decsmflem - Mathbox for Glauco Siliprandi

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Theorem decsmflem 47602

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

**Hypotheses**
Ref	Expression
decsmflem.x	⊢ Ⅎ𝑥𝜑
decsmflem.y	⊢ Ⅎ𝑦𝜑
decsmflem.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
decsmflem.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
decsmflem.i	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
decsmflem.j	⊢ 𝐽 = (topGen‘ran (,))
decsmflem.b	⊢ 𝐵 = (SalGen‘𝐽)
decsmflem.r	⊢ (𝜑 → 𝑅 ∈ ℝ^*)
decsmflem.l	⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)}
decsmflem.c	⊢ 𝐶 = sup(𝑌, ℝ^*, < )
decsmflem.d	⊢ 𝐷 = (-∞(,)𝐶)
decsmflem.e	⊢ 𝐸 = (-∞(,]𝐶)

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

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

**Proof of Theorem decsmflem**
Step	Hyp	Ref	Expression
1		decsmflem.e	. . . 4 ⊢ 𝐸 = (-∞(,]𝐶)
2		mnfxr 11294	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ^*)
4		decsmflem.l	. . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)}
5		ssrab2 4031	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} ⊆ 𝐴
6	4, 5	eqsstri 3980	. . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴)
8		decsmflem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	7, 8	sstrd 3944	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
10	9	sselda 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ ℝ)
11		decsmflem.j	. . . . 5 ⊢ 𝐽 = (topGen‘ran (,))
12		decsmflem.b	. . . . 5 ⊢ 𝐵 = (SalGen‘𝐽)
13	3, 10, 11, 12	iocborel 47192	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵)
14	1, 13	eqeltrid 2866	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵)
15		decsmflem.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
16		decsmflem.c	. . . . . . 7 ⊢ 𝐶 = sup(𝑌, ℝ^*, < )
17		nfrab1 3434	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)}
18	4, 17	nfcxfr 2922	. . . . . . . 8 ⊢ Ⅎ𝑥𝑌
19		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ^*
20		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑥 <
21	18, 19, 20	nfsup 9425	. . . . . . 7 ⊢ Ⅎ𝑥sup(𝑌, ℝ^*, < )
22	16, 21	nfcxfr 2922	. . . . . 6 ⊢ Ⅎ𝑥𝐶
23	22, 18	nfel 2938	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌
24	15, 23	nfan 1932	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌)
25	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
26		decsmflem.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
27	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ^*)
28		decsmflem.i	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
29	28	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
30		decsmflem.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
31	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
32		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌)
33	24, 25, 27, 29, 31, 4, 16, 32, 1	pimdecfgtioc 47551	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴))
34		ineq1 4162	. . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴))
35	34	rspceeqv 3602	. . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
36	14, 33, 35	syl2anc 596	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
37		decsmflem.d	. . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶)
38	11, 12	iooborel 47187	. . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵
39	37, 38	eqeltri 2858	. . . . 5 ⊢ 𝐷 ∈ 𝐵
40	39	a1i 11	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
41	40	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵)
42	23	nfn 1890	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌
43	15, 42	nfan 1932	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
44		decsmflem.y	. . . . 5 ⊢ Ⅎ𝑦𝜑
45		nfv 1947	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌
46	44, 45	nfan 1932	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
47	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
48	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ^*)
49	28	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
50	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
51		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌)
52	43, 46, 47, 48, 49, 50, 4, 16, 51, 37	pimdecfgtioo 47553	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴))
53		ineq1 4162	. . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴))
54	53	rspceeqv 3602	. . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
55	41, 52, 54	syl2anc 596	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
56	36, 55	pm2.61dan 825	1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2145 ∀wral 3078 ∃wrex 3088 {crab 3414 ∩ cin 3901 ⊆ wss 3902 class class class wbr 5107 ran crn 5660 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 supcsup 9414 ℝcr 11127 -∞cmnf 11269 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 (,)cioo 13402 (,]cioc 13403 topGenctg 17528 SalGencsalgen 47148

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-sup 9416 df-inf 9417 df-card 9948 df-acn 9951 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-n0 12533 df-z 12620 df-uz 12892 df-q 13002 df-rp 13047 df-ioo 13406 df-ioc 13407 df-fl 13857 df-topgen 17534 df-top 23125 df-bases 23177 df-salg 47145 df-salgen 47149

This theorem is used by: decsmf 47603

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