decsmflem - Mathbox for Glauco Siliprandi

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

Theorem decsmflem 47113

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 11201	. . . . . 6 ⊢ -∞ ∈ ℝ^*
3	2	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ^*)
4		decsmflem.l	. . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)}
5		ssrab2 4034	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} ⊆ 𝐴
6	4, 5	eqsstri 3982	. . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴)
8		decsmflem.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
9	7, 8	sstrd 3946	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ)
10	9	sselda 3935	. . . . 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 46703	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵)
14	1, 13	eqeltrid 2841	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵)
15		decsmflem.x	. . . . 5 ⊢ Ⅎ𝑥𝜑
16		decsmflem.c	. . . . . . 7 ⊢ 𝐶 = sup(𝑌, ℝ^*, < )
17		nfrab1 3421	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)}
18	4, 17	nfcxfr 2897	. . . . . . . 8 ⊢ Ⅎ𝑥𝑌
19		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ^*
20		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥 <
21	18, 19, 20	nfsup 9366	. . . . . . 7 ⊢ Ⅎ𝑥sup(𝑌, ℝ^*, < )
22	16, 21	nfcxfr 2897	. . . . . 6 ⊢ Ⅎ𝑥𝐶
23	22, 18	nfel 2914	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌
24	15, 23	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌)
25	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
26		decsmflem.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
27	26	adantr 480	. . . 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
30		decsmflem.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
31	30	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
32		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌)
33	24, 25, 27, 29, 31, 4, 16, 32, 1	pimdecfgtioc 47062	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴))
34		ineq1 4167	. . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴))
35	34	rspceeqv 3601	. . 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		decsmflem.d	. . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶)
38	11, 12	iooborel 46698	. . . . . 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	23	nfn 1859	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌
43	15, 42	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
44		decsmflem.y	. . . . 5 ⊢ Ⅎ𝑦𝜑
45		nfv 1916	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌
46	44, 45	nfan 1901	. . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌)
47	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ)
48	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ^*)
49	28	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
50	30	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ^*)
51		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌)
52	43, 46, 47, 48, 49, 50, 4, 16, 51, 37	pimdecfgtioo 47064	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴))
53		ineq1 4167	. . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴))
54	53	rspceeqv 3601	. . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
55	41, 52, 54	syl2anc 585	. 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴))
56	36, 55	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 3401 ∩ cin 3902 ⊆ wss 3903 class class class wbr 5100 ran crn 5633 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 supcsup 9355 ℝcr 11037 -∞cmnf 11176 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 (,)cioo 13273 (,]cioc 13274 topGenctg 17369 SalGencsalgen 46659

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-ioo 13277 df-ioc 13278 df-fl 13724 df-topgen 17375 df-top 22850 df-bases 22902 df-salg 46656 df-salgen 46660

This theorem is referenced by: decsmf 47114

W3C validator