elfz1b - Metamath Proof Explorer

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Theorem elfz1b 13561

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 13482	. . . 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 11182	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 1 ∈ ℝ)
4		zre 12540	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
5	4	3ad2ant3 1135	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℝ)
6		zre 12540	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
7	6	3ad2ant2 1134	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℝ)
8		letr 11275	. . . . . . 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 12566	. . . . 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 13490	. . . 4 ⊢ (𝑁 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
15		fznn 13560	. . . . 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 12557	. . . . . 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 2109 class class class wbr 5110 (class class class)co 7390 ℝcr 11074 1c1 11076 ≤ cle 11216 ℕcn 12193 ℤcz 12536 ...cfz 13475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-z 12537 df-uz 12801 df-fz 13476

This theorem is referenced by: ubmelfzo 13698 cshwidxm 14780 cshwidxn 14781 gausslemma2dlem1a 27283 gausslemma2dlem2 27285 gausslemma2dlem4 27287 dlwwlknondlwlknonf1olem1 30300 pmtrto1cl 33063 psgnfzto1stlem 33064 fzto1st 33067 psgnfzto1st 33069 hgt750lemb 34654 poimirlem32 37653

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