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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 43581 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193)) | |
2 | 5nn 11712 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
3 | 1nn0 11902 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
4 | 3nn 11705 | . . . . . . . . 9 ⊢ 3 ∈ ℕ | |
5 | 3, 4 | decnncl 12107 | . . . . . . . 8 ⊢ ;13 ∈ ℕ |
6 | 1lt5 11806 | . . . . . . . 8 ⊢ 1 < 5 | |
7 | 1nn 11638 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
8 | 3nn0 11904 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
9 | 1lt10 12226 | . . . . . . . . 9 ⊢ 1 < ;10 | |
10 | 7, 8, 3, 9 | declti 12125 | . . . . . . . 8 ⊢ 1 < ;13 |
11 | eqid 2821 | . . . . . . . 8 ⊢ (5 · ;13) = (5 · ;13) | |
12 | 2, 5, 6, 10, 11 | nprmi 16023 | . . . . . . 7 ⊢ ¬ (5 · ;13) ∈ ℙ |
13 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;65 → 𝑃 = ;65) | |
14 | 5nn0 11906 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
15 | eqid 2821 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
16 | 5cn 11714 | . . . . . . . . . . . . 13 ⊢ 5 ∈ ℂ | |
17 | 16 | mulid1i 10634 | . . . . . . . . . . . 12 ⊢ (5 · 1) = 5 |
18 | 17 | oveq1i 7155 | . . . . . . . . . . 11 ⊢ ((5 · 1) + 1) = (5 + 1) |
19 | 5p1e6 11773 | . . . . . . . . . . 11 ⊢ (5 + 1) = 6 | |
20 | 18, 19 | eqtri 2844 | . . . . . . . . . 10 ⊢ ((5 · 1) + 1) = 6 |
21 | 5t3e15 12188 | . . . . . . . . . 10 ⊢ (5 · 3) = ;15 | |
22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 12153 | . . . . . . . . 9 ⊢ (5 · ;13) = ;65 |
23 | 13, 22 | syl6eqr 2874 | . . . . . . . 8 ⊢ (𝑃 = ;65 → 𝑃 = (5 · ;13)) |
24 | 23 | eleq1d 2897 | . . . . . . 7 ⊢ (𝑃 = ;65 → (𝑃 ∈ ℙ ↔ (5 · ;13) ∈ ℙ)) |
25 | 12, 24 | mtbiri 328 | . . . . . 6 ⊢ (𝑃 = ;65 → ¬ 𝑃 ∈ ℙ) |
26 | 25 | pm2.21d 121 | . . . . 5 ⊢ (𝑃 = ;65 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) |
27 | 4nn0 11905 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
28 | 27, 4 | decnncl 12107 | . . . . . . . 8 ⊢ ;43 ∈ ℕ |
29 | 4nn 11709 | . . . . . . . . 9 ⊢ 4 ∈ ℕ | |
30 | 29, 8, 3, 9 | declti 12125 | . . . . . . . 8 ⊢ 1 < ;43 |
31 | 1lt3 11799 | . . . . . . . 8 ⊢ 1 < 3 | |
32 | eqid 2821 | . . . . . . . 8 ⊢ (;43 · 3) = (;43 · 3) | |
33 | 28, 4, 30, 31, 32 | nprmi 16023 | . . . . . . 7 ⊢ ¬ (;43 · 3) ∈ ℙ |
34 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;;129 → 𝑃 = ;;129) | |
35 | eqid 2821 | . . . . . . . . . 10 ⊢ ;43 = ;43 | |
36 | 4t3e12 12185 | . . . . . . . . . 10 ⊢ (4 · 3) = ;12 | |
37 | 3t3e9 11793 | . . . . . . . . . 10 ⊢ (3 · 3) = 9 | |
38 | 8, 27, 8, 35, 36, 37 | decmul1 12151 | . . . . . . . . 9 ⊢ (;43 · 3) = ;;129 |
39 | 34, 38 | syl6eqr 2874 | . . . . . . . 8 ⊢ (𝑃 = ;;129 → 𝑃 = (;43 · 3)) |
40 | 39 | eleq1d 2897 | . . . . . . 7 ⊢ (𝑃 = ;;129 → (𝑃 ∈ ℙ ↔ (;43 · 3) ∈ ℙ)) |
41 | 33, 40 | mtbiri 328 | . . . . . 6 ⊢ (𝑃 = ;;129 → ¬ 𝑃 ∈ ℙ) |
42 | 41 | pm2.21d 121 | . . . . 5 ⊢ (𝑃 = ;;129 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) |
43 | ax-1 6 | . . . . 5 ⊢ (𝑃 = ;;193 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) | |
44 | 26, 42, 43 | 3jaoi 1419 | . . . 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 1125 | . 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 1078 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 2c2 11681 3c3 11682 4c4 11683 5c5 11684 6c6 11685 9c9 11688 ;cdc 12087 ⌊cfl 13150 √csqrt 14582 ∥ cdvds 15597 ℙcprime 16005 FermatNocfmtno 43536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 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 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-xnn0 11957 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-ioo 12732 df-ico 12734 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-prod 15250 df-dvds 15598 df-gcd 15834 df-prm 16006 df-odz 16092 df-phi 16093 df-pc 16164 df-lgs 25799 df-fmtno 43537 |
This theorem is referenced by: fmtno4prm 43584 |
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