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 1914 | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | smfmul.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | elinel1 4134 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐴) | |
5 | 4 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐴) |
6 | 1, 5 | ssdf 42851 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
7 | eqid 2735 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
8 | smfmul.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
9 | 1, 7, 8 | dmmptdf 42997 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
10 | 9 | eqcomd 2741 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | smfmul.m | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
12 | eqid 2735 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
13 | 3, 11, 12 | smfdmss 44514 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
14 | 10, 13 | eqsstrd 3963 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
15 | 6, 14 | sstrd 3935 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ ∪ 𝑆) |
16 | 5, 8 | syldan 591 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
17 | elinel2 4135 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐶) | |
18 | 17 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐶) |
19 | smfmul.d | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) | |
20 | 18, 19 | syldan 591 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
21 | 16, 20 | remulcld 11065 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 · 𝐷) ∈ ℝ) |
22 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
23 | 1, 22 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
24 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
25 | smfmul.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
26 | 25 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
27 | 8 | adantlr 712 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
28 | 19 | adantlr 712 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
29 | 11 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
30 | smfmul.n | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) | |
31 | 30 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
32 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
33 | fveq1 6803 | . . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝‘2) = (𝑞‘2)) | |
34 | fveq1 6803 | . . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝‘3) = (𝑞‘3)) | |
35 | 33, 34 | oveq12d 7326 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → ((𝑝‘2)(,)(𝑝‘3)) = ((𝑞‘2)(,)(𝑞‘3))) |
36 | 35 | raleqdv 3308 | . . . . . 6 ⊢ (𝑝 = 𝑞 → (∀𝑣 ∈ ((𝑝‘2)(,)(𝑝‘3))(𝑢 · 𝑣) < 𝑎 ↔ ∀𝑣 ∈ ((𝑞‘2)(,)(𝑞‘3))(𝑢 · 𝑣) < 𝑎)) |
37 | 36 | ralbidv 3169 | . . . . 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 6803 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝‘0) = (𝑞‘0)) | |
39 | fveq1 6803 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝‘1) = (𝑞‘1)) | |
40 | 38, 39 | oveq12d 7326 | . . . . . 6 ⊢ (𝑝 = 𝑞 → ((𝑝‘0)(,)(𝑝‘1)) = ((𝑞‘0)(,)(𝑞‘1))) |
41 | 40 | raleqdv 3308 | . . . . 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 278 | . . . 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 3432 | . . 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 2735 | . . 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 44575 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 · 𝐷) < 𝑎} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
46 | 1, 2, 3, 15, 21, 45 | issmfdmpt 44529 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1538 Ⅎwnf 1782 ∈ wcel 2103 ∀wral 3060 {crab 3356 ∩ cin 3890 ∪ cuni 4843 class class class wbr 5080 ↦ cmpt 5163 dom cdm 5600 ‘cfv 6458 (class class class)co 7308 ↑m cmap 8651 ℝcr 10930 0cc0 10931 1c1 10932 · cmul 10936 < clt 11069 2c2 12088 3c3 12089 ℚcq 12748 (,)cioo 13139 ...cfz 13299 SAlgcsalg 44091 SMblFncsmblfn 44476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1968 ax-7 2008 ax-8 2105 ax-9 2113 ax-10 2134 ax-11 2151 ax-12 2168 ax-ext 2706 ax-rep 5217 ax-sep 5231 ax-nul 5238 ax-pow 5296 ax-pr 5360 ax-un 7621 ax-inf2 9457 ax-cc 10251 ax-ac2 10279 ax-cnex 10987 ax-resscn 10988 ax-1cn 10989 ax-icn 10990 ax-addcl 10991 ax-addrcl 10992 ax-mulcl 10993 ax-mulrcl 10994 ax-mulcom 10995 ax-addass 10996 ax-mulass 10997 ax-distr 10998 ax-i2m1 10999 ax-1ne0 11000 ax-1rid 11001 ax-rnegex 11002 ax-rrecex 11003 ax-cnre 11004 ax-pre-lttri 11005 ax-pre-lttrn 11006 ax-pre-ltadd 11007 ax-pre-mulgt0 11008 ax-pre-sup 11009 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1541 df-fal 1551 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2727 df-clel 2813 df-nfc 2885 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3339 df-reu 3340 df-rab 3357 df-v 3438 df-sbc 3721 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4844 df-int 4886 df-iun 4932 df-iin 4933 df-br 5081 df-opab 5143 df-mpt 5164 df-tr 5198 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-se 5556 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-isom 6467 df-riota 7265 df-ov 7311 df-oprab 7312 df-mpo 7313 df-om 7749 df-1st 7867 df-2nd 7868 df-frecs 8132 df-wrecs 8163 df-recs 8237 df-rdg 8276 df-1o 8332 df-oadd 8336 df-omul 8337 df-er 8534 df-map 8653 df-pm 8654 df-en 8770 df-dom 8771 df-sdom 8772 df-fin 8773 df-sup 9259 df-inf 9260 df-oi 9327 df-card 9755 df-acn 9758 df-ac 9932 df-pnf 11071 df-mnf 11072 df-xr 11073 df-ltxr 11074 df-le 11075 df-sub 11267 df-neg 11268 df-div 11693 df-nn 12034 df-2 12096 df-3 12097 df-4 12098 df-n0 12294 df-z 12380 df-uz 12643 df-q 12749 df-rp 12791 df-ioo 13143 df-ico 13145 df-icc 13146 df-fz 13300 df-fzo 13443 df-fl 13572 df-seq 13782 df-exp 13843 df-hash 14105 df-word 14277 df-concat 14333 df-s1 14360 df-s2 14620 df-s3 14621 df-s4 14622 df-cj 14869 df-re 14870 df-im 14871 df-sqrt 15005 df-abs 15006 df-rest 17192 df-salg 44092 df-smblfn 44477 |
This theorem is referenced by: smfmulc1 44577 smfdiv 44578 |
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