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Mirrors > Home > MPE Home > Th. List > elfzouz2 | Structured version Visualization version GIF version |
Description: The upper bound of a half-open range is greater than or equal to an element of the range. (Contributed by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
elfzouz2 | ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzoelz 13672 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ∈ ℤ) | |
2 | elfzoel2 13671 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ ℤ) | |
3 | elfzolt2 13681 | . . 3 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 < 𝑁) | |
4 | zre 12600 | . . . . 5 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
5 | zre 12600 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
6 | ltle 11340 | . . . . 5 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐾 < 𝑁 → 𝐾 ≤ 𝑁)) | |
7 | 4, 5, 6 | syl2an 594 | . . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 < 𝑁 → 𝐾 ≤ 𝑁)) |
8 | 1, 2, 7 | syl2anc 582 | . . 3 ⊢ (𝐾 ∈ (𝑀..^𝑁) → (𝐾 < 𝑁 → 𝐾 ≤ 𝑁)) |
9 | 3, 8 | mpd 15 | . 2 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝐾 ≤ 𝑁) |
10 | eluz2 12866 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) ↔ (𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ≤ 𝑁)) | |
11 | 1, 2, 9, 10 | syl3anbrc 1340 | 1 ⊢ (𝐾 ∈ (𝑀..^𝑁) → 𝑁 ∈ (ℤ≥‘𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 < clt 11286 ≤ cle 11287 ℤcz 12596 ℤ≥cuz 12860 ..^cfzo 13667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 |
This theorem is referenced by: elfzofz 13688 elfzo1 13722 seqcaopr2 14043 seqf1olem2a 14045 prodfrec 15881 gsumzaddlem 19883 wlkres 29504 trlreslem 29533 trlsegvdeg 30057 iundisjfi 32585 fz1nntr 32593 measiuns 33869 fmul01 44997 itgspltprt 45396 iccpartigtl 46792 |
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