incsmflem - Mathbox for Glauco Siliprandi

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Theorem incsmflem 46739

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 11231	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ^*)
4		incsmflem.l	. . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
5		ssrab2 4043	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴
6	4, 5	eqsstri 3993	. . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴)
8		incsmflem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	7, 8	sstrd 3957	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
10	9	sselda 3946	. . . . 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 46354	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵)
14	1, 13	eqeltrid 2832	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵)
15		incsmflem.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
16		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑥𝐶
17		nfrab1 3426	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
18	4, 17	nfcxfr 2889	. . . . . 6 ⊢ Ⅎ𝑥𝑌
19	16, 18	nfel 2906	. . . . 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 46714	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴))
34		ineq1 4176	. . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴))
35	34	rspceeqv 3611	. . 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 46349	. . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵
39	37, 38	eqeltri 2824	. . . . 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 46716	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴))
52		ineq1 4176	. . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴))
53	52	rspceeqv 3611	. . 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 812	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 2109 ∀wral 3044 ∃wrex 3053 {crab 3405 ∩ cin 3913 ⊆ wss 3914 class class class wbr 5107 ran crn 5639 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 supcsup 9391 ℝcr 11067 -∞cmnf 11206 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 (,)cioo 13306 (,]cioc 13307 topGenctg 17400 SalGencsalgen 46310

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-ioo 13310 df-ioc 13311 df-fl 13754 df-topgen 17406 df-top 22781 df-bases 22833 df-salg 46307 df-salgen 46311

This theorem is referenced by: incsmf 46740

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