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Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version |
Description: - Lemma for ostth 25128: 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 18652 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 25116 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 25109 | . . 3 ⊢ 𝑄 ∈ DivRing |
5 | 1 | qrngbas 25108 | . . . 4 ⊢ ℚ = (Base‘𝑄) |
6 | 1 | qrng0 25110 | . . . 4 ⊢ 0 = (0g‘𝑄) |
7 | ostth.k | . . . 4 ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) | |
8 | 2, 5, 6, 7 | abvtriv 18664 | . . 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 2916 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ (𝐹‘𝑛) < 1) |
12 | prmnn 15226 | . . . . 5 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ) | |
13 | ostth1.2 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) | |
14 | 13 | r19.21bi 2916 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ 1 < (𝐹‘𝑛)) |
15 | 12, 14 | sylan2 490 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ 1 < (𝐹‘𝑛)) |
16 | nnq 11677 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℚ) | |
17 | 12, 16 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℚ) |
18 | 2, 5 | abvcl 18647 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℚ) → (𝐹‘𝑛) ∈ ℝ) |
19 | 3, 17, 18 | syl2an 493 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) ∈ ℝ) |
20 | 1re 9918 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | lttri3 10000 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) | |
22 | 19, 20, 21 | sylancl 693 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) |
23 | 11, 15, 22 | mpbir2and 959 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = 1) |
24 | 12 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℕ) |
25 | eqeq1 2614 | . . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑥 = 0 ↔ 𝑛 = 0)) | |
26 | 25 | ifbid 4058 | . . . . . . 7 ⊢ (𝑥 = 𝑛 → if(𝑥 = 0, 0, 1) = if(𝑛 = 0, 0, 1)) |
27 | c0ex 9913 | . . . . . . . 8 ⊢ 0 ∈ V | |
28 | 1ex 9914 | . . . . . . . 8 ⊢ 1 ∈ V | |
29 | 27, 28 | ifex 4106 | . . . . . . 7 ⊢ if(𝑛 = 0, 0, 1) ∈ V |
30 | 26, 7, 29 | fvmpt 6191 | . . . . . 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 10930 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | |
33 | 32 | neneqd 2787 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0) |
34 | 33 | iffalsed 4047 | . . . . 5 ⊢ (𝑛 ∈ ℕ → if(𝑛 = 0, 0, 1) = 1) |
35 | 31, 34 | eqtrd 2644 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝐾‘𝑛) = 1) |
36 | 24, 35 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐾‘𝑛) = 1) |
37 | 23, 36 | eqtr4d 2647 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = (𝐾‘𝑛)) |
38 | 1, 2, 3, 9, 37 | ostthlem2 25117 | 1 ⊢ (𝜑 → 𝐹 = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 < clt 9953 -cneg 10146 ℕcn 10897 ℚcq 11664 ↑cexp 12722 ℙcprime 15223 pCnt cpc 15379 ↾s cress 15696 DivRingcdr 18570 AbsValcabv 18639 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-ico 12052 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-prm 15224 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-subrg 18601 df-abv 18640 df-cnfld 19568 |
This theorem is referenced by: ostth 25128 |
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