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

Theorem issmfge 46152
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iv) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
issmfge.s (𝜑𝑆 ∈ SAlg)
issmfge.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmfge (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))
Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎
Allowed substitution hints:   𝜑(𝑥,𝑎)   𝑆(𝑥)

Proof of Theorem issmfge
Dummy variables 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issmfge.s . . . . . . 7 (𝜑𝑆 ∈ SAlg)
21adantr 480 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 484 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfge.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 46115 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 46114 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1910 . . . . . . . . 9 𝑦𝜑
8 nfv 1910 . . . . . . . . 9 𝑦 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1895 . . . . . . . 8 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
10 nfv 1910 . . . . . . . 8 𝑦 𝑏 ∈ ℝ
119, 10nfan 1895 . . . . . . 7 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
12 nfv 1910 . . . . . . 7 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
131uniexd 7741 . . . . . . . . . . . 12 (𝜑 𝑆 ∈ V)
1413adantr 480 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
15 simpr 484 . . . . . . . . . . 11 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
1614, 15ssexd 5318 . . . . . . . . . 10 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
175, 16syldan 590 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
18 eqid 2728 . . . . . . . . 9 (𝑆t 𝐷) = (𝑆t 𝐷)
192, 17, 18subsalsal 45741 . . . . . . . 8 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2019adantr 480 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
216ffvelcdmda 7088 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ)
2221rexrd 11288 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
2322adantlr 714 . . . . . . 7 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦𝐷) → (𝐹𝑦) ∈ ℝ*)
242adantr 480 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑆 ∈ SAlg)
253adantr 480 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆))
26 simpr 484 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
2724, 25, 4, 26smfpreimagt 46144 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
2827adantlr 714 . . . . . . 7 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦𝐷𝑐 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
29 simpr 484 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
3011, 12, 20, 23, 28, 29salpreimagtge 46107 . . . . . 6 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
3130ralrimiva 3142 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
325, 6, 313jca 1126 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
3332ex 412 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
34 nfv 1910 . . . . . . 7 𝑦 𝐷 𝑆
35 nfv 1910 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
36 nfcv 2899 . . . . . . . 8 𝑦
37 nfrab1 3447 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)}
38 nfcv 2899 . . . . . . . . 9 𝑦(𝑆t 𝐷)
3937, 38nfel 2913 . . . . . . . 8 𝑦{𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4036, 39nfralw 3304 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4134, 35, 40nf3an 1897 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
427, 41nfan 1895 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
43 nfv 1910 . . . . . 6 𝑏𝜑
44 nfv 1910 . . . . . . 7 𝑏 𝐷 𝑆
45 nfv 1910 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
46 nfra1 3277 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)
4744, 45, 46nf3an 1897 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
4843, 47nfan 1895 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)))
491adantr 480 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
50 simpr1 1192 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
51 simpr2 1193 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
52 simpr3 1194 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))
5342, 48, 49, 4, 50, 51, 52issmfgelem 46151 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
5453ex 412 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
5533, 54impbid 211 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷))))
56 breq1 5145 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑦)))
5756rabbidv 3436 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑦𝐷𝑎 ≤ (𝐹𝑦)})
58 fveq2 6891 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
5958breq2d 5154 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 ≤ (𝐹𝑦) ↔ 𝑎 ≤ (𝐹𝑥)))
6059cbvrabv 3438 . . . . . . . 8 {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)}
6160a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6257, 61eqtrd 2768 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 ≤ (𝐹𝑦)} = {𝑥𝐷𝑎 ≤ (𝐹𝑥)})
6362eleq1d 2814 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷)))
6463cbvralvw 3230 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))
65643anbi3i 1157 . . 3 ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷)))
6665a1i 11 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 ≤ (𝐹𝑦)} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))
6755, 66bitrd 279 1 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 ≤ (𝐹𝑥)} ∈ (𝑆t 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3057  {crab 3428  Vcvv 3470  wss 3945   cuni 4903   class class class wbr 5142  dom cdm 5672  wf 6538  cfv 6542  (class class class)co 7414  cr 11131  *cxr 11271   < clt 11272  cle 11273  t crest 17395  SAlgcsalg 45690  SMblFncsmblfn 46077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-cc 10452  ax-ac2 10480  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209  ax-pre-sup 11210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9459  df-inf 9460  df-card 9956  df-acn 9959  df-ac 10133  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-div 11896  df-nn 12237  df-n0 12497  df-z 12583  df-uz 12847  df-q 12957  df-rp 13001  df-ioo 13354  df-ico 13356  df-fl 13783  df-rest 17397  df-salg 45691  df-smblfn 46078
This theorem is referenced by:  smfpreimage  46164
  Copyright terms: Public domain W3C validator