Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ssuzfz | Structured version Visualization version GIF version |
Description: A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
ssuzfz.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
ssuzfz.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝑍) |
ssuzfz.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
Ref | Expression |
---|---|
ssuzfz | ⊢ (𝜑 → 𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssuzfz.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝑍) | |
2 | 1 | sselda 3966 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍) |
3 | ssuzfz.1 | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrdi 2923 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (ℤ≥‘𝑀)) |
5 | eluzel2 12242 | . . . . . . . 8 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ ℤ) |
7 | uzssz 12258 | . . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
8 | 3, 7 | eqsstri 4000 | . . . . . . . . . . 11 ⊢ 𝑍 ⊆ ℤ |
9 | 8 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ⊆ ℤ) |
10 | 1, 9 | sstrd 3976 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℤ) |
11 | 10 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℤ) |
12 | ne0i 4299 | . . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐴 → 𝐴 ≠ ∅) | |
13 | 12 | adantl 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ≠ ∅) |
14 | ssuzfz.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
15 | 14 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ∈ Fin) |
16 | suprfinzcl 12091 | . . . . . . . . 9 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴) | |
17 | 11, 13, 15, 16 | syl3anc 1367 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
18 | 11, 17 | sseldd 3967 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ ℤ) |
19 | 10 | sselda 3966 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℤ) |
20 | 6, 18, 19 | 3jca 1124 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑀 ∈ ℤ ∧ sup(𝐴, ℝ, < ) ∈ ℤ ∧ 𝑘 ∈ ℤ)) |
21 | eluzle 12250 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑘) | |
22 | 4, 21 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝑘) |
23 | zssre 11982 | . . . . . . . . . 10 ⊢ ℤ ⊆ ℝ | |
24 | 23 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → ℤ ⊆ ℝ) |
25 | 10, 24 | sstrd 3976 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
26 | 25 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
27 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) | |
28 | eqidd 2822 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )) | |
29 | 26, 15, 27, 28 | supfirege 11621 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ≤ sup(𝐴, ℝ, < )) |
30 | 20, 22, 29 | jca32 518 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑀 ∈ ℤ ∧ sup(𝐴, ℝ, < ) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ sup(𝐴, ℝ, < )))) |
31 | elfz2 12893 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...sup(𝐴, ℝ, < )) ↔ ((𝑀 ∈ ℤ ∧ sup(𝐴, ℝ, < ) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ sup(𝐴, ℝ, < )))) | |
32 | 30, 31 | sylibr 236 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
33 | 32 | ex 415 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < )))) |
34 | 33 | ralrimiv 3181 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
35 | dfss3 3955 | . 2 ⊢ (𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < )) ↔ ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) | |
36 | 34, 35 | sylibr 236 | 1 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 supcsup 8898 ℝcr 10530 < clt 10669 ≤ cle 10670 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 |
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 ax-pre-sup 10609 |
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-rmo 3146 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-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-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 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-z 11976 df-uz 12238 df-fz 12887 |
This theorem is referenced by: sge0isum 42703 |
Copyright terms: Public domain | W3C validator |