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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 11298 | . . . 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 9467 | . 2 ⊢ (𝜑 → (𝐶 < 𝑆 ∨ 𝐶 = 𝑆)) |
8 | 3, 5 | sseldd 3978 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | 5 | ne0d 4330 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
10 | fisupcl 9466 | . . . . . 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 2827 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
13 | 3, 12 | sseldd 3978 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
14 | 8, 13 | leloed 11361 | . 2 ⊢ (𝜑 → (𝐶 ≤ 𝑆 ↔ (𝐶 < 𝑆 ∨ 𝐶 = 𝑆))) |
15 | 7, 14 | mpbird 257 | 1 ⊢ (𝜑 → 𝐶 ≤ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 Or wor 5580 Fincfn 8941 supcsup 9437 ℝcr 11111 < clt 11252 ≤ cle 11253 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-om 7853 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 |
This theorem is referenced by: fsuppmapnn0fiub 13962 ssuzfz 44631 |
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