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, nondecreasing 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 1915 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | incsmf.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | retop 23372 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
4 | 2, 3 | eqeltri 2911 | . . . 4 ⊢ 𝐽 ∈ Top |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | incsmf.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐽) | |
7 | 5, 6 | salgencld 42639 | . 2 ⊢ (𝜑 → 𝐵 ∈ SAlg) |
8 | incsmf.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 5, 6 | unisalgen2 42644 | . . . 4 ⊢ (𝜑 → ∪ 𝐵 = ∪ 𝐽) |
10 | 2 | unieqi 4853 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 = ∪ (topGen‘ran (,))) |
12 | uniretop 23373 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
13 | 12 | eqcomi 2832 | . . . . 5 ⊢ ∪ (topGen‘ran (,)) = ℝ |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ (topGen‘ran (,)) = ℝ) |
15 | 9, 11, 14 | 3eqtrrd 2863 | . . 3 ⊢ (𝜑 → ℝ = ∪ 𝐵) |
16 | 8, 15 | sseqtrd 4009 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵) |
17 | incsmf.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
18 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑎 ∈ ℝ) | |
19 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ ℝ) | |
20 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ⊆ ℝ) |
21 | 17 | frexr 41662 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
22 | 21 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝐴⟶ℝ*) |
23 | incsmf.i | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) | |
24 | breq1 5071 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦)) | |
25 | fveq2 6672 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) | |
26 | 25 | breq1d 5078 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑦))) |
27 | 24, 26 | imbi12d 347 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)))) |
28 | breq2 5072 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧)) | |
29 | fveq2 6672 | . . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) | |
30 | 29 | breq2d 5080 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
31 | 28, 30 | imbi12d 347 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) |
32 | 27, 31 | cbvral2vw 3463 | . . . . . 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 220 | . . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
34 | 33 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
35 | rexr 10689 | . . . . 5 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
37 | 25 | breq1d 5078 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑤) < 𝑎)) |
38 | 37 | cbvrabv 3493 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = {𝑤 ∈ 𝐴 ∣ (𝐹‘𝑤) < 𝑎} |
39 | eqid 2823 | . . . 4 ⊢ sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) = sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) | |
40 | eqid 2823 | . . . 4 ⊢ (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) = (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) | |
41 | eqid 2823 | . . . 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 43025 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴)) |
43 | reex 10630 | . . . . . . 7 ⊢ ℝ ∈ V | |
44 | 43 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
45 | 44, 8 | ssexd 5230 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
46 | elrest 16703 | . . . . 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 586 | . . . 4 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) |
48 | 47 | adantr 483 | . . 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 259 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴)) |
50 | 1, 7, 16, 17, 49 | issmfd 43019 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 {crab 3144 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∪ cuni 4840 class class class wbr 5068 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supcsup 8906 ℝcr 10538 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 (,)cioo 12741 (,]cioc 12742 ↾t crest 16696 topGenctg 16713 Topctop 21503 SAlgcsalg 42600 SalGencsalgen 42604 SMblFncsmblfn 42984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ioo 12745 df-ioc 12746 df-ico 12747 df-fl 13165 df-rest 16698 df-topgen 16719 df-top 21504 df-bases 21556 df-salg 42601 df-salgen 42605 df-smblfn 42985 |
This theorem is referenced by: smfid 43036 |
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