| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| 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 1914 | . 2 ⊢ Ⅎ𝑎𝜑 | |
| 3 | smfadd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 4 | elinel1 4201 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐴) | |
| 5 | 4 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐴) |
| 6 | 1, 5 | ssdf 45080 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
| 7 | eqid 2737 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 8 | smfadd.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
| 9 | 1, 7, 8 | dmmptdf 45229 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
| 10 | 9 | eqcomd 2743 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
| 11 | smfadd.m | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
| 12 | eqid 2737 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 13 | 3, 11, 12 | smfdmss 46748 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
| 14 | 10, 13 | eqsstrd 4018 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
| 15 | 6, 14 | sstrd 3994 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ ∪ 𝑆) |
| 16 | 5, 8 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
| 17 | elinel2 4202 | . . . . . . 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 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
| 21 | 16, 20 | readdcld 11290 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 + 𝐷) ∈ ℝ) |
| 22 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) = (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) | |
| 23 | 1, 21, 22 | fmptdf 7137 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)):(𝐴 ∩ 𝐶)⟶ℝ) |
| 24 | 23 | fvmptelcdm 7133 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 + 𝐷) ∈ ℝ) |
| 25 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
| 26 | 1, 25 | nfan 1899 | . . 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 715 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
| 31 | 19 | adantlr 715 | . . 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 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
| 36 | oveq2 7439 | . . . . . 6 ⊢ (𝑟 = 𝑞 → (𝑝 + 𝑟) = (𝑝 + 𝑞)) | |
| 37 | 36 | breq1d 5153 | . . . . 5 ⊢ (𝑟 = 𝑞 → ((𝑝 + 𝑟) < 𝑎 ↔ (𝑝 + 𝑞) < 𝑎)) |
| 38 | 37 | cbvrabv 3447 | . . . 4 ⊢ {𝑟 ∈ ℚ ∣ (𝑝 + 𝑟) < 𝑎} = {𝑞 ∈ ℚ ∣ (𝑝 + 𝑞) < 𝑎} |
| 39 | 38 | mpteq2i 5247 | . . 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 46779 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 + 𝐷) < 𝑎} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
| 41 | 1, 2, 3, 15, 24, 40 | issmfdmpt 46763 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 + 𝐷)) ∈ (SMblFn‘𝑆)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 {crab 3436 ∩ cin 3950 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 + caddc 11158 < clt 11295 ℚcq 12990 SAlgcsalg 46323 SMblFncsmblfn 46710 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 df-ico 13393 df-rest 17467 df-salg 46324 df-smblfn 46711 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |