Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfmul | Structured version Visualization version GIF version |
Description: The multiplication of two sigma-measurable functions is measurable. Proposition 121E (d) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfmul.x | ⊢ Ⅎ𝑥𝜑 |
smfmul.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfmul.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
smfmul.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
smfmul.d | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
smfmul.m | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
smfmul.n | ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
Ref | Expression |
---|---|
smfmul | ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfmul.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1922 | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | smfmul.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | elinel1 4095 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐴) | |
5 | 4 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐴) |
6 | 1, 5 | ssdf 42239 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
7 | eqid 2736 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
8 | smfmul.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
9 | 1, 7, 8 | dmmptdf 42377 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
10 | 9 | eqcomd 2742 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | smfmul.m | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
12 | eqid 2736 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
13 | 3, 11, 12 | smfdmss 43884 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
14 | 10, 13 | eqsstrd 3925 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
15 | 6, 14 | sstrd 3897 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ ∪ 𝑆) |
16 | 5, 8 | syldan 594 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
17 | elinel2 4096 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐶) | |
18 | 17 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐶) |
19 | smfmul.d | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) | |
20 | 18, 19 | syldan 594 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
21 | 16, 20 | remulcld 10828 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 · 𝐷) ∈ ℝ) |
22 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
23 | 1, 22 | nfan 1907 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
24 | 3 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
25 | smfmul.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
26 | 25 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
27 | 8 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
28 | 19 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
29 | 11 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
30 | smfmul.n | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) | |
31 | 30 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
32 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
33 | fveq1 6694 | . . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝‘2) = (𝑞‘2)) | |
34 | fveq1 6694 | . . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝‘3) = (𝑞‘3)) | |
35 | 33, 34 | oveq12d 7209 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → ((𝑝‘2)(,)(𝑝‘3)) = ((𝑞‘2)(,)(𝑞‘3))) |
36 | 35 | raleqdv 3315 | . . . . . 6 ⊢ (𝑝 = 𝑞 → (∀𝑣 ∈ ((𝑝‘2)(,)(𝑝‘3))(𝑢 · 𝑣) < 𝑎 ↔ ∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑎)) |
37 | 36 | ralbidv 3108 | . . . . 5 ⊢ (𝑝 = 𝑞 → (∀𝑢 ∈ ((𝑝‘0)(,)(𝑝‘1))∀𝑣 ∈ ((𝑝‘2)(,)(𝑝‘3))(𝑢 · 𝑣) < 𝑎 ↔ ∀𝑢 ∈ ((𝑝‘0)(,)(𝑝‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑎)) |
38 | fveq1 6694 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝‘0) = (𝑞‘0)) | |
39 | fveq1 6694 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝‘1) = (𝑞‘1)) | |
40 | 38, 39 | oveq12d 7209 | . . . . . 6 ⊢ (𝑝 = 𝑞 → ((𝑝‘0)(,)(𝑝‘1)) = ((𝑞‘0)(,)(𝑞‘1))) |
41 | 40 | raleqdv 3315 | . . . . 5 ⊢ (𝑝 = 𝑞 → (∀𝑢 ∈ ((𝑝‘0)(,)(𝑝‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑎 ↔ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑎)) |
42 | 37, 41 | bitrd 282 | . . . 4 ⊢ (𝑝 = 𝑞 → (∀𝑢 ∈ ((𝑝‘0)(,)(𝑝‘1))∀𝑣 ∈ ((𝑝‘2)(,)(𝑝‘3))(𝑢 · 𝑣) < 𝑎 ↔ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑎)) |
43 | 42 | cbvrabv 3392 | . . 3 ⊢ {𝑝 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑝‘0)(,)(𝑝‘1))∀𝑣 ∈ ((𝑝‘2)(,)(𝑝‘3))(𝑢 · 𝑣) < 𝑎} = {𝑞 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑞‘0)(,)(𝑞‘1))∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑎} |
44 | eqid 2736 | . . 3 ⊢ (𝑞 ∈ {𝑝 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑝‘0)(,)(𝑝‘1))∀𝑣 ∈ ((𝑝‘2)(,)(𝑝‘3))(𝑢 · 𝑣) < 𝑎} ↦ {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝐷 ∈ ((𝑞‘2)(,)(𝑞‘3)))}) = (𝑞 ∈ {𝑝 ∈ (ℚ ↑m (0...3)) ∣ ∀𝑢 ∈ ((𝑝‘0)(,)(𝑝‘1))∀𝑣 ∈ ((𝑝‘2)(,)(𝑝‘3))(𝑢 · 𝑣) < 𝑎} ↦ {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 ∈ ((𝑞‘0)(,)(𝑞‘1)) ∧ 𝐷 ∈ ((𝑞‘2)(,)(𝑞‘3)))}) | |
45 | 23, 24, 26, 27, 28, 29, 31, 32, 43, 44 | smfmullem4 43943 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 · 𝐷) < 𝑎} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
46 | 1, 2, 3, 15, 21, 45 | issmfdmpt 43899 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2112 ∀wral 3051 {crab 3055 ∩ cin 3852 ∪ cuni 4805 class class class wbr 5039 ↦ cmpt 5120 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 ℝcr 10693 0cc0 10694 1c1 10695 · cmul 10699 < clt 10832 2c2 11850 3c3 11851 ℚcq 12509 (,)cioo 12900 ...cfz 13060 SAlgcsalg 43467 SMblFncsmblfn 43851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 ax-ac2 10042 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-omul 8185 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-acn 9523 df-ac 9695 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-rp 12552 df-ioo 12904 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-seq 13540 df-exp 13601 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-s2 14378 df-s3 14379 df-s4 14380 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-rest 16881 df-salg 43468 df-smblfn 43852 |
This theorem is referenced by: smfmulc1 43945 smfdiv 43946 |
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