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Mirrors > Home > MPE Home > Th. List > elfz1b | Structured version Visualization version GIF version |
Description: Membership in a 1-based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) (Proof shortened by AV, 23-Jan-2022.) |
Ref | Expression |
---|---|
elfz1b | ⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2 13574 | . . . 4 ⊢ (𝑁 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))) | |
2 | simpl2 1192 | . . . . 5 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 𝑀 ∈ ℤ) | |
3 | 1red 11291 | . . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ) | |
4 | zre 12643 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
5 | 4 | 3ad2ant3 1135 | . . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ) |
6 | zre 12643 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
7 | 6 | 3ad2ant2 1134 | . . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ) |
8 | letr 11384 | . . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 1 ≤ 𝑀)) | |
9 | 3, 5, 7, 8 | syl3anc 1371 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 1 ≤ 𝑀)) |
10 | 9 | imp 406 | . . . . 5 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 1 ≤ 𝑀) |
11 | elnnz1 12669 | . . . . 5 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤ 𝑀)) | |
12 | 2, 10, 11 | sylanbrc 582 | . . . 4 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 𝑀 ∈ ℕ) |
13 | 1, 12 | sylbi 217 | . . 3 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℕ) |
14 | elfzel2 13582 | . . . 4 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℤ) | |
15 | fznn 13652 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀))) | |
16 | 15 | biimpd 229 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (1...𝑀) → (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀))) |
17 | 14, 16 | mpcom 38 | . . 3 ⊢ (𝑁 ∈ (1...𝑀) → (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀)) |
18 | 3anan12 1096 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀) ↔ (𝑀 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀))) | |
19 | 13, 17, 18 | sylanbrc 582 | . 2 ⊢ (𝑁 ∈ (1...𝑀) → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀)) |
20 | nnz 12660 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
21 | 20, 15 | syl 17 | . . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀))) |
22 | 21 | biimprd 248 | . . . 4 ⊢ (𝑀 ∈ ℕ → ((𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀) → 𝑁 ∈ (1...𝑀))) |
23 | 22 | expd 415 | . . 3 ⊢ (𝑀 ∈ ℕ → (𝑁 ∈ ℕ → (𝑁 ≤ 𝑀 → 𝑁 ∈ (1...𝑀)))) |
24 | 23 | 3imp21 1114 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀) → 𝑁 ∈ (1...𝑀)) |
25 | 19, 24 | impbii 209 | 1 ⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5166 (class class class)co 7448 ℝcr 11183 1c1 11185 ≤ cle 11325 ℕcn 12293 ℤcz 12639 ...cfz 13567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-z 12640 df-uz 12904 df-fz 13568 |
This theorem is referenced by: ubmelfzo 13781 cshwidxm 14856 cshwidxn 14857 gausslemma2dlem1a 27427 gausslemma2dlem2 27429 gausslemma2dlem4 27431 dlwwlknondlwlknonf1olem1 30396 pmtrto1cl 33092 psgnfzto1stlem 33093 fzto1st 33096 psgnfzto1st 33098 hgt750lemb 34633 poimirlem32 37612 |
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