Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > supxrleubrnmptf | Structured version Visualization version GIF version |
Description: The supremum of a nonempty bounded indexed set of extended reals is less than or equal to an upper bound. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
supxrleubrnmptf.x | ⊢ Ⅎ𝑥𝜑 |
supxrleubrnmptf.a | ⊢ Ⅎ𝑥𝐴 |
supxrleubrnmptf.n | ⊢ Ⅎ𝑥𝐶 |
supxrleubrnmptf.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
supxrleubrnmptf.c | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
Ref | Expression |
---|---|
supxrleubrnmptf | ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supxrleubrnmptf.a | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3906 | . . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3896 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 1, 2, 3, 4, 5 | cbvmptf 5157 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 6 | rneqi 5801 | . . . . 5 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 7 | supeq1i 8905 | . . . 4 ⊢ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = sup(ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵), ℝ*, < ) |
9 | 8 | breq1i 5065 | . . 3 ⊢ (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ sup(ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵), ℝ*, < ) ≤ 𝐶) |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ sup(ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵), ℝ*, < ) ≤ 𝐶)) |
11 | nfv 1911 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
12 | supxrleubrnmptf.x | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
13 | 1 | nfcri 2971 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
14 | 12, 13 | nfan 1896 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴) |
15 | 4 | nfel1 2994 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ* |
16 | 14, 15 | nfim 1893 | . . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ*) |
17 | eleq1w 2895 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | anbi2d 630 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))) |
19 | 5 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℝ* ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ*)) |
20 | 18, 19 | imbi12d 347 | . . . 4 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ*))) |
21 | supxrleubrnmptf.b | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
22 | 16, 20, 21 | chvarfv 2238 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ*) |
23 | supxrleubrnmptf.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
24 | 11, 22, 23 | supxrleubrnmpt 41672 | . 2 ⊢ (𝜑 → (sup(ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶)) |
25 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥 ≤ | |
26 | supxrleubrnmptf.n | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
27 | 4, 25, 26 | nfbr 5105 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶 |
28 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑦 𝐵 ≤ 𝐶 | |
29 | eqcom 2828 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
30 | 29 | imbi1i 352 | . . . . . . 7 ⊢ ((𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)) |
31 | eqcom 2828 | . . . . . . . 8 ⊢ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) | |
32 | 31 | imbi2i 338 | . . . . . . 7 ⊢ ((𝑦 = 𝑥 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)) |
33 | 30, 32 | bitri 277 | . . . . . 6 ⊢ ((𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)) |
34 | 5, 33 | mpbi 232 | . . . . 5 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) |
35 | 34 | breq1d 5068 | . . . 4 ⊢ (𝑦 = 𝑥 → (⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶 ↔ 𝐵 ≤ 𝐶)) |
36 | 2, 1, 27, 28, 35 | cbvralfw 3437 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶) |
37 | 36 | a1i 11 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) |
38 | 10, 24, 37 | 3bitrd 307 | 1 ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 ⦋csb 3882 class class class wbr 5058 ↦ cmpt 5138 ran crn 5550 supcsup 8898 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
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-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 |
This theorem is referenced by: liminflelimsuplem 42049 |
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