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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > supxrrernmpt | Structured version Visualization version GIF version |
Description: The real and extended real indexed suprema match when the indexed real supremum exists. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
supxrrernmpt.x | ⊢ Ⅎ𝑥𝜑 |
supxrrernmpt.a | ⊢ (𝜑 → 𝐴 ≠ ∅) |
supxrrernmpt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
supxrrernmpt.y | ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) |
Ref | Expression |
---|---|
supxrrernmpt | ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supxrrernmpt.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | eqid 2826 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | supxrrernmpt.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 1, 2, 3 | rnmptssd 40194 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ) |
5 | supxrrernmpt.a | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
6 | 1, 3, 2, 5 | rnmptn0 40220 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅) |
7 | supxrrernmpt.y | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) | |
8 | 1, 7 | rnmptbdd 40258 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) |
9 | supxrre 12446 | . 2 ⊢ ((ran (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑧 ≤ 𝑦) → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) | |
10 | 4, 6, 8, 9 | syl3anc 1496 | 1 ⊢ (𝜑 → sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 ≠ wne 3000 ∀wral 3118 ∃wrex 3119 ⊆ wss 3799 ∅c0 4145 class class class wbr 4874 ↦ cmpt 4953 ran crn 5344 supcsup 8616 ℝcr 10252 ℝ*cxr 10391 < clt 10392 ≤ cle 10393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 |
This theorem is referenced by: smfsupxr 41817 |
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