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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno4prmfac193 | Structured version Visualization version GIF version |
Description: If P was a (prime) factor of the fourth Fermat number, it would be 193. (Contributed by AV, 28-Jul-2021.) |
Ref | Expression |
---|---|
fmtno4prmfac193 | ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑃 = ;;193) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmtno4prmfac 43137 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193)) | |
2 | 5nn 11534 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
3 | 1nn0 11731 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
4 | 3nn 11525 | . . . . . . . . 9 ⊢ 3 ∈ ℕ | |
5 | 3, 4 | decnncl 11938 | . . . . . . . 8 ⊢ ;13 ∈ ℕ |
6 | 1lt5 11633 | . . . . . . . 8 ⊢ 1 < 5 | |
7 | 1nn 11458 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
8 | 3nn0 11733 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
9 | 1lt10 12058 | . . . . . . . . 9 ⊢ 1 < ;10 | |
10 | 7, 8, 3, 9 | declti 11956 | . . . . . . . 8 ⊢ 1 < ;13 |
11 | eqid 2780 | . . . . . . . 8 ⊢ (5 · ;13) = (5 · ;13) | |
12 | 2, 5, 6, 10, 11 | nprmi 15895 | . . . . . . 7 ⊢ ¬ (5 · ;13) ∈ ℙ |
13 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;65 → 𝑃 = ;65) | |
14 | 5nn0 11735 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
15 | eqid 2780 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
16 | 5cn 11536 | . . . . . . . . . . . . 13 ⊢ 5 ∈ ℂ | |
17 | 16 | mulid1i 10450 | . . . . . . . . . . . 12 ⊢ (5 · 1) = 5 |
18 | 17 | oveq1i 6992 | . . . . . . . . . . 11 ⊢ ((5 · 1) + 1) = (5 + 1) |
19 | 5p1e6 11600 | . . . . . . . . . . 11 ⊢ (5 + 1) = 6 | |
20 | 18, 19 | eqtri 2804 | . . . . . . . . . 10 ⊢ ((5 · 1) + 1) = 6 |
21 | 5t3e15 12020 | . . . . . . . . . 10 ⊢ (5 · 3) = ;15 | |
22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 11985 | . . . . . . . . 9 ⊢ (5 · ;13) = ;65 |
23 | 13, 22 | syl6eqr 2834 | . . . . . . . 8 ⊢ (𝑃 = ;65 → 𝑃 = (5 · ;13)) |
24 | 23 | eleq1d 2852 | . . . . . . 7 ⊢ (𝑃 = ;65 → (𝑃 ∈ ℙ ↔ (5 · ;13) ∈ ℙ)) |
25 | 12, 24 | mtbiri 319 | . . . . . 6 ⊢ (𝑃 = ;65 → ¬ 𝑃 ∈ ℙ) |
26 | 25 | pm2.21d 119 | . . . . 5 ⊢ (𝑃 = ;65 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) |
27 | 4nn0 11734 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
28 | 27, 4 | decnncl 11938 | . . . . . . . 8 ⊢ ;43 ∈ ℕ |
29 | 4nn 11530 | . . . . . . . . 9 ⊢ 4 ∈ ℕ | |
30 | 29, 8, 3, 9 | declti 11956 | . . . . . . . 8 ⊢ 1 < ;43 |
31 | 1lt3 11626 | . . . . . . . 8 ⊢ 1 < 3 | |
32 | eqid 2780 | . . . . . . . 8 ⊢ (;43 · 3) = (;43 · 3) | |
33 | 28, 4, 30, 31, 32 | nprmi 15895 | . . . . . . 7 ⊢ ¬ (;43 · 3) ∈ ℙ |
34 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;;129 → 𝑃 = ;;129) | |
35 | eqid 2780 | . . . . . . . . . 10 ⊢ ;43 = ;43 | |
36 | 4t3e12 12017 | . . . . . . . . . 10 ⊢ (4 · 3) = ;12 | |
37 | 3t3e9 11620 | . . . . . . . . . 10 ⊢ (3 · 3) = 9 | |
38 | 8, 27, 8, 35, 36, 37 | decmul1 11982 | . . . . . . . . 9 ⊢ (;43 · 3) = ;;129 |
39 | 34, 38 | syl6eqr 2834 | . . . . . . . 8 ⊢ (𝑃 = ;;129 → 𝑃 = (;43 · 3)) |
40 | 39 | eleq1d 2852 | . . . . . . 7 ⊢ (𝑃 = ;;129 → (𝑃 ∈ ℙ ↔ (;43 · 3) ∈ ℙ)) |
41 | 33, 40 | mtbiri 319 | . . . . . 6 ⊢ (𝑃 = ;;129 → ¬ 𝑃 ∈ ℙ) |
42 | 41 | pm2.21d 119 | . . . . 5 ⊢ (𝑃 = ;;129 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) |
43 | ax-1 6 | . . . . 5 ⊢ (𝑃 = ;;193 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) | |
44 | 26, 42, 43 | 3jaoi 1408 | . . . 4 ⊢ ((𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193) → (𝑃 ∈ ℙ → 𝑃 = ;;193)) |
45 | 44 | com12 32 | . . 3 ⊢ (𝑃 ∈ ℙ → ((𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193) → 𝑃 = ;;193)) |
46 | 45 | 3ad2ant1 1114 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → ((𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193) → 𝑃 = ;;193)) |
47 | 1, 46 | mpd 15 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑃 = ;;193) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1068 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 class class class wbr 4934 ‘cfv 6193 (class class class)co 6982 1c1 10342 + caddc 10344 · cmul 10346 ≤ cle 10481 2c2 11501 3c3 11502 4c4 11503 5c5 11504 6c6 11505 9c9 11508 ;cdc 11917 ⌊cfl 12981 √csqrt 14459 ∥ cdvds 15473 ℙcprime 15877 FermatNocfmtno 43092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 ax-inf2 8904 ax-cnex 10397 ax-resscn 10398 ax-1cn 10399 ax-icn 10400 ax-addcl 10401 ax-addrcl 10402 ax-mulcl 10403 ax-mulrcl 10404 ax-mulcom 10405 ax-addass 10406 ax-mulass 10407 ax-distr 10408 ax-i2m1 10409 ax-1ne0 10410 ax-1rid 10411 ax-rnegex 10412 ax-rrecex 10413 ax-cnre 10414 ax-pre-lttri 10415 ax-pre-lttrn 10416 ax-pre-ltadd 10417 ax-pre-mulgt0 10418 ax-pre-sup 10419 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-int 4755 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-se 5371 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-pred 5991 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-isom 6202 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-om 7403 df-1st 7507 df-2nd 7508 df-wrecs 7756 df-recs 7818 df-rdg 7856 df-1o 7911 df-2o 7912 df-oadd 7915 df-er 8095 df-map 8214 df-en 8313 df-dom 8314 df-sdom 8315 df-fin 8316 df-sup 8707 df-inf 8708 df-oi 8775 df-dju 9130 df-card 9168 df-pnf 10482 df-mnf 10483 df-xr 10484 df-ltxr 10485 df-le 10486 df-sub 10678 df-neg 10679 df-div 11105 df-nn 11446 df-2 11509 df-3 11510 df-4 11511 df-5 11512 df-6 11513 df-7 11514 df-8 11515 df-9 11516 df-n0 11714 df-xnn0 11786 df-z 11800 df-dec 11918 df-uz 12065 df-q 12169 df-rp 12211 df-ioo 12564 df-ico 12566 df-fz 12715 df-fzo 12856 df-fl 12983 df-mod 13059 df-seq 13191 df-exp 13251 df-fac 13455 df-hash 13512 df-cj 14325 df-re 14326 df-im 14327 df-sqrt 14461 df-abs 14462 df-clim 14712 df-prod 15126 df-dvds 15474 df-gcd 15710 df-prm 15878 df-odz 15964 df-phi 15965 df-pc 16036 df-lgs 25588 df-fmtno 43093 |
This theorem is referenced by: fmtno4prm 43140 |
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