Theorem mbfsub 23429

Description: The difference of two measurable functions is measurable. (Contributed by Mario Carneiro, 5-Sep-2014.)

Hypotheses

Ref	Expression
mbfadd.1	|- ( ph -> F e. MblFn )
mbfadd.2	|- ( ph -> G e. MblFn )

Assertion

Ref	Expression
mbfsub	|- ( ph -> ( F oF - G ) e. MblFn )

Proof of Theorem mbfsub

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfadd.1	. . . . . . . 8 |- ( ph -> F e. MblFn )
2		mbff 23394	. . . . . . . 8 |- ( F e. MblFn -> F : dom F --> CC )
3	1, 2	syl 17	. . . . . . 7 |- ( ph -> F : dom F --> CC )
4		elin 3796	. . . . . . . 8 |- ( x e. ( dom F i^i dom G ) <-> ( x e. dom F /\ x e. dom G ) )
5	4	simplbi 476	. . . . . . 7 |- ( x e. ( dom F i^i dom G ) -> x e. dom F )
6		ffvelrn 6357	. . . . . . 7 |- ( ( F : dom F --> CC /\ x e. dom F ) -> ( F ` x ) e. CC )
7	3, 5, 6	syl2an 494	. . . . . 6 |- ( ( ph /\ x e. ( dom F i^i dom G ) ) -> ( F ` x ) e. CC )
8		mbfadd.2	. . . . . . . 8 |- ( ph -> G e. MblFn )
9		mbff 23394	. . . . . . . 8 |- ( G e. MblFn -> G : dom G --> CC )
10	8, 9	syl 17	. . . . . . 7 |- ( ph -> G : dom G --> CC )
11	4	simprbi 480	. . . . . . 7 |- ( x e. ( dom F i^i dom G ) -> x e. dom G )
12		ffvelrn 6357	. . . . . . 7 |- ( ( G : dom G --> CC /\ x e. dom G ) -> ( G ` x ) e. CC )
13	10, 11, 12	syl2an 494	. . . . . 6 |- ( ( ph /\ x e. ( dom F i^i dom G ) ) -> ( G ` x ) e. CC )
14	7, 13	negsubd 10398	. . . . 5 |- ( ( ph /\ x e. ( dom F i^i dom G ) ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F ` x ) - ( G ` x ) ) )
15	14	eqcomd 2628	. . . 4 |- ( ( ph /\ x e. ( dom F i^i dom G ) ) -> ( ( F ` x ) - ( G ` x ) ) = ( ( F ` x ) + -u ( G ` x ) ) )
16	15	mpteq2dva 4744	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) - ( G ` x ) ) ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) + -u ( G ` x ) ) ) )
17		ffn 6045	. . . . 5 |- ( F : dom F --> CC -> F Fn dom F )
18	3, 17	syl 17	. . . 4 |- ( ph -> F Fn dom F )
19		ffn 6045	. . . . 5 |- ( G : dom G --> CC -> G Fn dom G )
20	10, 19	syl 17	. . . 4 |- ( ph -> G Fn dom G )
21		mbfdm 23395	. . . . 5 |- ( F e. MblFn -> dom F e. dom vol )
22	1, 21	syl 17	. . . 4 |- ( ph -> dom F e. dom vol )
23		mbfdm 23395	. . . . 5 |- ( G e. MblFn -> dom G e. dom vol )
24	8, 23	syl 17	. . . 4 |- ( ph -> dom G e. dom vol )
25		eqid 2622	. . . 4 |- ( dom F i^i dom G ) = ( dom F i^i dom G )
26		eqidd 2623	. . . 4 |- ( ( ph /\ x e. dom F ) -> ( F ` x ) = ( F ` x ) )
27		eqidd 2623	. . . 4 |- ( ( ph /\ x e. dom G ) -> ( G ` x ) = ( G ` x ) )
28	18, 20, 22, 24, 25, 26, 27	offval 6904	. . 3 |- ( ph -> ( F oF - G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) - ( G ` x ) ) ) )
29		inmbl 23310	. . . . 5 |- ( ( dom F e. dom vol /\ dom G e. dom vol ) -> ( dom F i^i dom G ) e. dom vol )
30	22, 24, 29	syl2anc 693	. . . 4 |- ( ph -> ( dom F i^i dom G ) e. dom vol )
31	13	negcld 10379	. . . 4 |- ( ( ph /\ x e. ( dom F i^i dom G ) ) -> -u ( G ` x ) e. CC )
32		eqidd 2623	. . . 4 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) = ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) )
33		eqidd 2623	. . . 4 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> -u ( G ` x ) ) = ( x e. ( dom F i^i dom G ) |-> -u ( G ` x ) ) )
34	30, 7, 31, 32, 33	offval2 6914	. . 3 |- ( ph -> ( ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) oF + ( x e. ( dom F i^i dom G ) |-> -u ( G ` x ) ) ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) + -u ( G ` x ) ) ) )
35	16, 28, 34	3eqtr4d 2666	. 2 |- ( ph -> ( F oF - G ) = ( ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) oF + ( x e. ( dom F i^i dom G ) |-> -u ( G ` x ) ) ) )
36		inss1 3833	. . . . 5 |- ( dom F i^i dom G ) C_ dom F
37		resmpt 5449	. . . . 5 |- ( ( dom F i^i dom G ) C_ dom F -> ( ( x e. dom F |-> ( F ` x ) ) |` ( dom F i^i dom G ) ) = ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) )
38	36, 37	mp1i 13	. . . 4 |- ( ph -> ( ( x e. dom F |-> ( F ` x ) ) |` ( dom F i^i dom G ) ) = ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) )
39	3	feqmptd 6249	. . . . . 6 |- ( ph -> F = ( x e. dom F |-> ( F ` x ) ) )
40	39, 1	eqeltrrd 2702	. . . . 5 |- ( ph -> ( x e. dom F |-> ( F ` x ) ) e. MblFn )
41		mbfres 23411	. . . . 5 |- ( ( ( x e. dom F |-> ( F ` x ) ) e. MblFn /\ ( dom F i^i dom G ) e. dom vol ) -> ( ( x e. dom F |-> ( F ` x ) ) |` ( dom F i^i dom G ) ) e. MblFn )
42	40, 30, 41	syl2anc 693	. . . 4 |- ( ph -> ( ( x e. dom F |-> ( F ` x ) ) |` ( dom F i^i dom G ) ) e. MblFn )
43	38, 42	eqeltrrd 2702	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) e. MblFn )
44		inss2 3834	. . . . . 6 |- ( dom F i^i dom G ) C_ dom G
45		resmpt 5449	. . . . . 6 |- ( ( dom F i^i dom G ) C_ dom G -> ( ( x e. dom G |-> ( G ` x ) ) |` ( dom F i^i dom G ) ) = ( x e. ( dom F i^i dom G ) |-> ( G ` x ) ) )
46	44, 45	mp1i 13	. . . . 5 |- ( ph -> ( ( x e. dom G |-> ( G ` x ) ) |` ( dom F i^i dom G ) ) = ( x e. ( dom F i^i dom G ) |-> ( G ` x ) ) )
47	10	feqmptd 6249	. . . . . . 7 |- ( ph -> G = ( x e. dom G |-> ( G ` x ) ) )
48	47, 8	eqeltrrd 2702	. . . . . 6 |- ( ph -> ( x e. dom G |-> ( G ` x ) ) e. MblFn )
49		mbfres 23411	. . . . . 6 |- ( ( ( x e. dom G |-> ( G ` x ) ) e. MblFn /\ ( dom F i^i dom G ) e. dom vol ) -> ( ( x e. dom G |-> ( G ` x ) ) |` ( dom F i^i dom G ) ) e. MblFn )
50	48, 30, 49	syl2anc 693	. . . . 5 |- ( ph -> ( ( x e. dom G |-> ( G ` x ) ) |` ( dom F i^i dom G ) ) e. MblFn )
51	46, 50	eqeltrrd 2702	. . . 4 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( G ` x ) ) e. MblFn )
52	13, 51	mbfneg 23417	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> -u ( G ` x ) ) e. MblFn )
53	43, 52	mbfadd 23428	. 2 |- ( ph -> ( ( x e. ( dom F i^i dom G ) |-> ( F ` x ) ) oF + ( x e. ( dom F i^i dom G ) |-> -u ( G ` x ) ) ) e. MblFn )
54	35, 53	eqeltrd 2701	1 |- ( ph -> ( F oF - G ) e. MblFn )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   + caddc 9939   - cmin 10266   -ucneg 10267   volcvol 23232  MblFncmbf 23383

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-cc 9257  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  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-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  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-rlim 14220  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233  df-vol 23234  df-mbf 23388

This theorem is referenced by:  mbfmul  23493  iblulm  24161