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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 11339 | . . . 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 9508 | . 2 ⊢ (𝜑 → (𝐶 < 𝑆 ∨ 𝐶 = 𝑆)) |
8 | 3, 5 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | 5 | ne0d 4348 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
10 | fisupcl 9507 | . . . . . 6 ⊢ (( < Or ℝ ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ ℝ)) → sup(𝐵, ℝ, < ) ∈ 𝐵) | |
11 | 2, 4, 9, 3, 10 | syl13anc 1371 | . . . . 5 ⊢ (𝜑 → sup(𝐵, ℝ, < ) ∈ 𝐵) |
12 | 6, 11 | eqeltrd 2839 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
13 | 3, 12 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
14 | 8, 13 | leloed 11402 | . 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 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 class class class wbr 5148 Or wor 5596 Fincfn 8984 supcsup 9478 ℝcr 11152 < clt 11293 ≤ cle 11294 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-om 7888 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 |
This theorem is referenced by: fsuppmapnn0fiub 14029 ssuzfz 45299 |
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