Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfmulc1 | Structured version Visualization version GIF version |
Description: A sigma-measurable function multiplied by a constant is sigma-measurable. Proposition 121E (c) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfmulc1.x | ⊢ Ⅎ𝑥𝜑 |
smfmulc1.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfmulc1.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
smfmulc1.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
smfmulc1.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
smfmulc1.m | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
Ref | Expression |
---|---|
smfmulc1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm 4158 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | 1 | eqcomi 2749 | . . . 4 ⊢ 𝐴 = (𝐴 ∩ 𝐴) |
3 | 2 | mpteq1i 5175 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑥 ∈ (𝐴 ∩ 𝐴) ↦ (𝐶 · 𝐵)) |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) = (𝑥 ∈ (𝐴 ∩ 𝐴) ↦ (𝐶 · 𝐵))) |
5 | smfmulc1.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
6 | smfmulc1.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
7 | smfmulc1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | smfmulc1.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
10 | smfmulc1.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
11 | eqid 2740 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 5, 11, 10 | dmmptdf 42731 | . . . . . 6 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
13 | 12 | eqcomd 2746 | . . . . 5 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
14 | smfmulc1.m | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
15 | eqid 2740 | . . . . . 6 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
16 | 6, 14, 15 | smfdmss 44235 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
17 | 13, 16 | eqsstrd 3964 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
18 | eqid 2740 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
19 | 5, 6, 17, 8, 18 | smfconst 44251 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (SMblFn‘𝑆)) |
20 | 5, 6, 7, 9, 10, 19, 14 | smfmul 44295 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐴) ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) |
21 | 4, 20 | eqeltrd 2841 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2110 ∩ cin 3891 ∪ cuni 4845 ↦ cmpt 5162 dom cdm 5589 ‘cfv 6431 (class class class)co 7269 ℝcr 10869 · cmul 10875 SAlgcsalg 43818 SMblFncsmblfn 44202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cc 10190 ax-ac2 10218 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-oadd 8290 df-omul 8291 df-er 8479 df-map 8598 df-pm 8599 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-sup 9177 df-inf 9178 df-oi 9245 df-card 9696 df-acn 9699 df-ac 9871 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-n0 12232 df-z 12318 df-uz 12580 df-q 12686 df-rp 12728 df-ioo 13080 df-ico 13082 df-icc 13083 df-fz 13237 df-fzo 13380 df-fl 13508 df-seq 13718 df-exp 13779 df-hash 14041 df-word 14214 df-concat 14270 df-s1 14297 df-s2 14557 df-s3 14558 df-s4 14559 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-rest 17129 df-salg 43819 df-smblfn 44203 |
This theorem is referenced by: smf2id 44301 smfneg 44303 |
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