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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpimgtxr | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.) |
Ref | Expression |
---|---|
smfpimgtxr.x | ⊢ Ⅎ𝑥𝐹 |
smfpimgtxr.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfpimgtxr.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smfpimgtxr.d | ⊢ 𝐷 = dom 𝐹 |
smfpimgtxr.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
Ref | Expression |
---|---|
smfpimgtxr | ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5145 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 < (𝐹‘𝑥) ↔ -∞ < (𝐹‘𝑥))) | |
2 | 1 | rabbidv 3436 | . . . 4 ⊢ (𝐴 = -∞ → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑥)}) |
3 | smfpimgtxr.d | . . . . . . 7 ⊢ 𝐷 = dom 𝐹 | |
4 | smfpimgtxr.x | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
5 | 4 | nfdm 5947 | . . . . . . 7 ⊢ Ⅎ𝑥dom 𝐹 |
6 | 3, 5 | nfcxfr 2897 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
7 | nfcv 2899 | . . . . . 6 ⊢ Ⅎ𝑦𝐷 | |
8 | nfv 1910 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
9 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
10 | nfcv 2899 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
11 | nfcv 2899 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
12 | 4, 11 | nffv 6901 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
13 | 9, 10, 12 | nfbr 5189 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
14 | fveq2 6891 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
15 | 14 | breq2d 5154 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (-∞ < (𝐹‘𝑥) ↔ -∞ < (𝐹‘𝑦))) |
16 | 6, 7, 8, 13, 15 | cbvrabw 3463 | . . . . 5 ⊢ {𝑥 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑦)} |
17 | nfv 1910 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
18 | smfpimgtxr.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
19 | smfpimgtxr.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
20 | 18, 19, 3 | smff 46114 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
21 | 20 | ffvelcdmda 7088 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ) |
22 | 17, 21 | pimgtmnf 46105 | . . . . 5 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑦)} = 𝐷) |
23 | 16, 22 | eqtrid 2780 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑥)} = 𝐷) |
24 | 2, 23 | sylan9eqr 2790 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = 𝐷) |
25 | 18, 19, 3 | smfdmss 46115 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
26 | 18, 25 | subsaluni 45742 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝑆 ↾t 𝐷)) |
27 | 26 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐷 ∈ (𝑆 ↾t 𝐷)) |
28 | 24, 27 | eqeltrd 2829 | . 2 ⊢ ((𝜑 ∧ 𝐴 = -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
29 | breq1 5145 | . . . . . . 7 ⊢ (𝐴 = +∞ → (𝐴 < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑥))) | |
30 | 29 | rabbidv 3436 | . . . . . 6 ⊢ (𝐴 = +∞ → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ +∞ < (𝐹‘𝑥)}) |
31 | 4, 6, 20 | pimgtpnf2f 46087 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ +∞ < (𝐹‘𝑥)} = ∅) |
32 | 30, 31 | sylan9eqr 2790 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = ∅) |
33 | 19 | dmexd 7905 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ∈ V) |
34 | 3, 33 | eqeltrid 2833 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V) |
35 | eqid 2728 | . . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
36 | 18, 34, 35 | subsalsal 45741 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) |
37 | 36 | 0sald 45732 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐷)) |
38 | 37 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = +∞) → ∅ ∈ (𝑆 ↾t 𝐷)) |
39 | 32, 38 | eqeltrd 2829 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
40 | 39 | adantlr 714 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
41 | simpll 766 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝜑) | |
42 | smfpimgtxr.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
43 | 41, 42 | syl 17 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ∈ ℝ*) |
44 | simplr 768 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ≠ -∞) | |
45 | neqne 2944 | . . . . . 6 ⊢ (¬ 𝐴 = +∞ → 𝐴 ≠ +∞) | |
46 | 45 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ≠ +∞) |
47 | 43, 44, 46 | xrred 44741 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ∈ ℝ) |
48 | 18 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝑆 ∈ SAlg) |
49 | 19 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐹 ∈ (SMblFn‘𝑆)) |
50 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ) | |
51 | 4, 48, 49, 3, 50 | smfpreimagtf 46150 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
52 | 41, 47, 51 | syl2anc 583 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
53 | 40, 52 | pm2.61dan 812 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
54 | 28, 53 | pm2.61dane 3025 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2879 ≠ wne 2936 {crab 3428 Vcvv 3470 ∅c0 4318 class class class wbr 5142 dom cdm 5672 ‘cfv 6542 (class class class)co 7414 ℝcr 11131 +∞cpnf 11269 -∞cmnf 11270 ℝ*cxr 11271 < clt 11272 ↾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: smfpimgtxrmptf 46166 smfpimne 46221 smfinfdmmbllem 46230 |
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