incsmflem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > incsmflem		Structured version Visualization version GIF version

Theorem incsmflem 47169

Description: A nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
incsmflem	⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))

Distinct variable groups: 𝐴,𝑏 𝑥,𝐴,𝑦 𝐵,𝑏 𝑥,𝐶,𝑦 𝐷,𝑏 𝑥,𝐷,𝑦 𝐸,𝑏 𝑥,𝐸,𝑦 𝑥,𝐹,𝑦 𝑥,𝑅,𝑦 𝑌,𝑏 𝑦,𝑌 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑏) 𝐵(𝑥,𝑦) 𝐶(𝑏) 𝑅(𝑏) 𝐹(𝑏) 𝐽(𝑥,𝑦,𝑏) 𝑌(𝑥)

**Proof of Theorem incsmflem**
Step	Hyp	Ref	Expression
1		incsmflem.e	. . . 4 ⊢ 𝐸 = (-∞(,]𝐶)
2		mnfxr 11202	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ^*)
4		incsmflem.l	. . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
5		ssrab2 4021	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴
6	4, 5	eqsstri 3969	. . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴)
8		incsmflem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	7, 8	sstrd 3933	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
10	9	sselda 3922	. . . . 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 46784	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵)
14	1, 13	eqeltrid 2841	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵)
15		incsmflem.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
16		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝐶
17		nfrab1 3410	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
18	4, 17	nfcxfr 2897	. . . . . 6 ⊢ Ⅎ𝑥𝑌
19	16, 18	nfel 2914	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌
20	15, 19	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌)
21		incsmflem.y	. . . . 5 ⊢ Ⅎ𝑦𝜑
22		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ 𝑌
23	21, 22	nfan 1901	. . . 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 47144	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴))
34		ineq1 4154	. . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴))
35	34	rspceeqv 3588	. . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
36	14, 33, 35	syl2anc 585	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
37		incsmflem.d	. . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶)
38	11, 12	iooborel 46779	. . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵
39	37, 38	eqeltri 2833	. . . . 5 ⊢ 𝐷 ∈ 𝐵
40	39	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
41	40	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵)
42	19	nfn 1859	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌
43	15, 42	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
44		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌
45	21, 44	nfan 1901	. . . 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 47146	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴))
52		ineq1 4154	. . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴))
53	52	rspceeqv 3588	. . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
54	41, 51, 53	syl2anc 585	. 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 1542 Ⅎwnf 1785 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {crab 3390 ∩ cin 3889 ⊆ wss 3890 class class class wbr 5086 ran crn 5632 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 supcsup 9353 ℝcr 11037 -∞cmnf 11177 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 (,)cioo 13298 (,]cioc 13299 topGenctg 17400 SalGencsalgen 46740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-card 9863 df-acn 9866 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-ioo 13302 df-ioc 13303 df-fl 13751 df-topgen 17406 df-top 22859 df-bases 22911 df-salg 46737 df-salgen 46741

This theorem is referenced by: incsmf 47170

W3C validator