elfz1b - Metamath Proof Explorer

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Theorem elfz1b 13493

Description: Membership in a 1-based finite set of sequential integers. (Contributed by AV, 30-Oct-2018.) (Proof shortened by AV, 23-Jan-2022.)

**Assertion**
Ref	Expression
elfz1b	⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))

**Proof of Theorem elfz1b**
Step	Hyp	Ref	Expression
1		elfz2 13414	. . . 4 ⊢ (𝑁 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)))
2		simpl2 1193	. . . . 5 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 𝑀 ∈ ℤ)
3		1red 11113	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ)
4		zre 12472	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
5	4	3ad2ant3 1135	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
6		zre 12472	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
7	6	3ad2ant2 1134	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
8		letr 11207	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 1 ≤ 𝑀))
9	3, 5, 7, 8	syl3anc 1373	. . . . . 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 12498	. . . . 5 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤ 𝑀))
12	2, 10, 11	sylanbrc 583	. . . 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 13422	. . . 4 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
15		fznn 13492	. . . . 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 1095	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀) ↔ (𝑀 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀)))
19	13, 17, 18	sylanbrc 583	. 2 ⊢ (𝑁 ∈ (1...𝑀) → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))
20		nnz 12489	. . . . . 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 1113	. 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 1086 ∈ wcel 2111 class class class wbr 5089 (class class class)co 7346 ℝcr 11005 1c1 11007 ≤ cle 11147 ℕcn 12125 ℤcz 12468 ...cfz 13407

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-z 12469 df-uz 12733 df-fz 13408

This theorem is referenced by: ubmelfzo 13630 cshwidxm 14715 cshwidxn 14716 gausslemma2dlem1a 27303 gausslemma2dlem2 27305 gausslemma2dlem4 27307 dlwwlknondlwlknonf1olem1 30344 pmtrto1cl 33068 psgnfzto1stlem 33069 fzto1st 33072 psgnfzto1st 33074 hgt750lemb 34669 poimirlem32 37691

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