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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 46691 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193)) | |
2 | 5nn 12294 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
3 | 1nn0 12484 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
4 | 3nn 12287 | . . . . . . . . 9 ⊢ 3 ∈ ℕ | |
5 | 3, 4 | decnncl 12693 | . . . . . . . 8 ⊢ ;13 ∈ ℕ |
6 | 1lt5 12388 | . . . . . . . 8 ⊢ 1 < 5 | |
7 | 1nn 12219 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
8 | 3nn0 12486 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
9 | 1lt10 12812 | . . . . . . . . 9 ⊢ 1 < ;10 | |
10 | 7, 8, 3, 9 | declti 12711 | . . . . . . . 8 ⊢ 1 < ;13 |
11 | eqid 2724 | . . . . . . . 8 ⊢ (5 · ;13) = (5 · ;13) | |
12 | 2, 5, 6, 10, 11 | nprmi 16622 | . . . . . . 7 ⊢ ¬ (5 · ;13) ∈ ℙ |
13 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;65 → 𝑃 = ;65) | |
14 | 5nn0 12488 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
15 | eqid 2724 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
16 | 5cn 12296 | . . . . . . . . . . . . 13 ⊢ 5 ∈ ℂ | |
17 | 16 | mulridi 11214 | . . . . . . . . . . . 12 ⊢ (5 · 1) = 5 |
18 | 17 | oveq1i 7411 | . . . . . . . . . . 11 ⊢ ((5 · 1) + 1) = (5 + 1) |
19 | 5p1e6 12355 | . . . . . . . . . . 11 ⊢ (5 + 1) = 6 | |
20 | 18, 19 | eqtri 2752 | . . . . . . . . . 10 ⊢ ((5 · 1) + 1) = 6 |
21 | 5t3e15 12774 | . . . . . . . . . 10 ⊢ (5 · 3) = ;15 | |
22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 12739 | . . . . . . . . 9 ⊢ (5 · ;13) = ;65 |
23 | 13, 22 | eqtr4di 2782 | . . . . . . . 8 ⊢ (𝑃 = ;65 → 𝑃 = (5 · ;13)) |
24 | 23 | eleq1d 2810 | . . . . . . 7 ⊢ (𝑃 = ;65 → (𝑃 ∈ ℙ ↔ (5 · ;13) ∈ ℙ)) |
25 | 12, 24 | mtbiri 327 | . . . . . 6 ⊢ (𝑃 = ;65 → ¬ 𝑃 ∈ ℙ) |
26 | 25 | pm2.21d 121 | . . . . 5 ⊢ (𝑃 = ;65 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) |
27 | 4nn0 12487 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
28 | 27, 4 | decnncl 12693 | . . . . . . . 8 ⊢ ;43 ∈ ℕ |
29 | 4nn 12291 | . . . . . . . . 9 ⊢ 4 ∈ ℕ | |
30 | 29, 8, 3, 9 | declti 12711 | . . . . . . . 8 ⊢ 1 < ;43 |
31 | 1lt3 12381 | . . . . . . . 8 ⊢ 1 < 3 | |
32 | eqid 2724 | . . . . . . . 8 ⊢ (;43 · 3) = (;43 · 3) | |
33 | 28, 4, 30, 31, 32 | nprmi 16622 | . . . . . . 7 ⊢ ¬ (;43 · 3) ∈ ℙ |
34 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;;129 → 𝑃 = ;;129) | |
35 | eqid 2724 | . . . . . . . . . 10 ⊢ ;43 = ;43 | |
36 | 4t3e12 12771 | . . . . . . . . . 10 ⊢ (4 · 3) = ;12 | |
37 | 3t3e9 12375 | . . . . . . . . . 10 ⊢ (3 · 3) = 9 | |
38 | 8, 27, 8, 35, 36, 37 | decmul1 12737 | . . . . . . . . 9 ⊢ (;43 · 3) = ;;129 |
39 | 34, 38 | eqtr4di 2782 | . . . . . . . 8 ⊢ (𝑃 = ;;129 → 𝑃 = (;43 · 3)) |
40 | 39 | eleq1d 2810 | . . . . . . 7 ⊢ (𝑃 = ;;129 → (𝑃 ∈ ℙ ↔ (;43 · 3) ∈ ℙ)) |
41 | 33, 40 | mtbiri 327 | . . . . . 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 1424 | . . . 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 1130 | . 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 1083 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 1c1 11106 + caddc 11108 · cmul 11110 ≤ cle 11245 2c2 12263 3c3 12264 4c4 12265 5c5 12266 6c6 12267 9c9 12270 ;cdc 12673 ⌊cfl 13751 √csqrt 15176 ∥ cdvds 16193 ℙcprime 16604 FermatNocfmtno 46646 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-inf 9433 df-oi 9500 df-dju 9891 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-ioo 13324 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-prod 15846 df-dvds 16194 df-gcd 16432 df-prm 16605 df-odz 16694 df-phi 16695 df-pc 16766 df-lgs 27132 df-fmtno 46647 |
This theorem is referenced by: fmtno4prm 46694 |
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