| 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 5146 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 < (𝐹‘𝑥) ↔ -∞ < (𝐹‘𝑥))) | |
| 2 | 1 | rabbidv 3444 | . . . 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 5962 | . . . . . . 7 ⊢ Ⅎ𝑥dom 𝐹 |
| 6 | 3, 5 | nfcxfr 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝐷 |
| 7 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑦𝐷 | |
| 8 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
| 9 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
| 10 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
| 11 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
| 12 | 4, 11 | nffv 6916 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
| 13 | 9, 10, 12 | nfbr 5190 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
| 14 | fveq2 6906 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
| 15 | 14 | breq2d 5155 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (-∞ < (𝐹‘𝑥) ↔ -∞ < (𝐹‘𝑦))) |
| 16 | 6, 7, 8, 13, 15 | cbvrabw 3473 | . . . . 5 ⊢ {𝑥 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑦)} |
| 17 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑦𝜑 | |
| 18 | smfpimgtxr.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 19 | smfpimgtxr.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
| 20 | 18, 19, 3 | smff 46747 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) |
| 21 | 20 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → (𝐹‘𝑦) ∈ ℝ) |
| 22 | 17, 21 | pimgtmnf 46738 | . . . . 5 ⊢ (𝜑 → {𝑦 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑦)} = 𝐷) |
| 23 | 16, 22 | eqtrid 2789 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ -∞ < (𝐹‘𝑥)} = 𝐷) |
| 24 | 2, 23 | sylan9eqr 2799 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = 𝐷) |
| 25 | 18, 19, 3 | smfdmss 46748 | . . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
| 26 | 18, 25 | subsaluni 46375 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝑆 ↾t 𝐷)) |
| 27 | 26 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = -∞) → 𝐷 ∈ (𝑆 ↾t 𝐷)) |
| 28 | 24, 27 | eqeltrd 2841 | . 2 ⊢ ((𝜑 ∧ 𝐴 = -∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) |
| 29 | breq1 5146 | . . . . . . 7 ⊢ (𝐴 = +∞ → (𝐴 < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑥))) | |
| 30 | 29 | rabbidv 3444 | . . . . . 6 ⊢ (𝐴 = +∞ → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ +∞ < (𝐹‘𝑥)}) |
| 31 | 4, 6, 20 | pimgtpnf2f 46720 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ +∞ < (𝐹‘𝑥)} = ∅) |
| 32 | 30, 31 | sylan9eqr 2799 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = +∞) → {𝑥 ∈ 𝐷 ∣ 𝐴 < (𝐹‘𝑥)} = ∅) |
| 33 | 19 | dmexd 7925 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ∈ V) |
| 34 | 3, 33 | eqeltrid 2845 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V) |
| 35 | eqid 2737 | . . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
| 36 | 18, 34, 35 | subsalsal 46374 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) |
| 37 | 36 | 0sald 46365 | . . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐷)) |
| 38 | 37 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = +∞) → ∅ ∈ (𝑆 ↾t 𝐷)) |
| 39 | 32, 38 | eqeltrd 2841 | . . . 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 2948 | . . . . . 6 ⊢ (¬ 𝐴 = +∞ → 𝐴 ≠ +∞) | |
| 46 | 45 | adantl 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≠ -∞) ∧ ¬ 𝐴 = +∞) → 𝐴 ≠ +∞) |
| 47 | 43, 44, 46 | xrred 45376 | . . . 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 46783 | . . . 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 3029 | 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 1540 ∈ wcel 2108 Ⅎwnfc 2890 ≠ wne 2940 {crab 3436 Vcvv 3480 ∅c0 4333 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 ↾t crest 17465 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-iin 4994 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-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-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-rp 13035 df-ioo 13391 df-ico 13393 df-fl 13832 df-rest 17467 df-salg 46324 df-smblfn 46711 |
| This theorem is referenced by: smfpimgtxrmptf 46799 smfpimne 46854 smfinfdmmbllem 46863 |
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