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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnfsmf | Structured version Visualization version GIF version |
Description: A continuous function is measurable. Proposition 121D (b) of [Fremlin1] p. 36 is a special case of this theorem, where the topology on the domain is induced by the standard topology on n-dimensional Real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
cnfsmf.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
cnfsmf.k | ⊢ 𝐾 = (topGen‘ran (,)) |
cnfsmf.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) |
cnfsmf.s | ⊢ 𝑆 = (SalGen‘𝐽) |
Ref | Expression |
---|---|
cnfsmf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | cnfsmf.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | cnfsmf.s | . . 3 ⊢ 𝑆 = (SalGen‘𝐽) | |
4 | 2, 3 | salgencld 45550 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) |
5 | cnfsmf.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) | |
6 | eqid 2724 | . . . . . 6 ⊢ ∪ (𝐽 ↾t dom 𝐹) = ∪ (𝐽 ↾t dom 𝐹) | |
7 | eqid 2724 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
8 | 6, 7 | cnf 23072 | . . . . 5 ⊢ (𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾) → 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶∪ 𝐾) |
9 | 5, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶∪ 𝐾) |
10 | 9 | fdmd 6718 | . . 3 ⊢ (𝜑 → dom 𝐹 = ∪ (𝐽 ↾t dom 𝐹)) |
11 | ovex 7434 | . . . . . . . 8 ⊢ (𝐽 ↾t dom 𝐹) ∈ V | |
12 | 11 | uniex 7724 | . . . . . . 7 ⊢ ∪ (𝐽 ↾t dom 𝐹) ∈ V |
13 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ∈ V) |
14 | 10, 13 | eqeltrd 2825 | . . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V) |
15 | 2, 14 | unirestss 44301 | . . . 4 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝐽) |
16 | 3 | sssalgen 45536 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ 𝑆) |
17 | 2, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ 𝑆) |
18 | 17 | unissd 4909 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 ⊆ ∪ 𝑆) |
19 | 15, 18 | sstrd 3984 | . . 3 ⊢ (𝜑 → ∪ (𝐽 ↾t dom 𝐹) ⊆ ∪ 𝑆) |
20 | 10, 19 | eqsstrd 4012 | . 2 ⊢ (𝜑 → dom 𝐹 ⊆ ∪ 𝑆) |
21 | uniretop 24601 | . . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
22 | cnfsmf.k | . . . . . . . 8 ⊢ 𝐾 = (topGen‘ran (,)) | |
23 | 22 | unieqi 4911 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ (topGen‘ran (,)) |
24 | 21, 23 | eqtr4i 2755 | . . . . . 6 ⊢ ℝ = ∪ 𝐾 |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → ℝ = ∪ 𝐾) |
26 | 25 | feq3d 6694 | . . . 4 ⊢ (𝜑 → (𝐹:∪ (𝐽 ↾t dom 𝐹)⟶ℝ ↔ 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶∪ 𝐾)) |
27 | 9, 26 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝐹:∪ (𝐽 ↾t dom 𝐹)⟶ℝ) |
28 | 27 | ffdmd 6738 | . 2 ⊢ (𝜑 → 𝐹:dom 𝐹⟶ℝ) |
29 | ssrest 23002 | . . . . 5 ⊢ ((𝑆 ∈ SAlg ∧ 𝐽 ⊆ 𝑆) → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) | |
30 | 4, 17, 29 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
31 | 30 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝐽 ↾t dom 𝐹) ⊆ (𝑆 ↾t dom 𝐹)) |
32 | 10 | rabeqdv 3439 | . . . . 5 ⊢ (𝜑 → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}) |
33 | 32 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎}) |
34 | nfcv 2895 | . . . . 5 ⊢ Ⅎ𝑥𝑎 | |
35 | nfcv 2895 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
36 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) | |
37 | eqid 2724 | . . . . 5 ⊢ {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} | |
38 | rexr 11257 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
39 | 38 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
40 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹 ∈ ((𝐽 ↾t dom 𝐹) Cn 𝐾)) |
41 | 34, 35, 36, 22, 6, 37, 39, 40 | rfcnpre2 44204 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝐽 ↾t dom 𝐹) ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
42 | 33, 41 | eqeltrd 2825 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐽 ↾t dom 𝐹)) |
43 | 31, 42 | sseldd 3975 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ dom 𝐹 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t dom 𝐹)) |
44 | 1, 4, 20, 28, 43 | issmfd 45936 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ⊆ wss 3940 ∪ cuni 4899 class class class wbr 5138 dom cdm 5666 ran crn 5667 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℝcr 11105 ℝ*cxr 11244 < clt 11245 (,)cioo 13321 ↾t crest 17365 topGenctg 17382 Topctop 22717 Cn ccn 23050 SAlgcsalg 45509 SalGencsalgen 45513 SMblFncsmblfn 45896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-ioo 13325 df-ico 13327 df-rest 17367 df-topgen 17388 df-top 22718 df-topon 22735 df-bases 22771 df-cn 23053 df-salg 45510 df-salgen 45514 df-smblfn 45897 |
This theorem is referenced by: cnfrrnsmf 45952 |
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