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 2764 | . . . . . . 7 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑦𝐵 | |
4 | nfcsb1v 3549 | . . . . . . 7 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
5 | csbeq1a 3542 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
6 | 1, 2, 3, 4, 5 | cbvmptf 4748 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
7 | 6 | rneqi 5352 | . . . . 5 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵) |
8 | 7 | supeq1i 8353 | . . . 4 ⊢ sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) = sup(ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵), ℝ*, < ) |
9 | 8 | breq1i 4660 | . . 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 1843 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
12 | supxrleubrnmptf.x | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
13 | 1 | nfcri 2758 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴 |
14 | 12, 13 | nfan 1828 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐴) |
15 | 4 | nfel1 2779 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ* |
16 | 14, 15 | nfim 1825 | . . . 4 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ*) |
17 | eleq1 2689 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | anbi2d 740 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))) |
19 | 5 | eleq1d 2686 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℝ* ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ*)) |
20 | 18, 19 | imbi12d 334 | . . . 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 | chvar 2262 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ*) |
23 | supxrleubrnmptf.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
24 | 11, 22, 23 | supxrleubrnmpt 39632 | . 2 ⊢ (𝜑 → (sup(ran (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶)) |
25 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑥 ≤ | |
26 | supxrleubrnmptf.n | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
27 | 4, 25, 26 | nfbr 4699 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶 |
28 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑦 𝐵 ≤ 𝐶 | |
29 | eqcom 2629 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
30 | 29 | imbi1i 339 | . . . . . . 7 ⊢ ((𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)) |
31 | eqcom 2629 | . . . . . . . 8 ⊢ (𝐵 = ⦋𝑦 / 𝑥⦌𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) | |
32 | 31 | imbi2i 326 | . . . . . . 7 ⊢ ((𝑦 = 𝑥 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)) |
33 | 30, 32 | bitri 264 | . . . . . 6 ⊢ ((𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) ↔ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵)) |
34 | 5, 33 | mpbi 220 | . . . . 5 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐵 = 𝐵) |
35 | 34 | breq1d 4663 | . . . 4 ⊢ (𝑦 = 𝑥 → (⦋𝑦 / 𝑥⦌𝐵 ≤ 𝐶 ↔ 𝐵 ≤ 𝐶)) |
36 | 2, 1, 27, 28, 35 | cbvralf 3165 | . . 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 294 | 1 ⊢ (𝜑 → (sup(ran (𝑥 ∈ 𝐴 ↦ 𝐵), ℝ*, < ) ≤ 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ⦋csb 3533 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 supcsup 8346 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 |
This theorem is referenced by: liminflelimsuplem 40007 |
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