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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 1910 | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | smfmul.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | elinel1 4196 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐴) | |
5 | 4 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐴) |
6 | 1, 5 | ssdf 44676 | . . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴) |
7 | eqid 2726 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
8 | smfmul.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
9 | 1, 7, 8 | dmmptdf 44831 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
10 | 9 | eqcomd 2732 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
11 | smfmul.m | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
12 | eqid 2726 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
13 | 3, 11, 12 | smfdmss 46354 | . . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
14 | 10, 13 | eqsstrd 4018 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
15 | 6, 14 | sstrd 3990 | . 2 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ ∪ 𝑆) |
16 | 5, 8 | syldan 589 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐵 ∈ ℝ) |
17 | elinel2 4197 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐶) → 𝑥 ∈ 𝐶) | |
18 | 17 | adantl 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝑥 ∈ 𝐶) |
19 | smfmul.d | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) | |
20 | 18, 19 | syldan 589 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → 𝐷 ∈ ℝ) |
21 | 16, 20 | remulcld 11294 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ 𝐶)) → (𝐵 · 𝐷) ∈ ℝ) |
22 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
23 | 1, 22 | nfan 1895 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
24 | 3 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
25 | smfmul.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
26 | 25 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ∈ 𝑉) |
27 | 8 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
28 | 19 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐶) → 𝐷 ∈ ℝ) |
29 | 11 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
30 | smfmul.n | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) | |
31 | 30 | adantr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑥 ∈ 𝐶 ↦ 𝐷) ∈ (SMblFn‘𝑆)) |
32 | simpr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
33 | fveq1 6900 | . . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝‘2) = (𝑞‘2)) | |
34 | fveq1 6900 | . . . . . . . 8 ⊢ (𝑝 = 𝑞 → (𝑝‘3) = (𝑞‘3)) | |
35 | 33, 34 | oveq12d 7442 | . . . . . . 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 3168 | . . . . 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 6900 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝‘0) = (𝑞‘0)) | |
39 | fveq1 6900 | . . . . . . 7 ⊢ (𝑝 = 𝑞 → (𝑝‘1) = (𝑞‘1)) | |
40 | 38, 39 | oveq12d 7442 | . . . . . 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 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 3430 | . . 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 2726 | . . 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 46415 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ (𝐴 ∩ 𝐶) ∣ (𝐵 · 𝐷) < 𝑎} ∈ (𝑆 ↾t (𝐴 ∩ 𝐶))) |
46 | 1, 2, 3, 15, 21, 45 | issmfdmpt 46369 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐶) ↦ (𝐵 · 𝐷)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∀wral 3051 {crab 3419 ∩ cin 3946 ∪ cuni 4913 class class class wbr 5153 ↦ cmpt 5236 dom cdm 5682 ‘cfv 6554 (class class class)co 7424 ↑m cmap 8855 ℝcr 11157 0cc0 11158 1c1 11159 · cmul 11163 < clt 11298 2c2 12319 3c3 12320 ℚcq 12984 (,)cioo 13378 ...cfz 13538 SAlgcsalg 45929 SMblFncsmblfn 46316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cc 10478 ax-ac2 10506 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-omul 8501 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-acn 9985 df-ac 10159 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-rp 13029 df-ioo 13382 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-seq 14022 df-exp 14082 df-hash 14348 df-word 14523 df-concat 14579 df-s1 14604 df-s2 14857 df-s3 14858 df-s4 14859 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-rest 17437 df-salg 45930 df-smblfn 46317 |
This theorem is referenced by: smfmulc1 46417 smfdiv 46418 |
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