elfzmlbm - Metamath Proof Explorer

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Theorem elfzmlbm 13679

Description: Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) (Proof shortened by OpenAI, 25-Mar-2020.)

**Assertion**
Ref	Expression
elfzmlbm	⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))

**Proof of Theorem elfzmlbm**
Step	Hyp	Ref	Expression
1		elfzuz 13561	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
2		uznn0sub 12918	. . 3 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾 − 𝑀) ∈ ℕ₀)
3	1, 2	syl 17	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ ℕ₀)
4		elfzuz2 13570	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
5		uznn0sub 12918	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ₀)
6	4, 5	syl 17	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
7		elfzelz 13565	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
8	7	zred 12724	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℝ)
9		elfzel2 13563	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
10	9	zred 12724	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℝ)
11		elfzel1 13564	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
12	11	zred 12724	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℝ)
13		elfzle2 13569	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁)
14	8, 10, 12, 13	lesub1dd 11880	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))
15		elfz2nn0 13659	. 2 ⊢ ((𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)) ↔ ((𝐾 − 𝑀) ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ₀ ∧ (𝐾 − 𝑀) ≤ (𝑁 − 𝑀)))
16	3, 6, 14, 15	syl3anbrc 1343	1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 0cc0 11156 ≤ cle 11297 − cmin 11493 ℕ₀cn0 12528 ℤ_≥cuz 12879 ...cfz 13548

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549

This theorem is referenced by: fz1fzo0m1 13751 bcm1k 14355 swrdccatin2 14768 poimirlem24 37652

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