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Mirrors > Home > MPE Home > Th. List > prmndvdsfaclt | Structured version Visualization version GIF version |
Description: A prime number does not divide the factorial of a nonnegative integer less than the prime number. (Contributed by AV, 13-Jul-2021.) |
Ref | Expression |
---|---|
prmndvdsfaclt | ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑁 < 𝑃 → ¬ 𝑃 ∥ (!‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 12488 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
2 | prmnn 16618 | . . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ) | |
3 | 2 | nnred 12234 | . . 3 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℝ) |
4 | ltnle 11300 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ 𝑃 ∈ ℝ) → (𝑁 < 𝑃 ↔ ¬ 𝑃 ≤ 𝑁)) | |
5 | 1, 3, 4 | syl2anr 596 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑁 < 𝑃 ↔ ¬ 𝑃 ≤ 𝑁)) |
6 | prmfac1 16665 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (!‘𝑁)) → 𝑃 ≤ 𝑁) | |
7 | 6 | 3exp 1118 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑃 ∈ ℙ → (𝑃 ∥ (!‘𝑁) → 𝑃 ≤ 𝑁))) |
8 | 7 | impcom 407 | . . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑃 ∥ (!‘𝑁) → 𝑃 ≤ 𝑁)) |
9 | 8 | con3d 152 | . 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (¬ 𝑃 ≤ 𝑁 → ¬ 𝑃 ∥ (!‘𝑁))) |
10 | 5, 9 | sylbid 239 | 1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ0) → (𝑁 < 𝑃 → ¬ 𝑃 ∥ (!‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2105 class class class wbr 5148 ‘cfv 6543 ℝcr 11115 < clt 11255 ≤ cle 11256 ℕ0cn0 12479 !cfa 14240 ∥ cdvds 16204 ℙcprime 16615 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-rp 12982 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-fac 14241 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-dvds 16205 df-gcd 16443 df-prm 16616 |
This theorem is referenced by: gausslemma2dlem0c 27204 prmdvdsbc 32455 |
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