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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 4482 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) | |
3 | 1, 2 | sylib 217 | . . . . 5 ⊢ (𝜑 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) |
4 | 3 | eqcomd 2734 | . . . 4 ⊢ (𝜑 → 𝐴 = (𝐵 ∪ (𝐴 ∖ 𝐵))) |
5 | 4 | fveq2d 6901 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘(𝐵 ∪ (𝐴 ∖ 𝐵)))) |
6 | meadif.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
7 | eqid 2728 | . . . 4 ⊢ dom 𝑀 = dom 𝑀 | |
8 | meadif.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀) | |
9 | 6, 7 | dmmeasal 45840 | . . . . 5 ⊢ (𝜑 → dom 𝑀 ∈ SAlg) |
10 | meadif.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ dom 𝑀) | |
11 | saldifcl2 45716 | . . . . 5 ⊢ ((dom 𝑀 ∈ SAlg ∧ 𝐴 ∈ dom 𝑀 ∧ 𝐵 ∈ dom 𝑀) → (𝐴 ∖ 𝐵) ∈ dom 𝑀) | |
12 | 9, 10, 8, 11 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ dom 𝑀) |
13 | disjdif 4472 | . . . . 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 45865 | . . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℝ) |
17 | difssd 4131 | . . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴) | |
18 | 6, 10, 15, 17, 12 | meassre 45865 | . . . 4 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
19 | 6, 7, 8, 12, 14, 16, 18 | meadjunre 45864 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵)))) |
20 | 5, 19 | eqtr2d 2769 | . 2 ⊢ (𝜑 → ((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵))) = (𝑀‘𝐴)) |
21 | 16 | recnd 11273 | . . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ ℂ) |
22 | 18 | recnd 11273 | . . 3 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) ∈ ℂ) |
23 | 15 | recnd 11273 | . . 3 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ℂ) |
24 | 21, 22, 23 | addrsub 11662 | . 2 ⊢ (𝜑 → (((𝑀‘𝐵) + (𝑀‘(𝐴 ∖ 𝐵))) = (𝑀‘𝐴) ↔ (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵)))) |
25 | 20, 24 | mpbid 231 | 1 ⊢ (𝜑 → (𝑀‘(𝐴 ∖ 𝐵)) = ((𝑀‘𝐴) − (𝑀‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 dom cdm 5678 ‘cfv 6548 (class class class)co 7420 ℝcr 11138 + caddc 11142 − cmin 11475 SAlgcsalg 45696 Meascmea 45837 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-xadd 13126 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 df-salg 45697 df-sumge0 45751 df-mea 45838 |
This theorem is referenced by: meaiininclem 45874 |
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