Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > incsmf | Structured version Visualization version GIF version |
Description: A real valued, non-decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
incsmf.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
incsmf.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
incsmf.i | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
incsmf.j | ⊢ 𝐽 = (topGen‘ran (,)) |
incsmf.b | ⊢ 𝐵 = (SalGen‘𝐽) |
Ref | Expression |
---|---|
incsmf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | incsmf.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | retop 22375 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
4 | 2, 3 | eqeltri 2684 | . . . 4 ⊢ 𝐽 ∈ Top |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | incsmf.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐽) | |
7 | 5, 6 | salgencld 39243 | . 2 ⊢ (𝜑 → 𝐵 ∈ SAlg) |
8 | incsmf.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 5, 6 | unisalgen2 39248 | . . . 4 ⊢ (𝜑 → ∪ 𝐵 = ∪ 𝐽) |
10 | 2 | unieqi 4381 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 = ∪ (topGen‘ran (,))) |
12 | uniretop 22376 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
13 | 12 | eqcomi 2619 | . . . . 5 ⊢ ∪ (topGen‘ran (,)) = ℝ |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ (topGen‘ran (,)) = ℝ) |
15 | 9, 11, 14 | 3eqtrrd 2649 | . . 3 ⊢ (𝜑 → ℝ = ∪ 𝐵) |
16 | 8, 15 | sseqtrd 3604 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵) |
17 | incsmf.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
18 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑎 ∈ ℝ) | |
19 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ ℝ) | |
20 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ⊆ ℝ) |
21 | 17 | frexr 38545 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝐴⟶ℝ*) |
23 | incsmf.i | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) | |
24 | breq1 4586 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦)) | |
25 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) | |
26 | 25 | breq1d 4593 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑦))) |
27 | 24, 26 | imbi12d 333 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)))) |
28 | breq2 4587 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧)) | |
29 | fveq2 6103 | . . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) | |
30 | 29 | breq2d 4595 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
31 | 28, 30 | imbi12d 333 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) |
32 | 27, 31 | cbvral2v 3155 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
33 | 23, 32 | sylib 207 | . . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
34 | 33 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
35 | rexr 9964 | . . . . 5 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
37 | 25 | breq1d 4593 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑤) < 𝑎)) |
38 | 37 | cbvrabv 3172 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = {𝑤 ∈ 𝐴 ∣ (𝐹‘𝑤) < 𝑎} |
39 | eqid 2610 | . . . 4 ⊢ sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) = sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) | |
40 | eqid 2610 | . . . 4 ⊢ (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) = (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) | |
41 | eqid 2610 | . . . 4 ⊢ (-∞(,]sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) = (-∞(,]sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) | |
42 | 18, 19, 20, 22, 34, 2, 6, 36, 38, 39, 40, 41 | incsmflem 39628 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴)) |
43 | reex 9906 | . . . . . . 7 ⊢ ℝ ∈ V | |
44 | 43 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
45 | 44, 8 | ssexd 4733 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
46 | elrest 15911 | . . . . 5 ⊢ ((𝐵 ∈ SAlg ∧ 𝐴 ∈ V) → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) | |
47 | 7, 45, 46 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) |
48 | 47 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) |
49 | 42, 48 | mpbird 246 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴)) |
50 | 1, 7, 16, 17, 49 | issmfd 39621 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℝcr 9814 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 (,]cioc 12047 ↾t crest 15904 topGenctg 15921 Topctop 20517 SAlgcsalg 39204 SalGencsalgen 39208 SMblFncsmblfn 39586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ioo 12050 df-ioc 12051 df-ico 12052 df-fl 12455 df-rest 15906 df-topgen 15927 df-top 20521 df-bases 20522 df-salg 39205 df-salgen 39209 df-smblfn 39587 |
This theorem is referenced by: smfid 39639 |
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