€˜π‘‹) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· 𝑐) = (π‘β€˜π‘‹)))
178177imbi2d 341 . . . . . . . . . . . 12 (𝑏 = (π‘’β€˜π‘‹) β†’ ((βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹)) ↔ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)((π‘’β€˜π‘‹) Β· 𝑐) = (π‘β€˜π‘‹))))
179174, 178syl5ibrcom 247 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑒 ∈ (Polyβ€˜π΅)) β†’ (𝑏 = (π‘’β€˜π‘‹) β†’ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹))))
180179rexlimdva 3149 . . . . . . . . . 10 (πœ‘ β†’ (βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹) β†’ (βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹))))
1811803imp 1112 . . . . . . . . 9 ((πœ‘ ∧ βˆƒπ‘’ ∈ (Polyβ€˜π΅)𝑏 = (π‘’β€˜π‘‹) ∧ βˆƒπ‘‘ ∈ (Polyβ€˜π΅)𝑐 = (π‘‘β€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹))
18249, 57, 65, 181syl3anb 1162 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ∧ 𝑐 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹))
183 ovex 7391 . . . . . . . . 9 (𝑏 Β· 𝑐) ∈ V
184 eqeq1 2737 . . . . . . . . . 10 (π‘Ž = (𝑏 Β· 𝑐) β†’ (π‘Ž = (π‘β€˜π‘‹) ↔ (𝑏 Β· 𝑐) = (π‘β€˜π‘‹)))
185184rexbidv 3172 . . . . . . . . 9 (π‘Ž = (𝑏 Β· 𝑐) β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹)))
186183, 185elab 3631 . . . . . . . 8 ((𝑏 Β· 𝑐) ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)(𝑏 Β· 𝑐) = (π‘β€˜π‘‹))
187182, 186sylibr 233 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ∧ 𝑐 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}) β†’ (𝑏 Β· 𝑐) ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
18815, 17, 19, 29, 48, 106, 158, 159, 160, 161, 187, 4issubrngd2 20674 . . . . . 6 (πœ‘ β†’ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ∈ (SubRingβ€˜β„‚fld))
189 plyid 25586 . . . . . . . . . . 11 ((𝐡 βŠ† β„‚ ∧ 1 ∈ 𝐡) β†’ Xp ∈ (Polyβ€˜π΅))
1909, 112, 189syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ Xp ∈ (Polyβ€˜π΅))
191 df-idp 25566 . . . . . . . . . . . 12 Xp = ( I β†Ύ β„‚)
192191fveq1i 6844 . . . . . . . . . . 11 (Xpβ€˜π‘‹) = (( I β†Ύ β„‚)β€˜π‘‹)
193 fvresi 7120 . . . . . . . . . . . 12 (𝑋 ∈ β„‚ β†’ (( I β†Ύ β„‚)β€˜π‘‹) = 𝑋)
19410, 193syl 17 . . . . . . . . . . 11 (πœ‘ β†’ (( I β†Ύ β„‚)β€˜π‘‹) = 𝑋)
195192, 194eqtr2id 2786 . . . . . . . . . 10 (πœ‘ β†’ 𝑋 = (Xpβ€˜π‘‹))
196 fveq1 6842 . . . . . . . . . . 11 (𝑝 = Xp β†’ (π‘β€˜π‘‹) = (Xpβ€˜π‘‹))
197196rspceeqv 3596 . . . . . . . . . 10 ((Xp ∈ (Polyβ€˜π΅) ∧ 𝑋 = (Xpβ€˜π‘‹)) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑋 = (π‘β€˜π‘‹))
198190, 195, 197syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑋 = (π‘β€˜π‘‹))
199 eqeq1 2737 . . . . . . . . . 10 (π‘Ž = 𝑋 β†’ (π‘Ž = (π‘β€˜π‘‹) ↔ 𝑋 = (π‘β€˜π‘‹)))
200199rexbidv 3172 . . . . . . . . 9 (π‘Ž = 𝑋 β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑋 = (π‘β€˜π‘‹)))
20110, 198, 200elabd 3634 . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
202201snssd 4770 . . . . . . 7 (πœ‘ β†’ {𝑋} βŠ† {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
20343, 202unssd 4147 . . . . . 6 (πœ‘ β†’ (𝐡 βˆͺ {𝑋}) βŠ† {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
2044, 6, 12, 13, 14, 188, 203rgspnmin 41541 . . . . 5 (πœ‘ β†’ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) βŠ† {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)})
205204sseld 3944 . . . 4 (πœ‘ β†’ (𝑉 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) β†’ 𝑉 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)}))
206 fvex 6856 . . . . . . 7 (π‘β€˜π‘‹) ∈ V
207 eleq1 2822 . . . . . . 7 (𝑉 = (π‘β€˜π‘‹) β†’ (𝑉 ∈ V ↔ (π‘β€˜π‘‹) ∈ V))
208206, 207mpbiri 258 . . . . . 6 (𝑉 = (π‘β€˜π‘‹) β†’ 𝑉 ∈ V)
209208rexlimivw 3145 . . . . 5 (βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹) β†’ 𝑉 ∈ V)
210 eqeq1 2737 . . . . . 6 (π‘Ž = 𝑉 β†’ (π‘Ž = (π‘β€˜π‘‹) ↔ 𝑉 = (π‘β€˜π‘‹)))
211210rexbidv 3172 . . . . 5 (π‘Ž = 𝑉 β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹)))
212209, 211elab3 3639 . . . 4 (𝑉 ∈ {π‘Ž ∣ βˆƒπ‘ ∈ (Polyβ€˜π΅)π‘Ž = (π‘β€˜π‘‹)} ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹))
213205, 212syl6ib 251 . . 3 (πœ‘ β†’ (𝑉 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) β†’ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹)))
2144, 6, 12, 13, 14rgspncl 41539 . . . . . . 7 (πœ‘ β†’ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) ∈ (SubRingβ€˜β„‚fld))
215214adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) ∈ (SubRingβ€˜β„‚fld))
216 simpr 486 . . . . . 6 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ 𝑝 ∈ (Polyβ€˜π΅))
2174, 6, 12, 13, 14rgspnssid 41540 . . . . . . . . 9 (πœ‘ β†’ (𝐡 βˆͺ {𝑋}) βŠ† ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
218217unssbd 4149 . . . . . . . 8 (πœ‘ β†’ {𝑋} βŠ† ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
219 snidg 4621 . . . . . . . . 9 (𝑋 ∈ β„‚ β†’ 𝑋 ∈ {𝑋})
22010, 219syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ {𝑋})
221218, 220sseldd 3946 . . . . . . 7 (πœ‘ β†’ 𝑋 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
222221adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ 𝑋 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
223217unssad 4148 . . . . . . 7 (πœ‘ β†’ 𝐡 βŠ† ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
224223adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ 𝐡 βŠ† ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
225215, 216, 222, 224cnsrplycl 41537 . . . . 5 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ (π‘β€˜π‘‹) ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})))
226 eleq1 2822 . . . . 5 (𝑉 = (π‘β€˜π‘‹) β†’ (𝑉 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) ↔ (π‘β€˜π‘‹) ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋}))))
227225, 226syl5ibrcom 247 . . . 4 ((πœ‘ ∧ 𝑝 ∈ (Polyβ€˜π΅)) β†’ (𝑉 = (π‘β€˜π‘‹) β†’ 𝑉 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋}))))
228227rexlimdva 3149 . . 3 (πœ‘ β†’ (βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹) β†’ 𝑉 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋}))))
229213, 228impbid 211 . 2 (πœ‘ β†’ (𝑉 ∈ ((RingSpanβ€˜β„‚fld)β€˜(𝐡 βˆͺ {𝑋})) ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹)))
2302, 229bitrd 279 1 (πœ‘ β†’ (𝑉 ∈ 𝑆 ↔ βˆƒπ‘ ∈ (Polyβ€˜π΅)𝑉 = (π‘β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070  Vcvv 3444   βˆͺ cun 3909   βŠ† wss 3911  {csn 4587   I cid 5531   Γ— cxp 5632   β†Ύ cres 5636   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∘f cof 7616  β„‚cc 11054  0cc0 11056  1c1 11057   + caddc 11059   Β· cmul 11061  -cneg 11391  Basecbs 17088   β†Ύs cress 17117  +gcplusg 17138  .rcmulr 17139  0gc0g 17326  invgcminusg 18754  SubGrpcsubg 18927  1rcur 19918  Ringcrg 19969  SubRingcsubrg 20232  RingSpancrgspn 20233  β„‚fldccnfld 20812  Polycply 25561  Xpcidp 25562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-fl 13703  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-rlim 15377  df-sum 15577  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-subg 18930  df-cmn 19569  df-mgp 19902  df-ur 19919  df-ring 19971  df-cring 19972  df-subrg 20234  df-rgspn 20235  df-cnfld 20813  df-0p 25050  df-ply 25565  df-idp 25566  df-coe 25567  df-dgr 25568
This theorem is referenced by: (None)
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