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Mirrors > Home > MPE Home > Th. List > fzm1ndvds | Structured version Visualization version GIF version |
Description: No number between 1 and 𝑀 − 1 divides 𝑀. (Contributed by Mario Carneiro, 24-Jan-2015.) |
Ref | Expression |
---|---|
fzm1ndvds | ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → ¬ 𝑀 ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzle2 12914 | . . . . 5 ⊢ (𝑁 ∈ (1...(𝑀 − 1)) → 𝑁 ≤ (𝑀 − 1)) | |
2 | 1 | adantl 484 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → 𝑁 ≤ (𝑀 − 1)) |
3 | elfzelz 12911 | . . . . . 6 ⊢ (𝑁 ∈ (1...(𝑀 − 1)) → 𝑁 ∈ ℤ) | |
4 | 3 | adantl 484 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → 𝑁 ∈ ℤ) |
5 | nnz 12007 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ) |
7 | zltlem1 12038 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ 𝑁 ≤ (𝑀 − 1))) | |
8 | 4, 6, 7 | syl2anc 586 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → (𝑁 < 𝑀 ↔ 𝑁 ≤ (𝑀 − 1))) |
9 | 2, 8 | mpbird 259 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → 𝑁 < 𝑀) |
10 | elfznn 12939 | . . . . . 6 ⊢ (𝑁 ∈ (1...(𝑀 − 1)) → 𝑁 ∈ ℕ) | |
11 | 10 | adantl 484 | . . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → 𝑁 ∈ ℕ) |
12 | 11 | nnred 11655 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → 𝑁 ∈ ℝ) |
13 | nnre 11647 | . . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ) |
15 | 12, 14 | ltnled 10789 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁)) |
16 | 9, 15 | mpbid 234 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → ¬ 𝑀 ≤ 𝑁) |
17 | dvdsle 15662 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 ∥ 𝑁 → 𝑀 ≤ 𝑁)) | |
18 | 6, 11, 17 | syl2anc 586 | . 2 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → (𝑀 ∥ 𝑁 → 𝑀 ≤ 𝑁)) |
19 | 16, 18 | mtod 200 | 1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (1...(𝑀 − 1))) → ¬ 𝑀 ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 1c1 10540 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 ℤcz 11984 ...cfz 12895 ∥ cdvds 15609 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-dvds 15610 |
This theorem is referenced by: prmdivdiv 16126 reumodprminv 16143 wilthlem1 25647 wilthlem2 25648 wilthlem3 25649 lgseisenlem1 25953 lgseisenlem2 25954 lgseisenlem3 25955 lgsquadlem3 25960 etransclem44 42570 |
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