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Mirrors > Home > MPE Home > Th. List > fvprmselelfz | Structured version Visualization version GIF version |
Description: The value of the prime selection function is in a finite sequence of integers if the argument is in this finite sequence of integers. (Contributed by AV, 19-Aug-2020.) |
Ref | Expression |
---|---|
fvprmselelfz.f | ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) |
Ref | Expression |
---|---|
fvprmselelfz | ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁)) → (𝐹‘𝑋) ∈ (1...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvprmselelfz.f | . . . 4 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1)) | |
2 | eleq1 2877 | . . . . . 6 ⊢ (𝑚 = 𝑋 → (𝑚 ∈ ℙ ↔ 𝑋 ∈ ℙ)) | |
3 | id 22 | . . . . . 6 ⊢ (𝑚 = 𝑋 → 𝑚 = 𝑋) | |
4 | 2, 3 | ifbieq1d 4448 | . . . . 5 ⊢ (𝑚 = 𝑋 → if(𝑚 ∈ ℙ, 𝑚, 1) = if(𝑋 ∈ ℙ, 𝑋, 1)) |
5 | iftrue 4431 | . . . . . 6 ⊢ (𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋) | |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → if(𝑋 ∈ ℙ, 𝑋, 1) = 𝑋) |
7 | 4, 6 | sylan9eqr 2855 | . . . 4 ⊢ (((𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 𝑋) |
8 | elfznn 12931 | . . . . . 6 ⊢ (𝑋 ∈ (1...𝑁) → 𝑋 ∈ ℕ) | |
9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁)) → 𝑋 ∈ ℕ) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → 𝑋 ∈ ℕ) |
11 | 1, 7, 10, 10 | fvmptd2 6753 | . . 3 ⊢ ((𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → (𝐹‘𝑋) = 𝑋) |
12 | simprr 772 | . . 3 ⊢ ((𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → 𝑋 ∈ (1...𝑁)) | |
13 | 11, 12 | eqeltrd 2890 | . 2 ⊢ ((𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → (𝐹‘𝑋) ∈ (1...𝑁)) |
14 | iffalse 4434 | . . . . . 6 ⊢ (¬ 𝑋 ∈ ℙ → if(𝑋 ∈ ℙ, 𝑋, 1) = 1) | |
15 | 14 | adantr 484 | . . . . 5 ⊢ ((¬ 𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → if(𝑋 ∈ ℙ, 𝑋, 1) = 1) |
16 | 4, 15 | sylan9eqr 2855 | . . . 4 ⊢ (((¬ 𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) ∧ 𝑚 = 𝑋) → if(𝑚 ∈ ℙ, 𝑚, 1) = 1) |
17 | 9 | adantl 485 | . . . 4 ⊢ ((¬ 𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → 𝑋 ∈ ℕ) |
18 | 1nn 11636 | . . . . 5 ⊢ 1 ∈ ℕ | |
19 | 18 | a1i 11 | . . . 4 ⊢ ((¬ 𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → 1 ∈ ℕ) |
20 | 1, 16, 17, 19 | fvmptd2 6753 | . . 3 ⊢ ((¬ 𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → (𝐹‘𝑋) = 1) |
21 | elnnuz 12270 | . . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
22 | eluzfz1 12909 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
23 | 21, 22 | sylbi 220 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁)) |
24 | 23 | adantr 484 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁)) → 1 ∈ (1...𝑁)) |
25 | 24 | adantl 485 | . . 3 ⊢ ((¬ 𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → 1 ∈ (1...𝑁)) |
26 | 20, 25 | eqeltrd 2890 | . 2 ⊢ ((¬ 𝑋 ∈ ℙ ∧ (𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁))) → (𝐹‘𝑋) ∈ (1...𝑁)) |
27 | 13, 26 | pm2.61ian 811 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁)) → (𝐹‘𝑋) ∈ (1...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ifcif 4425 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 1c1 10527 ℕcn 11625 ℤ≥cuz 12231 ...cfz 12885 ℙcprime 16005 |
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-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 |
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-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-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-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-z 11970 df-uz 12232 df-fz 12886 |
This theorem is referenced by: (None) |
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