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Mirrors > Home > MPE Home > Th. List > supfirege | Structured version Visualization version GIF version |
Description: The supremum of a finite set of real numbers is greater than or equal to all the real numbers of the set. (Contributed by AV, 1-Oct-2019.) |
Ref | Expression |
---|---|
supfirege.1 | ⊢ (𝜑 → 𝐵 ⊆ ℝ) |
supfirege.2 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
supfirege.3 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
supfirege.4 | ⊢ (𝜑 → 𝑆 = sup(𝐵, ℝ, < )) |
Ref | Expression |
---|---|
supfirege | ⊢ (𝜑 → 𝐶 ≤ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 10720 | . . . 4 ⊢ < Or ℝ | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → < Or ℝ) |
3 | supfirege.1 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ ℝ) | |
4 | supfirege.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
5 | supfirege.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
6 | supfirege.4 | . . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, ℝ, < )) | |
7 | 2, 3, 4, 5, 6 | supgtoreq 8933 | . 2 ⊢ (𝜑 → (𝐶 < 𝑆 ∨ 𝐶 = 𝑆)) |
8 | 3, 5 | sseldd 3967 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | 5 | ne0d 4300 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
10 | fisupcl 8932 | . . . . . 6 ⊢ (( < Or ℝ ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ ℝ)) → sup(𝐵, ℝ, < ) ∈ 𝐵) | |
11 | 2, 4, 9, 3, 10 | syl13anc 1368 | . . . . 5 ⊢ (𝜑 → sup(𝐵, ℝ, < ) ∈ 𝐵) |
12 | 6, 11 | eqeltrd 2913 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
13 | 3, 12 | sseldd 3967 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
14 | 8, 13 | leloed 10782 | . 2 ⊢ (𝜑 → (𝐶 ≤ 𝑆 ↔ (𝐶 < 𝑆 ∨ 𝐶 = 𝑆))) |
15 | 7, 14 | mpbird 259 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ∅c0 4290 class class class wbr 5065 Or wor 5472 Fincfn 8508 supcsup 8903 ℝcr 10535 < clt 10674 ≤ cle 10675 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-pre-lttri 10610 ax-pre-lttrn 10611 |
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 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-om 7580 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 |
This theorem is referenced by: fsuppmapnn0fiub 13358 ssuzfz 41615 |
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