| 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 1914 | . 2 ⊢ Ⅎ𝑎𝜑 | |
| 2 | incsmf.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
| 3 | retop 24782 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
| 4 | 2, 3 | eqeltri 2837 | . . . 4 ⊢ 𝐽 ∈ Top |
| 5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
| 6 | incsmf.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐽) | |
| 7 | 5, 6 | salgencld 46364 | . 2 ⊢ (𝜑 → 𝐵 ∈ SAlg) |
| 8 | incsmf.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
| 9 | 5, 6 | unisalgen2 46369 | . . . 4 ⊢ (𝜑 → ∪ 𝐵 = ∪ 𝐽) |
| 10 | 2 | unieqi 4919 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
| 11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 = ∪ (topGen‘ran (,))) |
| 12 | uniretop 24783 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 13 | 12 | eqcomi 2746 | . . . . 5 ⊢ ∪ (topGen‘ran (,)) = ℝ |
| 14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ (topGen‘ran (,)) = ℝ) |
| 15 | 9, 11, 14 | 3eqtrrd 2782 | . . 3 ⊢ (𝜑 → ℝ = ∪ 𝐵) |
| 16 | 8, 15 | sseqtrd 4020 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵) |
| 17 | incsmf.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
| 18 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑎 ∈ ℝ) | |
| 19 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ ℝ) | |
| 20 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ⊆ ℝ) |
| 21 | 17 | frexr 45396 | . . . . 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 5146 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦)) | |
| 25 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) | |
| 26 | 25 | breq1d 5153 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑦))) |
| 27 | 24, 26 | imbi12d 344 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)))) |
| 28 | breq2 5147 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧)) | |
| 29 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) | |
| 30 | 29 | breq2d 5155 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
| 31 | 28, 30 | imbi12d 344 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) |
| 32 | 27, 31 | cbvral2vw 3241 | . . . . . 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 218 | . . . . 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 11307 | . . . . 5 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
| 36 | 35 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
| 37 | 25 | breq1d 5153 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑤) < 𝑎)) |
| 38 | 37 | cbvrabv 3447 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = {𝑤 ∈ 𝐴 ∣ (𝐹‘𝑤) < 𝑎} |
| 39 | eqid 2737 | . . . 4 ⊢ sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) = sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) | |
| 40 | eqid 2737 | . . . 4 ⊢ (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) = (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) | |
| 41 | eqid 2737 | . . . 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 46756 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴)) |
| 43 | reex 11246 | . . . . . . 7 ⊢ ℝ ∈ V | |
| 44 | 43 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
| 45 | 44, 8 | ssexd 5324 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
| 46 | elrest 17472 | . . . . 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 584 | . . . 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 257 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴)) |
| 50 | 1, 7, 16, 17, 49 | issmfd 46750 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 supcsup 9480 ℝcr 11154 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 (,]cioc 13388 ↾t crest 17465 topGenctg 17482 Topctop 22899 SAlgcsalg 46323 SalGencsalgen 46327 SMblFncsmblfn 46710 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-ioo 13391 df-ioc 13392 df-ico 13393 df-fl 13832 df-rest 17467 df-topgen 17488 df-top 22900 df-bases 22953 df-salg 46324 df-salgen 46328 df-smblfn 46711 |
| This theorem is referenced by: smfid 46767 |
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