elfz1b - Metamath Proof Explorer

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Theorem elfz1b 13254

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 13175	. . . 4 ⊢ (𝑁 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)))
2		simpl2 1190	. . . . 5 ⊢ (((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)) → 𝑀 ∈ ℤ)
3		1red 10907	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ)
4		zre 12253	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
5	4	3ad2ant3 1133	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
6		zre 12253	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
7	6	3ad2ant2 1132	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
8		letr 10999	. . . . . . 7 ⊢ ((1 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((1 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 1 ≤ 𝑀))
9	3, 5, 7, 8	syl3anc 1369	. . . . . 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 12276	. . . . 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 216	. . 3 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℕ)
14		elfzel2 13183	. . . 4 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
15		fznn 13253	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀)))
16	15	biimpd 228	. . . 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 1094	. . 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 12272	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
21	20, 15	syl 17	. . . . 5 ⊢ (𝑀 ∈ ℕ → (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑁 ≤ 𝑀)))
22	21	biimprd 247	. . . 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 1112	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀) → 𝑁 ∈ (1...𝑀))
25	19, 24	impbii 208	1 ⊢ (𝑁 ∈ (1...𝑀) ↔ (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ≤ 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 ∈ wcel 2108 class class class wbr 5070 (class class class)co 7255 ℝcr 10801 1c1 10803 ≤ cle 10941 ℕcn 11903 ℤcz 12249 ...cfz 13168

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-z 12250 df-uz 12512 df-fz 13169

This theorem is referenced by: ubmelfzo 13380 cshwidxm 14449 cshwidxn 14450 gausslemma2dlem1a 26418 gausslemma2dlem2 26420 gausslemma2dlem4 26422 dlwwlknondlwlknonf1olem1 28629 pmtrto1cl 31268 psgnfzto1stlem 31269 fzto1st 31272 psgnfzto1st 31274 hgt750lemb 32536 poimirlem32 35736

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