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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfsuplem3 | Structured version Visualization version GIF version |
Description: The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
smfsuplem3.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
smfsuplem3.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
smfsuplem3.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfsuplem3.f | ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
smfsuplem3.d | ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} |
smfsuplem3.g | ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
Ref | Expression |
---|---|
smfsuplem3 | ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1912 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfsuplem3.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | smfsuplem3.d | . . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} | |
4 | ssrab2 4090 | . . . . 5 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) | |
5 | 3, 4 | eqsstri 4030 | . . . 4 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
7 | smfsuplem3.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | uzid 12891 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
10 | smfsuplem3.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
11 | 9, 10 | eleqtrrdi 2850 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
12 | fveq2 6907 | . . . . 5 ⊢ (𝑛 = 𝑀 → (𝐹‘𝑛) = (𝐹‘𝑀)) | |
13 | 12 | dmeqd 5919 | . . . 4 ⊢ (𝑛 = 𝑀 → dom (𝐹‘𝑛) = dom (𝐹‘𝑀)) |
14 | smfsuplem3.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
15 | 14, 11 | ffvelcdmd 7105 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ (SMblFn‘𝑆)) |
16 | eqid 2735 | . . . . 5 ⊢ dom (𝐹‘𝑀) = dom (𝐹‘𝑀) | |
17 | 2, 15, 16 | smfdmss 46689 | . . . 4 ⊢ (𝜑 → dom (𝐹‘𝑀) ⊆ ∪ 𝑆) |
18 | 11, 13, 17 | iinssd 45071 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ ∪ 𝑆) |
19 | 6, 18 | sstrd 4006 | . 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
20 | nfv 1912 | . . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷) | |
21 | 11 | ne0d 4348 | . . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅) |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅) |
23 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
24 | 14 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) |
25 | eqid 2735 | . . . . . . 7 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛) | |
26 | 23, 24, 25 | smff 46688 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
27 | 26 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
28 | iinss2 5062 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) | |
29 | 28 | adantl 481 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) |
30 | 5 | sseli 3991 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
31 | 30 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
32 | 29, 31 | sseldd 3996 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
33 | 32 | adantll 714 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
34 | 27, 33 | ffvelcdmd 7105 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
35 | 3 | reqabi 3457 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
36 | 35 | simprbi 496 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
37 | 36 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
38 | 20, 22, 34, 37 | suprclrnmpt 45196 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ) |
39 | smfsuplem3.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) | |
40 | 38, 39 | fmptd 7134 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) |
41 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑀 ∈ ℤ) |
42 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
43 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
44 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
45 | 41, 10, 42, 43, 3, 39, 44 | smfsuplem2 46768 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐺 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) |
46 | 1, 2, 19, 40, 45 | issmfle2d 46765 | 1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3433 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 ∩ ciin 4997 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ⟶wf 6559 ‘cfv 6563 supcsup 9478 ℝcr 11152 < clt 11293 ≤ cle 11294 ℤcz 12611 ℤ≥cuz 12876 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-2o 8506 df-oadd 8509 df-omul 8510 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-oi 9548 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-ioc 13389 df-ico 13390 df-fl 13829 df-rest 17469 df-topgen 17490 df-top 22916 df-bases 22969 df-salg 46265 df-salgen 46269 df-smblfn 46652 |
This theorem is referenced by: smfsup 46770 |
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