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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 4019 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | 1 | eqcomi 2809 | . . . 4 ⊢ 𝐴 = (𝐴 ∩ 𝐴) |
3 | 2 | mpteq1i 4933 | . . 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 473 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
10 | smfmulc1.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
11 | eqid 2800 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 5, 11, 10 | dmmptdf 40164 | . . . . . 6 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
13 | 12 | eqcomd 2806 | . . . . 5 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
14 | smfmulc1.m | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
15 | eqid 2800 | . . . . . 6 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
16 | 6, 14, 15 | smfdmss 41683 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
17 | 13, 16 | eqsstrd 3836 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
18 | eqid 2800 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
19 | 5, 6, 17, 8, 18 | smfconst 41699 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (SMblFn‘𝑆)) |
20 | 5, 6, 7, 9, 10, 19, 14 | smfmul 41743 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐴) ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) |
21 | 4, 20 | eqeltrd 2879 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 Ⅎwnf 1879 ∈ wcel 2157 ∩ cin 3769 ∪ cuni 4629 ↦ cmpt 4923 dom cdm 5313 ‘cfv 6102 (class class class)co 6879 ℝcr 10224 · cmul 10230 SAlgcsalg 41266 SMblFncsmblfn 41650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cc 9546 ax-ac2 9574 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-oadd 7804 df-omul 7805 df-er 7983 df-map 8098 df-pm 8099 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-sup 8591 df-inf 8592 df-oi 8658 df-card 9052 df-acn 9055 df-ac 9226 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-n0 11580 df-z 11666 df-uz 11930 df-q 12033 df-rp 12074 df-ioo 12427 df-ico 12429 df-icc 12430 df-fz 12580 df-fzo 12720 df-fl 12847 df-seq 13055 df-exp 13114 df-hash 13370 df-word 13534 df-concat 13590 df-s1 13615 df-s2 13932 df-s3 13933 df-s4 13934 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-rest 16397 df-salg 41267 df-smblfn 41651 |
This theorem is referenced by: smf2id 41749 smfneg 41751 |
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