incsmflem - Mathbox for Glauco Siliprandi

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Theorem incsmflem 44277

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 11032	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ^*)
4		incsmflem.l	. . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
5		ssrab2 4013	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴
6	4, 5	eqsstri 3955	. . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴)
8		incsmflem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	7, 8	sstrd 3931	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
10	9	sselda 3921	. . . . 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 43895	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵)
14	1, 13	eqeltrid 2843	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵)
15		incsmflem.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
16		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝐶
17		nfrab1 3317	. . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅}
18	4, 17	nfcxfr 2905	. . . . . 6 ⊢ Ⅎ𝑥𝑌
19	16, 18	nfel 2921	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌
20	15, 19	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌)
21		incsmflem.y	. . . . 5 ⊢ Ⅎ𝑦𝜑
22		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ 𝑌
23	21, 22	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐶 ∈ 𝑌)
24	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
25		incsmflem.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
26	25	adantr 481	. . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
29		incsmflem.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
30	29	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
31		incsmflem.c	. . . 4 ⊢ 𝐶 = sup(𝑌, ℝ^*, < )
32		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌)
33	20, 23, 24, 26, 28, 30, 4, 31, 32, 1	pimincfltioc 44253	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴))
34		ineq1 4139	. . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴))
35	34	rspceeqv 3575	. . 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 43890	. . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵
39	37, 38	eqeltri 2835	. . . . 5 ⊢ 𝐷 ∈ 𝐵
40	39	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
41	40	adantr 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵)
42	19	nfn 1860	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌
43	15, 42	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
44		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌
45	21, 44	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
46	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
47	25	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ^*)
48	27	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)))
49	29	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
50		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌)
51	43, 45, 46, 47, 48, 49, 4, 31, 50, 37	pimincfltioo 44255	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴))
52		ineq1 4139	. . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴))
53	52	rspceeqv 3575	. . 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 810	1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 ran crn 5590 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 supcsup 9199 ℝcr 10870 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 (,)cioo 13079 (,]cioc 13080 topGenctg 17148 SalGencsalgen 43853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-ioo 13083 df-ioc 13084 df-fl 13512 df-topgen 17154 df-top 22043 df-bases 22096 df-salg 43850 df-salgen 43854

This theorem is referenced by: incsmf 44278

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