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 4429 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) | |
3 | 1, 2 | sylib 220 | . . . . 5 ⊢ (𝜑 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) |
4 | 3 | eqcomd 2827 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) |
5 | 4 | fveq2d 6668 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘(𝐵 ∪ (𝐴 ∖ 𝐵)))) |
6 | meadif.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
7 | eqid 2821 | . . . 4 ⊢ dom 𝑀 = dom 𝑀 | |
8 | meadif.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) | |
9 | 6, 7 | dmmeasal 42728 | . . . . 5 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
10 | meadif.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) | |
11 | saldifcl2 42605 | . . . . 5 ⊢ ((dom 𝑀 ∈ SAlg ∧ 𝐴 ∈ dom 𝑀 ∧ 𝐵 ∈ dom 𝑀) → (𝐴 ∖ 𝐵) ∈ dom 𝑀) | |
12 | 9, 10, 8, 11 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ dom 𝑀) |
13 | disjdif 4420 | . . . . 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 42753 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) |
17 | difssd 4108 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | |
18 | 6, 10, 15, 17, 12 | meassre 42753 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
19 | 6, 7, 8, 12, 14, 16, 18 | meadjunre 42752 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵)))) |
20 | 5, 19 | eqtr2d 2857 | . 2 ⊢ (𝜑 → ((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵))) = (𝑀‘𝐴)) |
21 | 16 | recnd 10663 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℂ) |
22 | 18 | recnd 10663 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℂ) |
23 | 15 | recnd 10663 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℂ) |
24 | 21, 22, 23 | addrsub 11051 | . 2 ⊢ (𝜑 → (((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵))) = (𝑀‘𝐴) ↔ (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵)))) |
25 | 20, 24 | mpbid 234 | 1 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ∪ cun 3933 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 + caddc 10534 − cmin 10864 SAlgcsalg 42587 Meascmea 42725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-disj 5024 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-xadd 12502 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-salg 42588 df-sumge0 42639 df-mea 42726 |
This theorem is referenced by: meaiininclem 42762 |
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