| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > incsmflem | Structured version Visualization version GIF version | ||
| Description: A nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
| Ref | Expression |
|---|---|
| incsmflem.x | ⊢ Ⅎ𝑥𝜑 |
| incsmflem.y | ⊢ Ⅎ𝑦𝜑 |
| incsmflem.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| incsmflem.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
| incsmflem.i | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
| incsmflem.j | ⊢ 𝐽 = (topGen‘ran (,)) |
| incsmflem.b | ⊢ 𝐵 = (SalGen‘𝐽) |
| incsmflem.r | ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
| incsmflem.l | ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} |
| incsmflem.c | ⊢ 𝐶 = sup(𝑌, ℝ*, < ) |
| incsmflem.d | ⊢ 𝐷 = (-∞(,)𝐶) |
| incsmflem.e | ⊢ 𝐸 = (-∞(,]𝐶) |
| Ref | Expression |
|---|---|
| incsmflem | ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | incsmflem.e | . . . 4 ⊢ 𝐸 = (-∞(,]𝐶) | |
| 2 | mnfxr 11318 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
| 3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ*) |
| 4 | incsmflem.l | . . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} | |
| 5 | ssrab2 4080 | . . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴 | |
| 6 | 4, 5 | eqsstri 4030 | . . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴 |
| 7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴) |
| 8 | incsmflem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
| 9 | 7, 8 | sstrd 3994 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
| 10 | 9 | sselda 3983 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ ℝ) |
| 11 | incsmflem.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
| 12 | incsmflem.b | . . . . 5 ⊢ 𝐵 = (SalGen‘𝐽) | |
| 13 | 3, 10, 11, 12 | iocborel 46371 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵) |
| 14 | 1, 13 | eqeltrid 2845 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵) |
| 15 | incsmflem.x | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
| 16 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
| 17 | nfrab1 3457 | . . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} | |
| 18 | 4, 17 | nfcxfr 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝑌 |
| 19 | 16, 18 | nfel 2920 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌 |
| 20 | 15, 19 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌) |
| 21 | incsmflem.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
| 22 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ 𝑌 | |
| 23 | 21, 22 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐶 ∈ 𝑌) |
| 24 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
| 25 | incsmflem.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
| 26 | 25 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
| 27 | incsmflem.i | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) | |
| 28 | 27 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
| 29 | incsmflem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
| 30 | 29 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
| 31 | incsmflem.c | . . . 4 ⊢ 𝐶 = sup(𝑌, ℝ*, < ) | |
| 32 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌) | |
| 33 | 20, 23, 24, 26, 28, 30, 4, 31, 32, 1 | pimincfltioc 46731 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴)) |
| 34 | ineq1 4213 | . . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴)) | |
| 35 | 34 | rspceeqv 3645 | . . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
| 36 | 14, 33, 35 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
| 37 | incsmflem.d | . . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶) | |
| 38 | 11, 12 | iooborel 46366 | . . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵 |
| 39 | 37, 38 | eqeltri 2837 | . . . . 5 ⊢ 𝐷 ∈ 𝐵 |
| 40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵) |
| 41 | 40 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵) |
| 42 | 19 | nfn 1857 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌 |
| 43 | 15, 42 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
| 44 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌 | |
| 45 | 21, 44 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
| 46 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
| 47 | 25 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
| 48 | 27 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
| 49 | 29 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
| 50 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌) | |
| 51 | 43, 45, 46, 47, 48, 49, 4, 31, 50, 37 | pimincfltioo 46733 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴)) |
| 52 | ineq1 4213 | . . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴)) | |
| 53 | 52 | rspceeqv 3645 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
| 54 | 41, 51, 53 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
| 55 | 36, 54 | pm2.61dan 813 | 1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ∩ cin 3950 ⊆ wss 3951 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 topGenctg 17482 SalGencsalgen 46327 |
| 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-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-fl 13832 df-topgen 17488 df-top 22900 df-bases 22953 df-salg 46324 df-salgen 46328 |
| This theorem is referenced by: incsmf 46757 |
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