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Mathbox for Glauco Siliprandi |
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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 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝑍) | |
2 | 1 | sselda 3974 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑍) |
3 | ssuzfz.1 | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrdi 2835 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (ℤ≥‘𝑀)) |
5 | eluzel2 12824 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ ℤ) |
7 | uzssz 12840 | . . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) ⊆ ℤ | |
8 | 3, 7 | eqsstri 4008 | . . . . . . . . 9 ⊢ 𝑍 ⊆ ℤ |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 ⊆ ℤ) |
10 | 1, 9 | sstrd 3984 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℤ) |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℤ) |
12 | ne0i 4326 | . . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 → 𝐴 ≠ ∅) | |
13 | 12 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ≠ ∅) |
14 | ssuzfz.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
15 | 14 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ∈ Fin) |
16 | suprfinzcl 12673 | . . . . . . 7 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → sup(𝐴, ℝ, < ) ∈ 𝐴) | |
17 | 11, 13, 15, 16 | syl3anc 1368 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ 𝐴) |
18 | 11, 17 | sseldd 3975 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) ∈ ℤ) |
19 | 10 | sselda 3974 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℤ) |
20 | eluzle 12832 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑘) | |
21 | 4, 20 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝑘) |
22 | zssre 12562 | . . . . . . . . 9 ⊢ ℤ ⊆ ℝ | |
23 | 22 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → ℤ ⊆ ℝ) |
24 | 10, 23 | sstrd 3984 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
25 | 24 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐴 ⊆ ℝ) |
26 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) | |
27 | eqidd 2725 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → sup(𝐴, ℝ, < ) = sup(𝐴, ℝ, < )) | |
28 | 25, 15, 26, 27 | supfirege 12198 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ≤ sup(𝐴, ℝ, < )) |
29 | 6, 18, 19, 21, 28 | elfzd 13489 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
30 | 29 | ex 412 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < )))) |
31 | 30 | ralrimiv 3137 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) |
32 | dfss3 3962 | . 2 ⊢ (𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < )) ↔ ∀𝑘 ∈ 𝐴 𝑘 ∈ (𝑀...sup(𝐴, ℝ, < ))) | |
33 | 31, 32 | sylibr 233 | 1 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...sup(𝐴, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ⊆ wss 3940 ∅c0 4314 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 Fincfn 8935 supcsup 9431 ℝcr 11105 < clt 11245 ≤ cle 11246 ℤcz 12555 ℤ≥cuz 12819 ...cfz 13481 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-fz 13482 |
This theorem is referenced by: sge0isum 45628 |
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