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 3784 | . . . . 5 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | 1 | eqcomi 2619 | . . . 4 ⊢ 𝐴 = (𝐴 ∩ 𝐴) |
3 | 2 | mpteq1i 4667 | . . 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 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
10 | smfmulc1.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
11 | eqid 2610 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
12 | 5, 11, 10 | dmmptdf 38412 | . . . . . 6 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
13 | 12 | eqcomd 2616 | . . . . 5 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
14 | smfmulc1.m | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
15 | eqid 2610 | . . . . . 6 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
16 | 6, 14, 15 | smfdmss 39619 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ ∪ 𝑆) |
17 | 13, 16 | eqsstrd 3602 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
18 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
19 | 5, 6, 17, 8, 18 | smfconst 39636 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (SMblFn‘𝑆)) |
20 | 5, 6, 7, 9, 10, 19, 14 | smfmul 39680 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ 𝐴) ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) |
21 | 4, 20 | eqeltrd 2688 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∩ cin 3539 ∪ cuni 4372 ↦ cmpt 4643 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 · cmul 9820 SAlgcsalg 39204 SMblFncsmblfn 39586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-ac2 9168 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-s4 13446 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-rest 15906 df-salg 39205 df-smblfn 39587 |
This theorem is referenced by: smf2id 39686 |
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