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Mirrors > Home > MPE Home > Th. List > Mathboxes > prminf2 | Structured version Visualization version GIF version |
Description: The set of prime numbers is infinite. The proof of this variant of prminf 16949 is based on Goldbach's theorem goldbachth 47472 (via prmdvdsfmtnof1 47512 and prmdvdsfmtnof1lem2 47510), see Wikipedia "Fermat number", 4-Aug-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties 47510. (Contributed by AV, 4-Aug-2021.) |
Ref | Expression |
---|---|
prminf2 | ⊢ ℙ ∉ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . 3 ⊢ (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )) = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )) | |
2 | 1 | prmdvdsfmtnof1 47512 | . 2 ⊢ (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )):ran FermatNo–1-1→ℙ |
3 | ax-1 6 | . . 3 ⊢ (ℙ ∉ Fin → ((𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )):ran FermatNo–1-1→ℙ → ℙ ∉ Fin)) | |
4 | nnel 3054 | . . . 4 ⊢ (¬ ℙ ∉ Fin ↔ ℙ ∈ Fin) | |
5 | fmtnoinf 47461 | . . . . . 6 ⊢ ran FermatNo ∉ Fin | |
6 | f1fi 9350 | . . . . . 6 ⊢ ((ℙ ∈ Fin ∧ (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )):ran FermatNo–1-1→ℙ) → ran FermatNo ∈ Fin) | |
7 | df-nel 3045 | . . . . . . 7 ⊢ (ran FermatNo ∉ Fin ↔ ¬ ran FermatNo ∈ Fin) | |
8 | pm2.21 123 | . . . . . . 7 ⊢ (¬ ran FermatNo ∈ Fin → (ran FermatNo ∈ Fin → ℙ ∉ Fin)) | |
9 | 7, 8 | sylbi 217 | . . . . . 6 ⊢ (ran FermatNo ∉ Fin → (ran FermatNo ∈ Fin → ℙ ∉ Fin)) |
10 | 5, 6, 9 | mpsyl 68 | . . . . 5 ⊢ ((ℙ ∈ Fin ∧ (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )):ran FermatNo–1-1→ℙ) → ℙ ∉ Fin) |
11 | 10 | ex 412 | . . . 4 ⊢ (ℙ ∈ Fin → ((𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )):ran FermatNo–1-1→ℙ → ℙ ∉ Fin)) |
12 | 4, 11 | sylbi 217 | . . 3 ⊢ (¬ ℙ ∉ Fin → ((𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )):ran FermatNo–1-1→ℙ → ℙ ∉ Fin)) |
13 | 3, 12 | pm2.61i 182 | . 2 ⊢ ((𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < )):ran FermatNo–1-1→ℙ → ℙ ∉ Fin) |
14 | 2, 13 | ax-mp 5 | 1 ⊢ ℙ ∉ Fin |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2106 ∉ wnel 3044 {crab 3433 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 –1-1→wf1 6560 Fincfn 8984 infcinf 9479 ℝcr 11152 < clt 11293 ∥ cdvds 16287 ℙcprime 16705 FermatNocfmtno 47452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 df-dvds 16288 df-gcd 16529 df-prm 16706 df-fmtno 47453 |
This theorem is referenced by: (None) |
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