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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 47559 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193)) | |
| 2 | 5nn 12352 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
| 3 | 1nn0 12542 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
| 4 | 3nn 12345 | . . . . . . . . 9 ⊢ 3 ∈ ℕ | |
| 5 | 3, 4 | decnncl 12753 | . . . . . . . 8 ⊢ ;13 ∈ ℕ |
| 6 | 1lt5 12446 | . . . . . . . 8 ⊢ 1 < 5 | |
| 7 | 1nn 12277 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
| 8 | 3nn0 12544 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
| 9 | 1lt10 12872 | . . . . . . . . 9 ⊢ 1 < ;10 | |
| 10 | 7, 8, 3, 9 | declti 12771 | . . . . . . . 8 ⊢ 1 < ;13 |
| 11 | eqid 2737 | . . . . . . . 8 ⊢ (5 · ;13) = (5 · ;13) | |
| 12 | 2, 5, 6, 10, 11 | nprmi 16726 | . . . . . . 7 ⊢ ¬ (5 · ;13) ∈ ℙ |
| 13 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;65 → 𝑃 = ;65) | |
| 14 | 5nn0 12546 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
| 15 | eqid 2737 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
| 16 | 5cn 12354 | . . . . . . . . . . . . 13 ⊢ 5 ∈ ℂ | |
| 17 | 16 | mulridi 11265 | . . . . . . . . . . . 12 ⊢ (5 · 1) = 5 |
| 18 | 17 | oveq1i 7441 | . . . . . . . . . . 11 ⊢ ((5 · 1) + 1) = (5 + 1) |
| 19 | 5p1e6 12413 | . . . . . . . . . . 11 ⊢ (5 + 1) = 6 | |
| 20 | 18, 19 | eqtri 2765 | . . . . . . . . . 10 ⊢ ((5 · 1) + 1) = 6 |
| 21 | 5t3e15 12834 | . . . . . . . . . 10 ⊢ (5 · 3) = ;15 | |
| 22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 12799 | . . . . . . . . 9 ⊢ (5 · ;13) = ;65 |
| 23 | 13, 22 | eqtr4di 2795 | . . . . . . . 8 ⊢ (𝑃 = ;65 → 𝑃 = (5 · ;13)) |
| 24 | 23 | eleq1d 2826 | . . . . . . 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 12545 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
| 28 | 27, 4 | decnncl 12753 | . . . . . . . 8 ⊢ ;43 ∈ ℕ |
| 29 | 4nn 12349 | . . . . . . . . 9 ⊢ 4 ∈ ℕ | |
| 30 | 29, 8, 3, 9 | declti 12771 | . . . . . . . 8 ⊢ 1 < ;43 |
| 31 | 1lt3 12439 | . . . . . . . 8 ⊢ 1 < 3 | |
| 32 | eqid 2737 | . . . . . . . 8 ⊢ (;43 · 3) = (;43 · 3) | |
| 33 | 28, 4, 30, 31, 32 | nprmi 16726 | . . . . . . 7 ⊢ ¬ (;43 · 3) ∈ ℙ |
| 34 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;;129 → 𝑃 = ;;129) | |
| 35 | eqid 2737 | . . . . . . . . . 10 ⊢ ;43 = ;43 | |
| 36 | 4t3e12 12831 | . . . . . . . . . 10 ⊢ (4 · 3) = ;12 | |
| 37 | 3t3e9 12433 | . . . . . . . . . 10 ⊢ (3 · 3) = 9 | |
| 38 | 8, 27, 8, 35, 36, 37 | decmul1 12797 | . . . . . . . . 9 ⊢ (;43 · 3) = ;;129 |
| 39 | 34, 38 | eqtr4di 2795 | . . . . . . . 8 ⊢ (𝑃 = ;;129 → 𝑃 = (;43 · 3)) |
| 40 | 39 | eleq1d 2826 | . . . . . . 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 1430 | . . . 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 1134 | . 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 1086 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 2c2 12321 3c3 12322 4c4 12323 5c5 12324 6c6 12325 9c9 12328 ;cdc 12733 ⌊cfl 13830 √csqrt 15272 ∥ cdvds 16290 ℙcprime 16708 FermatNocfmtno 47514 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-ioo 13391 df-ico 13393 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-prod 15940 df-dvds 16291 df-gcd 16532 df-prm 16709 df-odz 16802 df-phi 16803 df-pc 16875 df-lgs 27339 df-fmtno 47515 |
| This theorem is referenced by: fmtno4prm 47562 |
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