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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 43754 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193)) | |
2 | 5nn 11724 | . . . . . . . 8 ⊢ 5 ∈ ℕ | |
3 | 1nn0 11914 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
4 | 3nn 11717 | . . . . . . . . 9 ⊢ 3 ∈ ℕ | |
5 | 3, 4 | decnncl 12119 | . . . . . . . 8 ⊢ ;13 ∈ ℕ |
6 | 1lt5 11818 | . . . . . . . 8 ⊢ 1 < 5 | |
7 | 1nn 11649 | . . . . . . . . 9 ⊢ 1 ∈ ℕ | |
8 | 3nn0 11916 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
9 | 1lt10 12238 | . . . . . . . . 9 ⊢ 1 < ;10 | |
10 | 7, 8, 3, 9 | declti 12137 | . . . . . . . 8 ⊢ 1 < ;13 |
11 | eqid 2821 | . . . . . . . 8 ⊢ (5 · ;13) = (5 · ;13) | |
12 | 2, 5, 6, 10, 11 | nprmi 16033 | . . . . . . 7 ⊢ ¬ (5 · ;13) ∈ ℙ |
13 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;65 → 𝑃 = ;65) | |
14 | 5nn0 11918 | . . . . . . . . . 10 ⊢ 5 ∈ ℕ0 | |
15 | eqid 2821 | . . . . . . . . . 10 ⊢ ;13 = ;13 | |
16 | 5cn 11726 | . . . . . . . . . . . . 13 ⊢ 5 ∈ ℂ | |
17 | 16 | mulid1i 10645 | . . . . . . . . . . . 12 ⊢ (5 · 1) = 5 |
18 | 17 | oveq1i 7166 | . . . . . . . . . . 11 ⊢ ((5 · 1) + 1) = (5 + 1) |
19 | 5p1e6 11785 | . . . . . . . . . . 11 ⊢ (5 + 1) = 6 | |
20 | 18, 19 | eqtri 2844 | . . . . . . . . . 10 ⊢ ((5 · 1) + 1) = 6 |
21 | 5t3e15 12200 | . . . . . . . . . 10 ⊢ (5 · 3) = ;15 | |
22 | 14, 3, 8, 15, 14, 3, 20, 21 | decmul2c 12165 | . . . . . . . . 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 329 | . . . . . 6 ⊢ (𝑃 = ;65 → ¬ 𝑃 ∈ ℙ) |
26 | 25 | pm2.21d 121 | . . . . 5 ⊢ (𝑃 = ;65 → (𝑃 ∈ ℙ → 𝑃 = ;;193)) |
27 | 4nn0 11917 | . . . . . . . . 9 ⊢ 4 ∈ ℕ0 | |
28 | 27, 4 | decnncl 12119 | . . . . . . . 8 ⊢ ;43 ∈ ℕ |
29 | 4nn 11721 | . . . . . . . . 9 ⊢ 4 ∈ ℕ | |
30 | 29, 8, 3, 9 | declti 12137 | . . . . . . . 8 ⊢ 1 < ;43 |
31 | 1lt3 11811 | . . . . . . . 8 ⊢ 1 < 3 | |
32 | eqid 2821 | . . . . . . . 8 ⊢ (;43 · 3) = (;43 · 3) | |
33 | 28, 4, 30, 31, 32 | nprmi 16033 | . . . . . . 7 ⊢ ¬ (;43 · 3) ∈ ℙ |
34 | id 22 | . . . . . . . . 9 ⊢ (𝑃 = ;;129 → 𝑃 = ;;129) | |
35 | eqid 2821 | . . . . . . . . . 10 ⊢ ;43 = ;43 | |
36 | 4t3e12 12197 | . . . . . . . . . 10 ⊢ (4 · 3) = ;12 | |
37 | 3t3e9 11805 | . . . . . . . . . 10 ⊢ (3 · 3) = 9 | |
38 | 8, 27, 8, 35, 36, 37 | decmul1 12163 | . . . . . . . . 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 329 | . . . . . 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 1423 | . . . 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 1129 | . 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 1082 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 1c1 10538 + caddc 10540 · cmul 10542 ≤ cle 10676 2c2 11693 3c3 11694 4c4 11695 5c5 11696 6c6 11697 9c9 11700 ;cdc 12099 ⌊cfl 13161 √csqrt 14592 ∥ cdvds 15607 ℙcprime 16015 FermatNocfmtno 43709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-ioo 12743 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-dvds 15608 df-gcd 15844 df-prm 16016 df-odz 16102 df-phi 16103 df-pc 16174 df-lgs 25871 df-fmtno 43710 |
This theorem is referenced by: fmtno4prm 43757 |
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