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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 12247 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐵 ∈ ℤ) | |
2 | 1 | ad2antrr 724 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℤ) |
3 | 2 | zred 12081 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ) |
4 | eluzel2 12242 | . . . . . . 7 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → 𝐴 ∈ ℤ) | |
5 | 4 | ad2antrr 724 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℤ) |
6 | 5 | zred 12081 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐴 ∈ ℝ) |
7 | 3, 6 | resubcld 11062 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐵 − 𝐴) ∈ ℝ) |
8 | simplr 767 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℤ) | |
9 | 8 | zred 12081 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ ℝ) |
10 | 9, 6 | resubcld 11062 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐶 − 𝐴) ∈ ℝ) |
11 | 1red 10636 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 1 ∈ ℝ) | |
12 | simpr 487 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 < 𝐶) | |
13 | 3, 9, 6, 12 | ltsub1dd 11246 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐵 − 𝐴) < (𝐶 − 𝐴)) |
14 | 7, 10, 11, 13 | ltadd1dd 11245 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → ((𝐵 − 𝐴) + 1) < ((𝐶 − 𝐴) + 1)) |
15 | hashfz 13782 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) | |
16 | 15 | ad2antrr 724 | . . 3 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐵)) = ((𝐵 − 𝐴) + 1)) |
17 | 3, 9, 12 | ltled 10782 | . . . . . 6 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ≤ 𝐶) |
18 | eluz2 12243 | . . . . . 6 ⊢ (𝐶 ∈ (ℤ≥‘𝐵) ↔ (𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ ∧ 𝐵 ≤ 𝐶)) | |
19 | 2, 8, 17, 18 | syl3anbrc 1339 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ (ℤ≥‘𝐵)) |
20 | simpll 765 | . . . . 5 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐵 ∈ (ℤ≥‘𝐴)) | |
21 | uztrn 12255 | . . . . 5 ⊢ ((𝐶 ∈ (ℤ≥‘𝐵) ∧ 𝐵 ∈ (ℤ≥‘𝐴)) → 𝐶 ∈ (ℤ≥‘𝐴)) | |
22 | 19, 20, 21 | syl2anc 586 | . . . 4 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → 𝐶 ∈ (ℤ≥‘𝐴)) |
23 | hashfz 13782 | . . . 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 5090 | . 2 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶))) |
26 | fzfi 13334 | . . 3 ⊢ (𝐴...𝐵) ∈ Fin | |
27 | fzfi 13334 | . . 3 ⊢ (𝐴...𝐶) ∈ Fin | |
28 | hashsdom 13736 | . . 3 ⊢ (((𝐴...𝐵) ∈ Fin ∧ (𝐴...𝐶) ∈ Fin) → ((♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶)) ↔ (𝐴...𝐵) ≺ (𝐴...𝐶))) | |
29 | 26, 27, 28 | mp2an 690 | . 2 ⊢ ((♯‘(𝐴...𝐵)) < (♯‘(𝐴...𝐶)) ↔ (𝐴...𝐵) ≺ (𝐴...𝐶)) |
30 | 25, 29 | sylib 220 | 1 ⊢ (((𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐶 ∈ ℤ) ∧ 𝐵 < 𝐶) → (𝐴...𝐵) ≺ (𝐴...𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ≺ csdm 8502 Fincfn 8503 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − cmin 10864 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 ♯chash 13684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 |
This theorem is referenced by: irrapxlem1 39412 |
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