elfzmlbm - Metamath Proof Explorer

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Theorem elfzmlbm 13589

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 13471	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
2		uznn0sub 12820	. . 3 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾 − 𝑀) ∈ ℕ₀)
3	1, 2	syl 17	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ ℕ₀)
4		elfzuz2 13480	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
5		uznn0sub 12820	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ₀)
6	4, 5	syl 17	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
7		elfzelz 13475	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
8	7	zred 12630	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℝ)
9		elfzel2 13473	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
10	9	zred 12630	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℝ)
11		elfzel1 13474	. . . 4 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
12	11	zred 12630	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℝ)
13		elfzle2 13479	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ≤ 𝑁)
14	8, 10, 12, 13	lesub1dd 11763	. 2 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))
15		elfz2nn0 13569	. 2 ⊢ ((𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)) ↔ ((𝐾 − 𝑀) ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ₀ ∧ (𝐾 − 𝑀) ≤ (𝑁 − 𝑀)))
16	3, 6, 14, 15	syl3anbrc 1345	1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2114 class class class wbr 5086 ‘cfv 6496 (class class class)co 7364 0cc0 11035 ≤ cle 11177 − cmin 11374 ℕ₀cn0 12434 ℤ_≥cuz 12785 ...cfz 13458

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-nn 12172 df-n0 12435 df-z 12522 df-uz 12786 df-fz 13459

This theorem is referenced by: fz1fzo0m1 13662 bcm1k 14274 swrdccatin2 14688 poimirlem24 37987

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