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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 47497 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193)) | |
2 | 5nn 12350 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
3 | 1nn0 12540 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
4 | 3nn 12343 | . . . . . . . . 9 ⊢ 3 ∈ ℕ | |
5 | 3, 4 | decnncl 12751 | . . . . . . . 8 ⊢ ;13 ∈ ℕ |
6 | 1lt5 12444 | . . . . . . . 8 ⊢ 1 < 5 | |
7 | 1nn 12275 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
8 | 3nn0 12542 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
9 | 1lt10 12870 | . . . . . . . . 9 ⊢ 1 < ;10 | |
10 | 7, 8, 3, 9 | declti 12769 | . . . . . . . 8 ⊢ 1 < ;13 |
11 | eqid 2735 | . . . . . . . 8 ⊢ (5 · ;13) = (5 · ;13) | |
12 | 2, 5, 6, 10, 11 | nprmi 16723 | . . . . . . 7 ⊢ ¬ (5 · ;13) ∈ ℙ |
13 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;65 → 𝑃 = ;65) | |
14 | 5nn0 12544 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
15 | eqid 2735 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
16 | 5cn 12352 | . . . . . . . . . . . . 13 ⊢ 5 ∈ ℂ | |
17 | 16 | mulridi 11263 | . . . . . . . . . . . 12 ⊢ (5 · 1) = 5 |
18 | 17 | oveq1i 7441 | . . . . . . . . . . 11 ⊢ ((5 · 1) + 1) = (5 + 1) |
19 | 5p1e6 12411 | . . . . . . . . . . 11 ⊢ (5 + 1) = 6 | |
20 | 18, 19 | eqtri 2763 | . . . . . . . . . 10 ⊢ ((5 · 1) + 1) = 6 |
21 | 5t3e15 12832 | . . . . . . . . . 10 ⊢ (5 · 3) = ;15 | |
22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 12797 | . . . . . . . . 9 ⊢ (5 · ;13) = ;65 |
23 | 13, 22 | eqtr4di 2793 | . . . . . . . 8 ⊢ (𝑃 = ;65 → 𝑃 = (5 · ;13)) |
24 | 23 | eleq1d 2824 | . . . . . . 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 12543 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
28 | 27, 4 | decnncl 12751 | . . . . . . . 8 ⊢ ;43 ∈ ℕ |
29 | 4nn 12347 | . . . . . . . . 9 ⊢ 4 ∈ ℕ | |
30 | 29, 8, 3, 9 | declti 12769 | . . . . . . . 8 ⊢ 1 < ;43 |
31 | 1lt3 12437 | . . . . . . . 8 ⊢ 1 < 3 | |
32 | eqid 2735 | . . . . . . . 8 ⊢ (;43 · 3) = (;43 · 3) | |
33 | 28, 4, 30, 31, 32 | nprmi 16723 | . . . . . . 7 ⊢ ¬ (;43 · 3) ∈ ℙ |
34 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;;129 → 𝑃 = ;;129) | |
35 | eqid 2735 | . . . . . . . . . 10 ⊢ ;43 = ;43 | |
36 | 4t3e12 12829 | . . . . . . . . . 10 ⊢ (4 · 3) = ;12 | |
37 | 3t3e9 12431 | . . . . . . . . . 10 ⊢ (3 · 3) = 9 | |
38 | 8, 27, 8, 35, 36, 37 | decmul1 12795 | . . . . . . . . 9 ⊢ (;43 · 3) = ;;129 |
39 | 34, 38 | eqtr4di 2793 | . . . . . . . 8 ⊢ (𝑃 = ;;129 → 𝑃 = (;43 · 3)) |
40 | 39 | eleq1d 2824 | . . . . . . 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 1427 | . . . 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 1132 | . 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 1085 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 2c2 12319 3c3 12320 4c4 12321 5c5 12322 6c6 12323 9c9 12326 ;cdc 12731 ⌊cfl 13827 √csqrt 15269 ∥ 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-tp 4636 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-oadd 8509 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-dju 9939 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-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-ioo 13388 df-ico 13390 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 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-odz 16799 df-phi 16800 df-pc 16871 df-lgs 27354 df-fmtno 47453 |
This theorem is referenced by: fmtno4prm 47500 |
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