elfz1b - Metamath Proof Explorer

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Theorem elfz1b 12979

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 12902	. . . 4 ⊢ (𝑁 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)))
2		simpl2 1188	. . . . 5 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 𝑀 ∈ ℤ)
3		1red 10644	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ)
4		zre 11988	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
5	4	3ad2ant3 1131	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
6		zre 11988	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
7	6	3ad2ant2 1130	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
8		letr 10736	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 1 ≤ 𝑀))
9	3, 5, 7, 8	syl3anc 1367	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 1 ≤ 𝑀))
10	9	imp 409	. . . . 5 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 1 ≤ 𝑀)
11		elnnz1 12011	. . . . 5 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 1 ≤ 𝑀))
12	2, 10, 11	sylanbrc 585	. . . 4 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 𝑀 ∈ ℕ)
13	1, 12	sylbi 219	. . 3 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℕ)
14		elfzel2 12909	. . . 4 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
15		fznn 12978	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀)))
16	15	biimpd 231	. . . 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 1092	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀) ↔ (𝑀 ∈ ℕ ∧ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀)))
19	13, 17, 18	sylanbrc 585	. 2 ⊢ (𝑁 ∈ (1...𝑀) → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))
20		nnz 12007	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
21	20, 15	syl 17	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀)))
22	21	biimprd 250	. . . 4 ⊢ (𝑀 ∈ ℕ → ((𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀) → 𝑁 ∈ (1...𝑀)))
23	22	expd 418	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑁 ∈ ℕ → (𝑁 ≤ 𝑀 → 𝑁 ∈ (1...𝑀))))
24	23	3imp21 1110	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀) → 𝑁 ∈ (1...𝑀))
25	19, 24	impbii 211	1 ⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 ℝcr 10538 1c1 10540 ≤ cle 10678 ℕcn 11640 ℤcz 11984 ...cfz 12895

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-z 11985 df-uz 12247 df-fz 12896

This theorem is referenced by: ubmelfzo 13105 cshwidxm 14172 cshwidxn 14173 gausslemma2dlem1a 25943 gausslemma2dlem2 25945 gausslemma2dlem4 25947 dlwwlknondlwlknonf1olem1 28145 pmtrto1cl 30743 psgnfzto1stlem 30744 fzto1st 30747 psgnfzto1st 30749 hgt750lemb 31929 poimirlem32 34926

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