elfzmlbm - Intuitionistic Logic Explorer

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Theorem elfzmlbm 10327

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 10217	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
2		uznn0sub 9754	. . 3 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾 − 𝑀) ∈ ℕ₀)
3	1, 2	syl 14	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ ℕ₀)
4		elfzuz2 10225	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
5		uznn0sub 9754	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ₀)
6	4, 5	syl 14	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
7		elfzelz 10221	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
8	7	zred 9569	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℝ)
9		elfzel2 10219	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
10	9	zred 9569	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℝ)
11		elfzel1 10220	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
12	11	zred 9569	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℝ)
13		elfzle2 10224	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁)
14	8, 10, 12, 13	lesub1dd 8708	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))
15		elfz2nn0 10308	. 2 ⊢ ((𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)) ↔ ((𝐾 − 𝑀) ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ₀ ∧ (𝐾 − 𝑀) ≤ (𝑁 − 𝑀)))
16	3, 6, 14, 15	syl3anbrc 1205	1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2200 class class class wbr 4083 ‘cfv 5318 (class class class)co 6001 0cc0 7999 ≤ cle 8182 − cmin 8317 ℕ₀cn0 9369 ℤ_≥cuz 9722 ...cfz 10204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-addcom 8099 ax-addass 8101 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-0id 8107 ax-rnegex 8108 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-ltadd 8115

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-inn 9111 df-n0 9370 df-z 9447 df-uz 9723 df-fz 10205

This theorem is referenced by: fz1fzo0m1 10389 bcm1k 10982 swrdccatin2 11261

W3C validator