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Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version |
Description: - Lemma for ostth 26223: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If 𝐹 is equal to 1 on the primes, then by complete induction and the multiplicative property abvmul 19593 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 26211 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.) |
Ref | Expression |
---|---|
qrng.q | ⊢ 𝑄 = (ℂfld ↾s ℚ) |
qabsabv.a | ⊢ 𝐴 = (AbsVal‘𝑄) |
padic.j | ⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) |
ostth.k | ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) |
ostth.1 | ⊢ (𝜑 → 𝐹 ∈ 𝐴) |
ostth1.2 | ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) |
ostth1.3 | ⊢ (𝜑 → ∀𝑛 ∈ ℙ ¬ (𝐹‘𝑛) < 1) |
Ref | Expression |
---|---|
ostth1 | ⊢ (𝜑 → 𝐹 = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qrng.q | . 2 ⊢ 𝑄 = (ℂfld ↾s ℚ) | |
2 | qabsabv.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑄) | |
3 | ostth.1 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐴) | |
4 | 1 | qdrng 26204 | . . 3 ⊢ 𝑄 ∈ DivRing |
5 | 1 | qrngbas 26203 | . . . 4 ⊢ ℚ = (Base‘𝑄) |
6 | 1 | qrng0 26205 | . . . 4 ⊢ 0 = (0g‘𝑄) |
7 | ostth.k | . . . 4 ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) | |
8 | 2, 5, 6, 7 | abvtriv 19605 | . . 3 ⊢ (𝑄 ∈ DivRing → 𝐾 ∈ 𝐴) |
9 | 4, 8 | mp1i 13 | . 2 ⊢ (𝜑 → 𝐾 ∈ 𝐴) |
10 | ostth1.3 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ℙ ¬ (𝐹‘𝑛) < 1) | |
11 | 10 | r19.21bi 3173 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ (𝐹‘𝑛) < 1) |
12 | prmnn 16008 | . . . . 5 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ) | |
13 | ostth1.2 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) | |
14 | 13 | r19.21bi 3173 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ 1 < (𝐹‘𝑛)) |
15 | 12, 14 | sylan2 595 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ 1 < (𝐹‘𝑛)) |
16 | nnq 12349 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℚ) | |
17 | 12, 16 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℚ) |
18 | 2, 5 | abvcl 19588 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℚ) → (𝐹‘𝑛) ∈ ℝ) |
19 | 3, 17, 18 | syl2an 598 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) ∈ ℝ) |
20 | 1re 10630 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | lttri3 10713 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) | |
22 | 19, 20, 21 | sylancl 589 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) |
23 | 11, 15, 22 | mpbir2and 712 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = 1) |
24 | 12 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℕ) |
25 | eqeq1 2802 | . . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑥 = 0 ↔ 𝑛 = 0)) | |
26 | 25 | ifbid 4447 | . . . . . . 7 ⊢ (𝑥 = 𝑛 → if(𝑥 = 0, 0, 1) = if(𝑛 = 0, 0, 1)) |
27 | c0ex 10624 | . . . . . . . 8 ⊢ 0 ∈ V | |
28 | 1ex 10626 | . . . . . . . 8 ⊢ 1 ∈ V | |
29 | 27, 28 | ifex 4473 | . . . . . . 7 ⊢ if(𝑛 = 0, 0, 1) ∈ V |
30 | 26, 7, 29 | fvmpt 6745 | . . . . . 6 ⊢ (𝑛 ∈ ℚ → (𝐾‘𝑛) = if(𝑛 = 0, 0, 1)) |
31 | 16, 30 | syl 17 | . . . . 5 ⊢ (𝑛 ∈ ℕ → (𝐾‘𝑛) = if(𝑛 = 0, 0, 1)) |
32 | nnne0 11659 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | |
33 | 32 | neneqd 2992 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0) |
34 | 33 | iffalsed 4436 | . . . . 5 ⊢ (𝑛 ∈ ℕ → if(𝑛 = 0, 0, 1) = 1) |
35 | 31, 34 | eqtrd 2833 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝐾‘𝑛) = 1) |
36 | 24, 35 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐾‘𝑛) = 1) |
37 | 23, 36 | eqtr4d 2836 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = (𝐾‘𝑛)) |
38 | 1, 2, 3, 9, 37 | ostthlem2 26212 | 1 ⊢ (𝜑 → 𝐹 = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 < clt 10664 -cneg 10860 ℕcn 11625 ℚcq 12336 ↑cexp 13425 ℙcprime 16005 pCnt cpc 16163 ↾s cress 16476 DivRingcdr 19495 AbsValcabv 19580 ℂfldccnfld 20091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-ico 12732 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-dvds 15600 df-prm 16006 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-subg 18268 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-subrg 19526 df-abv 19581 df-cnfld 20092 |
This theorem is referenced by: ostth 26223 |
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