Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfadd | Structured version Visualization version GIF version |
Description: The sum of two sigma-measurable functions is measurable. Proposition 121E (b) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfadd.x | ⊢ Ⅎ𝑥𝜑 |
smfadd.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfadd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
smfadd.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
smfadd.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
smfadd.m | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
smfadd.n | ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
Ref | Expression |
---|---|
smfadd | ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfadd.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1830 | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | smfadd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | elinel1 3761 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐴) | |
5 | 4 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐴) |
6 | 1, 5 | ssdf 38273 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
7 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
8 | smfadd.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
9 | 1, 7, 8 | dmmptdf 38412 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
10 | 9 | eqcomd 2616 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | smfadd.m | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
12 | eqid 2610 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
13 | 3, 11, 12 | smfdmss 39619 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
14 | 10, 13 | eqsstrd 3602 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
15 | 6, 14 | sstrd 3578 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ ∪ 𝑆) |
16 | 5, 8 | syldan 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
17 | elinel2 3762 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐶) | |
18 | 17 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐶) |
19 | smfadd.d | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) | |
20 | 18, 19 | syldan 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
21 | 16, 20 | readdcld 9948 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 + 𝐷) ∈ ℝ) |
22 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) | |
23 | 1, 21, 22 | fmptdf 6294 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)):(𝐴 ∩ 𝐶)⟶ℝ) |
24 | 23 | mptex2 38344 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 + 𝐷) ∈ ℝ) |
25 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
26 | 1, 25 | nfan 1816 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
27 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
28 | smfadd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
29 | 28 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
30 | 8 | adantlr 747 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
31 | 19 | adantlr 747 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
32 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
33 | smfadd.n | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) | |
34 | 33 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
35 | simpr 476 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
36 | oveq2 6557 | . . . . . 6 ⊢ (𝑟 = 𝑞 → (𝑝 + 𝑟) = (𝑝 + 𝑞)) | |
37 | 36 | breq1d 4593 | . . . . 5 ⊢ (𝑟 = 𝑞 → ((𝑝 + 𝑟) < 𝑎 ↔ (𝑝 + 𝑞) < 𝑎)) |
38 | 37 | cbvrabv 3172 | . . . 4 ⊢ {𝑟 ∈ ℚ ∣ (𝑝 + 𝑟) < 𝑎} = {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑎} |
39 | 38 | mpteq2i 4669 | . . 3 ⊢ (𝑝 ∈ ℚ ↦ {𝑟 ∈ ℚ ∣ (𝑝 + 𝑟) < 𝑎}) = (𝑝 ∈ ℚ ↦ {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑎}) |
40 | 26, 27, 29, 30, 31, 32, 34, 35, 39 | smfaddlem2 39650 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑎} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
41 | 1, 2, 3, 15, 24, 40 | issmfdmpt 39635 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 {crab 2900 ∩ cin 3539 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 + caddc 9818 < clt 9953 ℚcq 11664 SAlgcsalg 39204 SMblFncsmblfn 39586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050 df-ico 12052 df-rest 15906 df-salg 39205 df-smblfn 39587 |
This theorem is referenced by: (None) |
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