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| Mirrors > Home > MPE Home > Th. List > ostth1 | Structured version Visualization version GIF version | ||
| Description: - Lemma for ostth 27700: 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 20867 of the absolute value, 𝐹 is equal to 1 on all the integers, and ostthlem1 27688 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 27681 | . . 3 ⊢ 𝑄 ∈ DivRing |
| 5 | 1 | qrngbas 27680 | . . . 4 ⊢ ℚ = (Base‘𝑄) |
| 6 | 1 | qrng0 27682 | . . . 4 ⊢ 0 = (0g‘𝑄) |
| 7 | ostth.k | . . . 4 ⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) | |
| 8 | 2, 5, 6, 7 | abvtriv 20880 | . . 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 3254 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ (𝐹‘𝑛) < 1) |
| 12 | prmnn 16708 | . . . . 5 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℕ) | |
| 13 | ostth1.2 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ¬ 1 < (𝐹‘𝑛)) | |
| 14 | 13 | r19.21bi 3254 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ 1 < (𝐹‘𝑛)) |
| 15 | 12, 14 | sylan2 602 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ¬ 1 < (𝐹‘𝑛)) |
| 16 | nnq 12963 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℚ) | |
| 17 | 12, 16 | syl 17 | . . . . . 6 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ ℚ) |
| 18 | 2, 5 | abvcl 20862 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℚ) → (𝐹‘𝑛) ∈ ℝ) |
| 19 | 3, 17, 18 | syl2an 605 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) ∈ ℝ) |
| 20 | 1re 11181 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 21 | lttri3 11266 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℝ ∧ 1 ∈ ℝ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) | |
| 22 | 19, 20, 21 | sylancl 595 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → ((𝐹‘𝑛) = 1 ↔ (¬ (𝐹‘𝑛) < 1 ∧ ¬ 1 < (𝐹‘𝑛)))) |
| 23 | 11, 15, 22 | mpbir2and 723 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = 1) |
| 24 | 12 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℕ) |
| 25 | eqeq1 2766 | . . . . . . . 8 ⊢ (𝑥 = 𝑛 → (𝑥 = 0 ↔ 𝑛 = 0)) | |
| 26 | 25 | ifbid 4504 | . . . . . . 7 ⊢ (𝑥 = 𝑛 → if(𝑥 = 0, 0, 1) = if(𝑛 = 0, 0, 1)) |
| 27 | c0ex 11173 | . . . . . . . 8 ⊢ 0 ∈ V | |
| 28 | 1ex 11176 | . . . . . . . 8 ⊢ 1 ∈ V | |
| 29 | 27, 28 | ifex 4531 | . . . . . . 7 ⊢ if(𝑛 = 0, 0, 1) ∈ V |
| 30 | 26, 7, 29 | fvmpt 6975 | . . . . . 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 12247 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0) | |
| 33 | 32 | neneqd 2962 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0) |
| 34 | 33 | iffalsed 4491 | . . . . 5 ⊢ (𝑛 ∈ ℕ → if(𝑛 = 0, 0, 1) = 1) |
| 35 | 31, 34 | eqtrd 2797 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝐾‘𝑛) = 1) |
| 36 | 24, 35 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐾‘𝑛) = 1) |
| 37 | 23, 36 | eqtr4d 2800 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℙ) → (𝐹‘𝑛) = (𝐾‘𝑛)) |
| 38 | 1, 2, 3, 9, 37 | ostthlem2 27689 | 1 ⊢ (𝜑 → 𝐹 = 𝐾) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ifcif 4480 class class class wbr 5100 ↦ cmpt 5181 ‘cfv 6521 (class class class)co 7396 ℝcr 11072 0cc0 11073 1c1 11074 < clt 11216 -cneg 11415 ℕcn 12210 ℚcq 12949 ↑cexp 14074 ℙcprime 16705 pCnt cpc 16872 ↾s cress 17266 DivRingcdr 20775 AbsValcabv 20854 ℂfldccnfld 21421 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152 ax-mulf 11153 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9388 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-ico 13355 df-fz 13513 df-seq 14015 df-exp 14075 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-dvds 16287 df-prm 16706 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-0g 17470 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-grp 18978 df-minusg 18979 df-subg 19165 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20228 df-ring 20281 df-cring 20282 df-oppr 20382 df-dvdsr 20402 df-unit 20403 df-invr 20433 df-dvr 20446 df-nzr 20559 df-subrng 20592 df-subrg 20616 df-rlreg 20740 df-domn 20741 df-drng 20777 df-abv 20855 df-cnfld 21422 |
| This theorem is referenced by: ostth 27700 |
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