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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 11334 | . . . 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 9503 | . 2 ⊢ (𝜑 → (𝐶 < 𝑆 ∨ 𝐶 = 𝑆)) |
8 | 3, 5 | sseldd 3983 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | 5 | ne0d 4339 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
10 | fisupcl 9502 | . . . . . 6 ⊢ (( < Or ℝ ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ ℝ)) → sup(𝐵, ℝ, < ) ∈ 𝐵) | |
11 | 2, 4, 9, 3, 10 | syl13anc 1369 | . . . . 5 ⊢ (𝜑 → sup(𝐵, ℝ, < ) ∈ 𝐵) |
12 | 6, 11 | eqeltrd 2829 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
13 | 3, 12 | sseldd 3983 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
14 | 8, 13 | leloed 11397 | . 2 ⊢ (𝜑 → (𝐶 ≤ 𝑆 ↔ (𝐶 < 𝑆 ∨ 𝐶 = 𝑆))) |
15 | 7, 14 | mpbird 256 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ⊆ wss 3949 ∅c0 4326 class class class wbr 5152 Or wor 5593 Fincfn 8972 supcsup 9473 ℝcr 11147 < clt 11288 ≤ cle 11289 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-resscn 11205 ax-pre-lttri 11222 ax-pre-lttrn 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-om 7879 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 |
This theorem is referenced by: fsuppmapnn0fiub 13998 ssuzfz 44778 |
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