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Mirrors > Home > MPE Home > Th. List > fzsdom2 | Structured version Visualization version GIF version |
Description: Condition for finite ranges to have a strict dominance relation. (Contributed by Stefan O'Rear, 12-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2015.) |
Ref | Expression |
---|---|
fzsdom2 | ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 12780 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
2 | 1 | ad2antrr 725 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℤ) |
3 | 2 | zred 12614 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ) |
4 | eluzel2 12775 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
5 | 4 | ad2antrr 725 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℤ) |
6 | 5 | zred 12614 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℝ) |
7 | 3, 6 | resubcld 11590 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐵 − 𝐴) ∈ ℝ) |
8 | simplr 768 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℤ) | |
9 | 8 | zred 12614 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℝ) |
10 | 9, 6 | resubcld 11590 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐶 − 𝐴) ∈ ℝ) |
11 | 1red 11163 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 1 ∈ ℝ) | |
12 | simpr 486 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 < 𝐶) | |
13 | 3, 9, 6, 12 | ltsub1dd 11774 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐵 − 𝐴) < (𝐶 − 𝐴)) |
14 | 7, 10, 11, 13 | ltadd1dd 11773 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → ((𝐵 − 𝐴) + 1) < ((𝐶 − 𝐴) + 1)) |
15 | hashfz 14334 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) | |
16 | 15 | ad2antrr 725 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) |
17 | 3, 9, 12 | ltled 11310 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ≤ 𝐶) |
18 | eluz2 12776 | . . . . . 6 ⊢ (𝐶 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ≤ 𝐶)) | |
19 | 2, 8, 17, 18 | syl3anbrc 1344 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ (ℤ≥‘𝐵)) |
20 | simpll 766 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ (ℤ≥‘𝐴)) | |
21 | uztrn 12788 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐵) ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐶 ∈ (ℤ≥‘𝐴)) | |
22 | 19, 20, 21 | syl2anc 585 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ (ℤ≥‘𝐴)) |
23 | hashfz 14334 | . . . 4 ⊢ (𝐶 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐶)) = ((𝐶 − 𝐴) + 1)) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐶)) = ((𝐶 − 𝐴) + 1)) |
25 | 14, 16, 24 | 3brtr4d 5142 | . 2 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶))) |
26 | fzfi 13884 | . . 3 ⊢ (𝐴...𝐵) ∈ Fin | |
27 | fzfi 13884 | . . 3 ⊢ (𝐴...𝐶) ∈ Fin | |
28 | hashsdom 14288 | . . 3 ⊢ (((𝐴...𝐵) ∈ Fin ∧ (𝐴...𝐶) ∈ Fin) → ((♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶)) ↔ (𝐴...𝐵) ≺ (𝐴...𝐶))) | |
29 | 26, 27, 28 | mp2an 691 | . 2 ⊢ ((♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶)) ↔ (𝐴...𝐵) ≺ (𝐴...𝐶)) |
30 | 25, 29 | sylib 217 | 1 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ≺ csdm 8889 Fincfn 8890 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 − cmin 11392 ℤcz 12506 ℤ≥cuz 12770 ...cfz 13431 ♯chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-xnn0 12493 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: irrapxlem1 41174 |
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