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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 5151 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 < (𝐹‘𝑥) ↔ -∞ < (𝐹‘𝑥))) | |
2 | 1 | rabbidv 3441 | . . . 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 5965 | . . . . . . 7 ⊢ Ⅎ𝑥dom 𝐹 |
6 | 3, 5 | nfcxfr 2901 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
7 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑦𝐷 | |
8 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
9 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
10 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
11 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
12 | 4, 11 | nffv 6917 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
13 | 9, 10, 12 | nfbr 5195 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
14 | fveq2 6907 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
15 | 14 | breq2d 5160 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (-∞ < (𝐹‘𝑥) ↔ -∞ < (𝐹‘𝑦))) |
16 | 6, 7, 8, 13, 15 | cbvrabw 3471 | . . . . 5 ⊢ {𝑥 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑦)} |
17 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
18 | smfpimgtxr.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
19 | smfpimgtxr.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
20 | 18, 19, 3 | smff 46688 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
21 | 20 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ) |
22 | 17, 21 | pimgtmnf 46679 | . . . . 5 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑦)} = 𝐷) |
23 | 16, 22 | eqtrid 2787 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑥)} = 𝐷) |
24 | 2, 23 | sylan9eqr 2797 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = 𝐷) |
25 | 18, 19, 3 | smfdmss 46689 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
26 | 18, 25 | subsaluni 46316 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝑆 ↾t 𝐷)) |
27 | 26 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐷 ∈ (𝑆 ↾t 𝐷)) |
28 | 24, 27 | eqeltrd 2839 | . 2 ⊢ ((𝜑 ∧ 𝐴 = -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
29 | breq1 5151 | . . . . . . 7 ⊢ (𝐴 = +∞ → (𝐴 < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑥))) | |
30 | 29 | rabbidv 3441 | . . . . . 6 ⊢ (𝐴 = +∞ → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ +∞ < (𝐹‘𝑥)}) |
31 | 4, 6, 20 | pimgtpnf2f 46661 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ +∞ < (𝐹‘𝑥)} = ∅) |
32 | 30, 31 | sylan9eqr 2797 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = ∅) |
33 | 19 | dmexd 7926 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ∈ V) |
34 | 3, 33 | eqeltrid 2843 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V) |
35 | eqid 2735 | . . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
36 | 18, 34, 35 | subsalsal 46315 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) |
37 | 36 | 0sald 46306 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐷)) |
38 | 37 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = +∞) → ∅ ∈ (𝑆 ↾t 𝐷)) |
39 | 32, 38 | eqeltrd 2839 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
40 | 39 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
41 | simpll 767 | . . . 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 769 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ≠ -∞) | |
45 | neqne 2946 | . . . . . 6 ⊢ (¬ 𝐴 = +∞ → 𝐴 ≠ +∞) | |
46 | 45 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ≠ +∞) |
47 | 43, 44, 46 | xrred 45315 | . . . 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 46724 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
52 | 41, 47, 51 | syl2anc 584 | . . 3 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
53 | 40, 52 | pm2.61dan 813 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
54 | 28, 53 | pm2.61dane 3027 | 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 1537 ∈ wcel 2106 Ⅎwnfc 2888 ≠ wne 2938 {crab 3433 Vcvv 3478 ∅c0 4339 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 +∞cpnf 11290 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 ↾t crest 17467 SAlgcsalg 46264 SMblFncsmblfn 46651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-ioo 13388 df-ico 13390 df-fl 13829 df-rest 17469 df-salg 46265 df-smblfn 46652 |
This theorem is referenced by: smfpimgtxrmptf 46740 smfpimne 46795 smfinfdmmbllem 46804 |
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