Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfconst | Structured version Visualization version GIF version |
Description: A constant function is measurable. Proposition 121E (a) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfconst.x | ⊢ Ⅎ𝑥𝜑 |
smfconst.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfconst.a | ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
smfconst.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
smfconst.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
smfconst | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfconst.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | nfmpt1 4675 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | nfcxfr 2749 | . 2 ⊢ Ⅎ𝑥𝐹 |
4 | nfv 1830 | . 2 ⊢ Ⅎ𝑎𝜑 | |
5 | smfconst.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
6 | smfconst.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) | |
7 | smfconst.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
8 | smfconst.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
10 | 7, 9, 1 | fmptdf 6294 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
11 | nfv 1830 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
12 | 7, 11 | nfan 1816 | . . . . . . 7 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
13 | nfv 1830 | . . . . . . 7 ⊢ Ⅎ𝑥 𝐵 < 𝑎 | |
14 | 12, 13 | nfan 1816 | . . . . . 6 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) |
15 | 8 | ad2antrr 758 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
16 | simpr 476 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐵 < 𝑎) | |
17 | 14, 15, 1, 16 | pimconstlt1 39592 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = 𝐴) |
18 | eqidd 2611 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = 𝐴) | |
19 | sseqin2 3779 | . . . . . . . 8 ⊢ (𝐴 ⊆ ∪ 𝑆 ↔ (∪ 𝑆 ∩ 𝐴) = 𝐴) | |
20 | 6, 19 | sylib 207 | . . . . . . 7 ⊢ (𝜑 → (∪ 𝑆 ∩ 𝐴) = 𝐴) |
21 | 20 | eqcomd 2616 | . . . . . 6 ⊢ (𝜑 → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
22 | 21 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 = (∪ 𝑆 ∩ 𝐴)) |
23 | 17, 18, 22 | 3eqtrd 2648 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (∪ 𝑆 ∩ 𝐴)) |
24 | 5 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝑆 ∈ SAlg) |
25 | 5 | uniexd 38310 | . . . . . . 7 ⊢ (𝜑 → ∪ 𝑆 ∈ V) |
26 | 25, 6 | ssexd 4733 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V) |
27 | 26 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → 𝐴 ∈ V) |
28 | 24 | salunid 39247 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → ∪ 𝑆 ∈ 𝑆) |
29 | eqid 2610 | . . . . 5 ⊢ (∪ 𝑆 ∩ 𝐴) = (∪ 𝑆 ∩ 𝐴) | |
30 | 24, 27, 28, 29 | elrestd 38322 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → (∪ 𝑆 ∩ 𝐴) ∈ (𝑆 ↾t 𝐴)) |
31 | 23, 30 | eqeltrd 2688 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
32 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑥 ¬ 𝐵 < 𝑎 | |
33 | 12, 32 | nfan 1816 | . . . . 5 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) |
34 | 8 | ad2antrr 758 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝐵 ∈ ℝ) |
35 | rexr 9964 | . . . . . 6 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | ad2antlr 759 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ*) |
37 | simpr 476 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ¬ 𝐵 < 𝑎) | |
38 | simplr 788 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ∈ ℝ) | |
39 | 38, 34 | lenltd 10062 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → (𝑎 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑎)) |
40 | 37, 39 | mpbird 246 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → 𝑎 ≤ 𝐵) |
41 | 33, 34, 1, 36, 40 | pimconstlt0 39591 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = ∅) |
42 | eqid 2610 | . . . . . . 7 ⊢ (𝑆 ↾t 𝐴) = (𝑆 ↾t 𝐴) | |
43 | 5, 26, 42 | subsalsal 39253 | . . . . . 6 ⊢ (𝜑 → (𝑆 ↾t 𝐴) ∈ SAlg) |
44 | 43 | 0sald 39244 | . . . . 5 ⊢ (𝜑 → ∅ ∈ (𝑆 ↾t 𝐴)) |
45 | 44 | ad2antrr 758 | . . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → ∅ ∈ (𝑆 ↾t 𝐴)) |
46 | 41, 45 | eqeltrd 2688 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ ¬ 𝐵 < 𝑎) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
47 | 31, 46 | pm2.61dan 828 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
48 | 3, 4, 5, 6, 10, 47 | issmfdf 39624 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 ↾t crest 15904 SAlgcsalg 39204 SMblFncsmblfn 39586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-acn 8651 df-ac 8822 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-ioo 12050 df-ico 12052 df-rest 15906 df-salg 39205 df-smblfn 39587 |
This theorem is referenced by: smfmbfcex 39646 smfmulc1 39681 |
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