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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 1995 | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | smfadd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | elinel1 3950 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐴) | |
5 | 4 | adantl 467 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐴) |
6 | 1, 5 | ssdf 39766 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
7 | eqid 2771 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
8 | smfadd.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
9 | 1, 7, 8 | dmmptdf 39932 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
10 | 9 | eqcomd 2777 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | smfadd.m | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
12 | eqid 2771 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
13 | 3, 11, 12 | smfdmss 41457 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
14 | 10, 13 | eqsstrd 3788 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
15 | 6, 14 | sstrd 3762 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ ∪ 𝑆) |
16 | 5, 8 | syldan 579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
17 | elinel2 3951 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐶) | |
18 | 17 | adantl 467 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐶) |
19 | smfadd.d | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) | |
20 | 18, 19 | syldan 579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
21 | 16, 20 | readdcld 10275 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 + 𝐷) ∈ ℝ) |
22 | eqid 2771 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) | |
23 | 1, 21, 22 | fmptdf 6532 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)):(𝐴 ∩ 𝐶)⟶ℝ) |
24 | 23 | mptex2 6528 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 + 𝐷) ∈ ℝ) |
25 | nfv 1995 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
26 | 1, 25 | nfan 1980 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
27 | 3 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
28 | smfadd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
29 | 28 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
30 | 8 | adantlr 694 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
31 | 19 | adantlr 694 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
32 | 11 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
33 | smfadd.n | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) | |
34 | 33 | adantr 466 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
35 | simpr 471 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
36 | oveq2 6804 | . . . . . 6 ⊢ (𝑟 = 𝑞 → (𝑝 + 𝑟) = (𝑝 + 𝑞)) | |
37 | 36 | breq1d 4797 | . . . . 5 ⊢ (𝑟 = 𝑞 → ((𝑝 + 𝑟) < 𝑎 ↔ (𝑝 + 𝑞) < 𝑎)) |
38 | 37 | cbvrabv 3349 | . . . 4 ⊢ {𝑟 ∈ ℚ ∣ (𝑝 + 𝑟) < 𝑎} = {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑎} |
39 | 38 | mpteq2i 4876 | . . 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 41487 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑎} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
41 | 1, 2, 3, 15, 24, 40 | issmfdmpt 41472 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 Ⅎwnf 1856 ∈ wcel 2145 {crab 3065 ∩ cin 3722 ∪ cuni 4575 class class class wbr 4787 ↦ cmpt 4864 dom cdm 5250 ‘cfv 6030 (class class class)co 6796 ℝcr 10141 + caddc 10145 < clt 10280 ℚcq 11996 SAlgcsalg 41040 SMblFncsmblfn 41424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-inf2 8706 ax-cc 9463 ax-ac2 9491 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-omul 7722 df-er 7900 df-map 8015 df-pm 8016 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-sup 8508 df-inf 8509 df-oi 8575 df-card 8969 df-acn 8972 df-ac 9143 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-n0 11500 df-z 11585 df-uz 11894 df-q 11997 df-ioo 12384 df-ico 12386 df-rest 16291 df-salg 41041 df-smblfn 41425 |
This theorem is referenced by: (None) |
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