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 4430 | . . . . . 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 6674 | . . 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 42754 | . . . . 5 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
10 | meadif.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) | |
11 | saldifcl2 42631 | . . . . 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 4421 | . . . . 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 42779 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) |
17 | difssd 4109 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | |
18 | 6, 10, 15, 17, 12 | meassre 42779 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
19 | 6, 7, 8, 12, 14, 16, 18 | meadjunre 42778 | . . 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 10669 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℂ) |
22 | 18 | recnd 10669 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℂ) |
23 | 15 | recnd 10669 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℂ) |
24 | 21, 22, 23 | addrsub 11057 | . 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 1537 ∈ wcel 2114 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 + caddc 10540 − cmin 10870 SAlgcsalg 42613 Meascmea 42751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-xadd 12509 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-salg 42614 df-sumge0 42665 df-mea 42752 |
This theorem is referenced by: meaiininclem 42788 |
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