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Theorem ostth 27010
Description: Ostrowski's theorem, which classifies all absolute values on β„š. Any such absolute value must either be the trivial absolute value 𝐾, a constant exponent 0 < π‘Ž ≀ 1 times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
Hypotheses
Ref Expression
qrng.q 𝑄 = (β„‚fld β†Ύs β„š)
qabsabv.a 𝐴 = (AbsValβ€˜π‘„)
padic.j 𝐽 = (π‘ž ∈ β„™ ↦ (π‘₯ ∈ β„š ↦ if(π‘₯ = 0, 0, (π‘žβ†‘-(π‘ž pCnt π‘₯)))))
ostth.k 𝐾 = (π‘₯ ∈ β„š ↦ if(π‘₯ = 0, 0, 1))
Assertion
Ref Expression
ostth (𝐹 ∈ 𝐴 ↔ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
Distinct variable groups:   π‘ž,π‘Ž,π‘₯,𝑦   𝑔,π‘Ž,𝐽,𝑦   𝐴,π‘Ž,π‘ž,π‘₯,𝑦   π‘₯,𝑄,𝑦   𝐹,π‘Ž   𝑔,π‘ž,𝐹,𝑦   π‘₯,𝐹
Allowed substitution hints:   𝐴(𝑔)   𝑄(𝑔,π‘ž,π‘Ž)   𝐽(π‘₯,π‘ž)   𝐾(π‘₯,𝑦,𝑔,π‘ž,π‘Ž)

Proof of Theorem ostth
Dummy variables π‘˜ 𝑛 𝑝 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qrng.q . . . . . 6 𝑄 = (β„‚fld β†Ύs β„š)
2 qabsabv.a . . . . . 6 𝐴 = (AbsValβ€˜π‘„)
3 padic.j . . . . . 6 𝐽 = (π‘ž ∈ β„™ ↦ (π‘₯ ∈ β„š ↦ if(π‘₯ = 0, 0, (π‘žβ†‘-(π‘ž pCnt π‘₯)))))
4 ostth.k . . . . . 6 𝐾 = (π‘₯ ∈ β„š ↦ if(π‘₯ = 0, 0, 1))
5 simpl 484 . . . . . 6 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ 𝐹 ∈ 𝐴)
6 1re 11163 . . . . . . . . . . 11 1 ∈ ℝ
76ltnri 11272 . . . . . . . . . 10 Β¬ 1 < 1
8 ax-1ne0 11128 . . . . . . . . . . . 12 1 β‰  0
91qrng1 26993 . . . . . . . . . . . . 13 1 = (1rβ€˜π‘„)
101qrng0 26992 . . . . . . . . . . . . 13 0 = (0gβ€˜π‘„)
112, 9, 10abv1z 20334 . . . . . . . . . . . 12 ((𝐹 ∈ 𝐴 ∧ 1 β‰  0) β†’ (πΉβ€˜1) = 1)
128, 11mpan2 690 . . . . . . . . . . 11 (𝐹 ∈ 𝐴 β†’ (πΉβ€˜1) = 1)
1312breq2d 5121 . . . . . . . . . 10 (𝐹 ∈ 𝐴 β†’ (1 < (πΉβ€˜1) ↔ 1 < 1))
147, 13mtbiri 327 . . . . . . . . 9 (𝐹 ∈ 𝐴 β†’ Β¬ 1 < (πΉβ€˜1))
1514adantr 482 . . . . . . . 8 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ Β¬ 1 < (πΉβ€˜1))
16 simprr 772 . . . . . . . . 9 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ 1 < (πΉβ€˜π‘›))
17 fveq2 6846 . . . . . . . . . 10 (𝑛 = 1 β†’ (πΉβ€˜π‘›) = (πΉβ€˜1))
1817breq2d 5121 . . . . . . . . 9 (𝑛 = 1 β†’ (1 < (πΉβ€˜π‘›) ↔ 1 < (πΉβ€˜1)))
1916, 18syl5ibcom 244 . . . . . . . 8 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ (𝑛 = 1 β†’ 1 < (πΉβ€˜1)))
2015, 19mtod 197 . . . . . . 7 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ Β¬ 𝑛 = 1)
21 simprl 770 . . . . . . . . 9 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ 𝑛 ∈ β„•)
22 elnn1uz2 12858 . . . . . . . . 9 (𝑛 ∈ β„• ↔ (𝑛 = 1 ∨ 𝑛 ∈ (β„€β‰₯β€˜2)))
2321, 22sylib 217 . . . . . . . 8 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ (𝑛 = 1 ∨ 𝑛 ∈ (β„€β‰₯β€˜2)))
2423ord 863 . . . . . . 7 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ (Β¬ 𝑛 = 1 β†’ 𝑛 ∈ (β„€β‰₯β€˜2)))
2520, 24mpd 15 . . . . . 6 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ 𝑛 ∈ (β„€β‰₯β€˜2))
26 eqid 2733 . . . . . 6 ((logβ€˜(πΉβ€˜π‘›)) / (logβ€˜π‘›)) = ((logβ€˜(πΉβ€˜π‘›)) / (logβ€˜π‘›))
271, 2, 3, 4, 5, 25, 16, 26ostth2 27008 . . . . 5 ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ β„• ∧ 1 < (πΉβ€˜π‘›))) β†’ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)))
2827rexlimdvaa 3150 . . . 4 (𝐹 ∈ 𝐴 β†’ (βˆƒπ‘› ∈ β„• 1 < (πΉβ€˜π‘›) β†’ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž))))
29 3mix2 1332 . . . 4 (βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
3028, 29syl6 35 . . 3 (𝐹 ∈ 𝐴 β†’ (βˆƒπ‘› ∈ β„• 1 < (πΉβ€˜π‘›) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)))))
31 ralnex 3072 . . . 4 (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ↔ Β¬ βˆƒπ‘› ∈ β„• 1 < (πΉβ€˜π‘›))
32 simpll 766 . . . . . . . . . 10 (((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) ∧ (𝑝 ∈ β„™ ∧ (πΉβ€˜π‘) < 1)) β†’ 𝐹 ∈ 𝐴)
33 simplr 768 . . . . . . . . . . 11 (((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) ∧ (𝑝 ∈ β„™ ∧ (πΉβ€˜π‘) < 1)) β†’ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›))
34 fveq2 6846 . . . . . . . . . . . . . 14 (𝑛 = π‘˜ β†’ (πΉβ€˜π‘›) = (πΉβ€˜π‘˜))
3534breq2d 5121 . . . . . . . . . . . . 13 (𝑛 = π‘˜ β†’ (1 < (πΉβ€˜π‘›) ↔ 1 < (πΉβ€˜π‘˜)))
3635notbid 318 . . . . . . . . . . . 12 (𝑛 = π‘˜ β†’ (Β¬ 1 < (πΉβ€˜π‘›) ↔ Β¬ 1 < (πΉβ€˜π‘˜)))
3736cbvralvw 3224 . . . . . . . . . . 11 (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ↔ βˆ€π‘˜ ∈ β„• Β¬ 1 < (πΉβ€˜π‘˜))
3833, 37sylib 217 . . . . . . . . . 10 (((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) ∧ (𝑝 ∈ β„™ ∧ (πΉβ€˜π‘) < 1)) β†’ βˆ€π‘˜ ∈ β„• Β¬ 1 < (πΉβ€˜π‘˜))
39 simprl 770 . . . . . . . . . 10 (((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) ∧ (𝑝 ∈ β„™ ∧ (πΉβ€˜π‘) < 1)) β†’ 𝑝 ∈ β„™)
40 simprr 772 . . . . . . . . . 10 (((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) ∧ (𝑝 ∈ β„™ ∧ (πΉβ€˜π‘) < 1)) β†’ (πΉβ€˜π‘) < 1)
41 eqid 2733 . . . . . . . . . 10 -((logβ€˜(πΉβ€˜π‘)) / (logβ€˜π‘)) = -((logβ€˜(πΉβ€˜π‘)) / (logβ€˜π‘))
42 eqid 2733 . . . . . . . . . 10 if((πΉβ€˜π‘) ≀ (πΉβ€˜π‘§), (πΉβ€˜π‘§), (πΉβ€˜π‘)) = if((πΉβ€˜π‘) ≀ (πΉβ€˜π‘§), (πΉβ€˜π‘§), (πΉβ€˜π‘))
431, 2, 3, 4, 32, 38, 39, 40, 41, 42ostth3 27009 . . . . . . . . 9 (((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) ∧ (𝑝 ∈ β„™ ∧ (πΉβ€˜π‘) < 1)) β†’ βˆƒπ‘Ž ∈ ℝ+ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)))
4443expr 458 . . . . . . . 8 (((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) ∧ 𝑝 ∈ β„™) β†’ ((πΉβ€˜π‘) < 1 β†’ βˆƒπ‘Ž ∈ ℝ+ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž))))
4544reximdva 3162 . . . . . . 7 ((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) β†’ (βˆƒπ‘ ∈ β„™ (πΉβ€˜π‘) < 1 β†’ βˆƒπ‘ ∈ β„™ βˆƒπ‘Ž ∈ ℝ+ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž))))
461, 2, 3padicabvf 27002 . . . . . . . . . . 11 𝐽:β„™βŸΆπ΄
47 ffn 6672 . . . . . . . . . . 11 (𝐽:β„™βŸΆπ΄ β†’ 𝐽 Fn β„™)
48 fveq1 6845 . . . . . . . . . . . . . . 15 (𝑔 = (π½β€˜π‘) β†’ (π‘”β€˜π‘¦) = ((π½β€˜π‘)β€˜π‘¦))
4948oveq1d 7376 . . . . . . . . . . . . . 14 (𝑔 = (π½β€˜π‘) β†’ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž) = (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž))
5049mpteq2dv 5211 . . . . . . . . . . . . 13 (𝑔 = (π½β€˜π‘) β†’ (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)))
5150eqeq2d 2744 . . . . . . . . . . . 12 (𝑔 = (π½β€˜π‘) β†’ (𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) ↔ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž))))
5251rexrn 7041 . . . . . . . . . . 11 (𝐽 Fn β„™ β†’ (βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) ↔ βˆƒπ‘ ∈ β„™ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž))))
5346, 47, 52mp2b 10 . . . . . . . . . 10 (βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) ↔ βˆƒπ‘ ∈ β„™ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)))
5453rexbii 3094 . . . . . . . . 9 (βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) ↔ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘ ∈ β„™ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)))
55 rexcom 3272 . . . . . . . . 9 (βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘ ∈ β„™ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) ↔ βˆƒπ‘ ∈ β„™ βˆƒπ‘Ž ∈ ℝ+ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)))
5654, 55bitri 275 . . . . . . . 8 (βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) ↔ βˆƒπ‘ ∈ β„™ βˆƒπ‘Ž ∈ ℝ+ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)))
57 3mix3 1333 . . . . . . . 8 (βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
5856, 57sylbir 234 . . . . . . 7 (βˆƒπ‘ ∈ β„™ βˆƒπ‘Ž ∈ ℝ+ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
5945, 58syl6 35 . . . . . 6 ((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) β†’ (βˆƒπ‘ ∈ β„™ (πΉβ€˜π‘) < 1 β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)))))
60 ralnex 3072 . . . . . . 7 (βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1 ↔ Β¬ βˆƒπ‘ ∈ β„™ (πΉβ€˜π‘) < 1)
61 simpl 484 . . . . . . . . . 10 ((𝐹 ∈ 𝐴 ∧ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ∧ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)) β†’ 𝐹 ∈ 𝐴)
62 simprl 770 . . . . . . . . . . 11 ((𝐹 ∈ 𝐴 ∧ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ∧ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)) β†’ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›))
6362, 37sylib 217 . . . . . . . . . 10 ((𝐹 ∈ 𝐴 ∧ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ∧ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)) β†’ βˆ€π‘˜ ∈ β„• Β¬ 1 < (πΉβ€˜π‘˜))
64 simprr 772 . . . . . . . . . . 11 ((𝐹 ∈ 𝐴 ∧ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ∧ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)) β†’ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)
65 fveq2 6846 . . . . . . . . . . . . . 14 (𝑝 = π‘˜ β†’ (πΉβ€˜π‘) = (πΉβ€˜π‘˜))
6665breq1d 5119 . . . . . . . . . . . . 13 (𝑝 = π‘˜ β†’ ((πΉβ€˜π‘) < 1 ↔ (πΉβ€˜π‘˜) < 1))
6766notbid 318 . . . . . . . . . . . 12 (𝑝 = π‘˜ β†’ (Β¬ (πΉβ€˜π‘) < 1 ↔ Β¬ (πΉβ€˜π‘˜) < 1))
6867cbvralvw 3224 . . . . . . . . . . 11 (βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1 ↔ βˆ€π‘˜ ∈ β„™ Β¬ (πΉβ€˜π‘˜) < 1)
6964, 68sylib 217 . . . . . . . . . 10 ((𝐹 ∈ 𝐴 ∧ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ∧ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)) β†’ βˆ€π‘˜ ∈ β„™ Β¬ (πΉβ€˜π‘˜) < 1)
701, 2, 3, 4, 61, 63, 69ostth1 27004 . . . . . . . . 9 ((𝐹 ∈ 𝐴 ∧ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ∧ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)) β†’ 𝐹 = 𝐾)
71703mix1d 1337 . . . . . . . 8 ((𝐹 ∈ 𝐴 ∧ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) ∧ βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1)) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
7271expr 458 . . . . . . 7 ((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) β†’ (βˆ€π‘ ∈ β„™ Β¬ (πΉβ€˜π‘) < 1 β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)))))
7360, 72biimtrrid 242 . . . . . 6 ((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) β†’ (Β¬ βˆƒπ‘ ∈ β„™ (πΉβ€˜π‘) < 1 β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)))))
7459, 73pm2.61d 179 . . . . 5 ((𝐹 ∈ 𝐴 ∧ βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›)) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
7574ex 414 . . . 4 (𝐹 ∈ 𝐴 β†’ (βˆ€π‘› ∈ β„• Β¬ 1 < (πΉβ€˜π‘›) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)))))
7631, 75biimtrrid 242 . . 3 (𝐹 ∈ 𝐴 β†’ (Β¬ βˆƒπ‘› ∈ β„• 1 < (πΉβ€˜π‘›) β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)))))
7730, 76pm2.61d 179 . 2 (𝐹 ∈ 𝐴 β†’ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
78 id 22 . . . 4 (𝐹 = 𝐾 β†’ 𝐹 = 𝐾)
791qdrng 26991 . . . . 5 𝑄 ∈ DivRing
801qrngbas 26990 . . . . . 6 β„š = (Baseβ€˜π‘„)
812, 80, 10, 4abvtriv 20343 . . . . 5 (𝑄 ∈ DivRing β†’ 𝐾 ∈ 𝐴)
8279, 81ax-mp 5 . . . 4 𝐾 ∈ 𝐴
8378, 82eqeltrdi 2842 . . 3 (𝐹 = 𝐾 β†’ 𝐹 ∈ 𝐴)
841, 2qabsabv 27000 . . . . . 6 (abs β†Ύ β„š) ∈ 𝐴
85 fvres 6865 . . . . . . . . . 10 (𝑦 ∈ β„š β†’ ((abs β†Ύ β„š)β€˜π‘¦) = (absβ€˜π‘¦))
8685oveq1d 7376 . . . . . . . . 9 (𝑦 ∈ β„š β†’ (((abs β†Ύ β„š)β€˜π‘¦)β†‘π‘π‘Ž) = ((absβ€˜π‘¦)β†‘π‘π‘Ž))
8786mpteq2ia 5212 . . . . . . . 8 (𝑦 ∈ β„š ↦ (((abs β†Ύ β„š)β€˜π‘¦)β†‘π‘π‘Ž)) = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž))
8887eqcomi 2742 . . . . . . 7 (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) = (𝑦 ∈ β„š ↦ (((abs β†Ύ β„š)β€˜π‘¦)β†‘π‘π‘Ž))
892, 80, 88abvcxp 26986 . . . . . 6 (((abs β†Ύ β„š) ∈ 𝐴 ∧ π‘Ž ∈ (0(,]1)) β†’ (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∈ 𝐴)
9084, 89mpan 689 . . . . 5 (π‘Ž ∈ (0(,]1) β†’ (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∈ 𝐴)
91 eleq1 2822 . . . . 5 (𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) β†’ (𝐹 ∈ 𝐴 ↔ (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∈ 𝐴))
9290, 91syl5ibrcom 247 . . . 4 (π‘Ž ∈ (0(,]1) β†’ (𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) β†’ 𝐹 ∈ 𝐴))
9392rexlimiv 3142 . . 3 (βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) β†’ 𝐹 ∈ 𝐴)
941, 2, 3padicabvcxp 27003 . . . . . . 7 ((𝑝 ∈ β„™ ∧ π‘Ž ∈ ℝ+) β†’ (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) ∈ 𝐴)
9594ancoms 460 . . . . . 6 ((π‘Ž ∈ ℝ+ ∧ 𝑝 ∈ β„™) β†’ (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) ∈ 𝐴)
96 eleq1 2822 . . . . . 6 (𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) β†’ (𝐹 ∈ 𝐴 ↔ (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) ∈ 𝐴))
9795, 96syl5ibrcom 247 . . . . 5 ((π‘Ž ∈ ℝ+ ∧ 𝑝 ∈ β„™) β†’ (𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) β†’ 𝐹 ∈ 𝐴))
9897rexlimivv 3193 . . . 4 (βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘ ∈ β„™ 𝐹 = (𝑦 ∈ β„š ↦ (((π½β€˜π‘)β€˜π‘¦)β†‘π‘π‘Ž)) β†’ 𝐹 ∈ 𝐴)
9954, 98sylbi 216 . . 3 (βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž)) β†’ 𝐹 ∈ 𝐴)
10083, 93, 993jaoi 1428 . 2 ((𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))) β†’ 𝐹 ∈ 𝐴)
10177, 100impbii 208 1 (𝐹 ∈ 𝐴 ↔ (𝐹 = 𝐾 ∨ βˆƒπ‘Ž ∈ (0(,]1)𝐹 = (𝑦 ∈ β„š ↦ ((absβ€˜π‘¦)β†‘π‘π‘Ž)) ∨ βˆƒπ‘Ž ∈ ℝ+ βˆƒπ‘” ∈ ran 𝐽 𝐹 = (𝑦 ∈ β„š ↦ ((π‘”β€˜π‘¦)β†‘π‘π‘Ž))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  ifcif 4490   class class class wbr 5109   ↦ cmpt 5192  ran crn 5638   β†Ύ cres 5639   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  0cc0 11059  1c1 11060   < clt 11197   ≀ cle 11198  -cneg 11394   / cdiv 11820  β„•cn 12161  2c2 12216  β„€β‰₯cuz 12771  β„šcq 12881  β„+crp 12923  (,]cioc 13274  β†‘cexp 13976  abscabs 15128  β„™cprime 16555   pCnt cpc 16716   β†Ύs cress 17120  DivRingcdr 20219  AbsValcabv 20318  β„‚fldccnfld 20819  logclog 25933  β†‘𝑐ccxp 25934
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-addf 11138  ax-mulf 11139
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 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-tpos 8161  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-map 8773  df-pm 8774  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-fi 9355  df-sup 9386  df-inf 9387  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-ioo 13277  df-ioc 13278  df-ico 13279  df-icc 13280  df-fz 13434  df-fzo 13577  df-fl 13706  df-mod 13784  df-seq 13916  df-exp 13977  df-fac 14183  df-bc 14212  df-hash 14240  df-shft 14961  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-limsup 15362  df-clim 15379  df-rlim 15380  df-sum 15580  df-ef 15958  df-sin 15960  df-cos 15961  df-pi 15963  df-dvds 16145  df-gcd 16383  df-prm 16556  df-pc 16717  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-starv 17156  df-sca 17157  df-vsca 17158  df-ip 17159  df-tset 17160  df-ple 17161  df-ds 17163  df-unif 17164  df-hom 17165  df-cco 17166  df-rest 17312  df-topn 17313  df-0g 17331  df-gsum 17332  df-topgen 17333  df-pt 17334  df-prds 17337  df-xrs 17392  df-qtop 17397  df-imas 17398  df-xps 17400  df-mre 17474  df-mrc 17475  df-acs 17477  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-submnd 18610  df-grp 18759  df-minusg 18760  df-mulg 18881  df-subg 18933  df-cntz 19105  df-cmn 19572  df-mgp 19905  df-ur 19922  df-ring 19974  df-cring 19975  df-oppr 20057  df-dvdsr 20078  df-unit 20079  df-invr 20109  df-dvr 20120  df-drng 20221  df-subrg 20262  df-abv 20319  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-fbas 20816  df-fg 20817  df-cnfld 20820  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-lp 22510  df-perf 22511  df-cn 22601  df-cnp 22602  df-haus 22689  df-tx 22936  df-hmeo 23129  df-fil 23220  df-fm 23312  df-flim 23313  df-flf 23314  df-xms 23696  df-ms 23697  df-tms 23698  df-cncf 24264  df-limc 25253  df-dv 25254  df-log 25935  df-cxp 25936
This theorem is referenced by: (None)
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