Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meadif | Structured version Visualization version GIF version |
Description: The measure of the difference of two sets. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
meadif.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meadif.a | ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) |
meadif.r | ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) |
meadif.b | ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) |
meadif.s | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
Ref | Expression |
---|---|
meadif | ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meadif.s | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
2 | undif 4049 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) | |
3 | 1, 2 | sylib 208 | . . . . 5 ⊢ (𝜑 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) |
4 | 3 | eqcomd 2628 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) |
5 | 4 | fveq2d 6195 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘(𝐵 ∪ (𝐴 ∖ 𝐵)))) |
6 | meadif.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
7 | eqid 2622 | . . . 4 ⊢ dom 𝑀 = dom 𝑀 | |
8 | meadif.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) | |
9 | 6, 7 | dmmeasal 40669 | . . . . 5 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
10 | meadif.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) | |
11 | saldifcl2 40546 | . . . . 5 ⊢ ((dom 𝑀 ∈ SAlg ∧ 𝐴 ∈ dom 𝑀 ∧ 𝐵 ∈ dom 𝑀) → (𝐴 ∖ 𝐵) ∈ dom 𝑀) | |
12 | 9, 10, 8, 11 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ dom 𝑀) |
13 | disjdif 4040 | . . . . 5 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅ | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅) |
15 | meadif.r | . . . . 5 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℝ) | |
16 | 6, 10, 15, 1, 8 | meassre 40694 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) |
17 | difssd 3738 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | |
18 | 6, 10, 15, 17, 12 | meassre 40694 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
19 | 6, 7, 8, 12, 14, 16, 18 | meadjunre 40693 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵)))) |
20 | 5, 19 | eqtr2d 2657 | . 2 ⊢ (𝜑 → ((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵))) = (𝑀‘𝐴)) |
21 | 16 | recnd 10068 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℂ) |
22 | 18 | recnd 10068 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℂ) |
23 | 15 | recnd 10068 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℂ) |
24 | 21, 22, 23 | addrsub 10448 | . 2 ⊢ (𝜑 → (((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵))) = (𝑀‘𝐴) ↔ (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵)))) |
25 | 20, 24 | mpbid 222 | 1 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 + caddc 9939 − cmin 10266 SAlgcsalg 40528 Meascmea 40666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-salg 40529 df-sumge0 40580 df-mea 40667 |
This theorem is referenced by: meaiininclem 40700 |
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