iblsplit - Mathbox for Glauco Siliprandi

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Theorem iblsplit 40182

Description: The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
iblsplit.1
iblsplit.2
iblsplit.3	$/\$
iblsplit.4
iblsplit.5

**Assertion**
Ref	Expression
iblsplit

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem iblsplit Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iblsplit.3	. . . 4 $/\$
2		eqid 2622	. . . 4
3	1, 2	fmptd 6385	. . 3
4		ssun1 3776	. . . . . 6
5		iblsplit.2	. . . . . 6
6	4, 5	syl5sseqr 3654	. . . . 5
7	6	resmptd 5452	. . . 4
8		iblsplit.4	. . . . . 6
9		eqidd 2623	. . . . . . 7 $/\$ $/\$
10		eqidd 2623	. . . . . . 7 $/\$
11	6	sseld 3602	. . . . . . . . 9
12	11	imdistani 726	. . . . . . . 8 $/\$ $/\$
13	12, 1	syl 17	. . . . . . 7 $/\$
14	9, 10, 13	isibl2 23533	. . . . . 6 MblFn $/\$ $/\$
15	8, 14	mpbid 222	. . . . 5 MblFn $/\$ $/\$
16	15	simpld 475	. . . 4 MblFn
17	7, 16	eqeltrd 2701	. . 3 MblFn
18		ssun2 3777	. . . . . 6
19	18, 5	syl5sseqr 3654	. . . . 5
20	19	resmptd 5452	. . . 4
21		iblsplit.5	. . . . . 6
22		eqidd 2623	. . . . . . 7 $/\$ $/\$
23		eqidd 2623	. . . . . . 7 $/\$
24	19	sseld 3602	. . . . . . . . 9
25	24	imdistani 726	. . . . . . . 8 $/\$ $/\$
26	25, 1	syl 17	. . . . . . 7 $/\$
27	22, 23, 26	isibl2 23533	. . . . . 6 MblFn $/\$ $/\$
28	21, 27	mpbid 222	. . . . 5 MblFn $/\$ $/\$
29	28	simpld 475	. . . 4 MblFn
30	20, 29	eqeltrd 2701	. . 3 MblFn
31	5	eqcomd 2628	. . 3
32	3, 17, 30, 31	mbfres2cn 40174	. 2 MblFn
33	16, 13	mbfdm2 23405	. . . . . 6
34	33	adantr 481	. . . . 5 $/\$
35	29, 26	mbfdm2 23405	. . . . . 6
36	35	adantr 481	. . . . 5 $/\$
37		iblsplit.1	. . . . . 6
38	37	adantr 481	. . . . 5 $/\$
39	5	adantr 481	. . . . 5 $/\$
40	1	adantlr 751	. . . . . . . . . . 11 $/\$ $/\$
41		ax-icn 9995	. . . . . . . . . . . . . 14
42	41	a1i 11	. . . . . . . . . . . . 13
43		elfznn0 12433	. . . . . . . . . . . . 13
44	42, 43	expcld 13008	. . . . . . . . . . . 12
45	44	ad2antlr 763	. . . . . . . . . . 11 $/\$ $/\$
46	41	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$
47		ine0 10465	. . . . . . . . . . . . 13
48	47	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$
49		elfzelz 12342	. . . . . . . . . . . . 13
50	49	ad2antlr 763	. . . . . . . . . . . 12 $/\$ $/\$
51	46, 48, 50	expne0d 13014	. . . . . . . . . . 11 $/\$ $/\$
52	40, 45, 51	divcld 10801	. . . . . . . . . 10 $/\$ $/\$
53	52	recld 13934	. . . . . . . . 9 $/\$ $/\$
54	53	rexrd 10089	. . . . . . . 8 $/\$ $/\$
55	54	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$
56		simpr 477	. . . . . . 7 $/\$ $/\$ $/\$
57		pnfge 11964	. . . . . . . 8
58	55, 57	syl 17	. . . . . . 7 $/\$ $/\$ $/\$
59		0xr 10086	. . . . . . . 8
60		pnfxr 10092	. . . . . . . 8
61		elicc1 12219	. . . . . . . 8 $/\$ $/\$ $/\$
62	59, 60, 61	mp2an 708	. . . . . . 7 $/\$ $/\$
63	55, 56, 58, 62	syl3anbrc 1246	. . . . . 6 $/\$ $/\$ $/\$
64		0e0iccpnf 12283	. . . . . . 7
65	64	a1i 11	. . . . . 6 $/\$ $/\$ $/\$
66	63, 65	ifclda 4120	. . . . 5 $/\$ $/\$
67		eqid 2622	. . . . 5
68		eqid 2622	. . . . 5
69		ifan 4134	. . . . . 6 $/\$
70	69	mpteq2i 4741	. . . . 5 $/\$
71		ifan 4134	. . . . . . . . . 10 $/\$
72	71	eqcomi 2631	. . . . . . . . 9 $/\$
73	72	mpteq2i 4741	. . . . . . . 8 $/\$
74	73	a1i 11	. . . . . . 7 $/\$ $/\$
75	74	fveq2d 6195	. . . . . 6 $/\$ $/\$
76		eqidd 2623	. . . . . . . . . 10 $/\$ $/\$
77		eqidd 2623	. . . . . . . . . 10 $/\$
78	76, 77, 13	isibl2 23533	. . . . . . . . 9 MblFn $/\$ $/\$
79	8, 78	mpbid 222	. . . . . . . 8 MblFn $/\$ $/\$
80	79	simprd 479	. . . . . . 7 $/\$
81	80	r19.21bi 2932	. . . . . 6 $/\$ $/\$
82	75, 81	eqeltrd 2701	. . . . 5 $/\$
83		ifan 4134	. . . . . . . . 9 $/\$
84	83	eqcomi 2631	. . . . . . . 8 $/\$
85	84	mpteq2i 4741	. . . . . . 7 $/\$
86	85	fveq2i 6194	. . . . . 6 $/\$
87		eqidd 2623	. . . . . . . . . 10 $/\$ $/\$
88		eqidd 2623	. . . . . . . . . 10 $/\$
89	87, 88, 26	isibl2 23533	. . . . . . . . 9 MblFn $/\$ $/\$
90	21, 89	mpbid 222	. . . . . . . 8 MblFn $/\$ $/\$
91	90	simprd 479	. . . . . . 7 $/\$
92	91	r19.21bi 2932	. . . . . 6 $/\$ $/\$
93	86, 92	syl5eqel 2705	. . . . 5 $/\$
94	34, 36, 38, 39, 66, 67, 68, 70, 82, 93	itg2split 23516	. . . 4 $/\$ $/\$
95	82, 93	readdcld 10069	. . . 4 $/\$
96	94, 95	eqeltrd 2701	. . 3 $/\$ $/\$
97	96	ralrimiva 2966	. 2 $/\$
98		eqidd 2623	. . 3 $/\$ $/\$
99		eqidd 2623	. . 3 $/\$
100	98, 99, 1	isibl2 23533	. 2 MblFn $/\$ $/\$
101	32, 97, 100	mpbir2and 957	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

cun 3572

cin 3573

cif 4086 class class class wbr 4653

cmpt 4729

cdm 5114

cres 5116

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

ci 9938

caddc 9939

cpnf 10071

cxr 10073

cle 10075

cdiv 10684

c3 11071

cz 11377

cicc 12178

cfz 12326

cexp 12860

cre 13837

covol 23231

cvol 23232 MblFncmbf 23383

citg2 23385

cibl 23386

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 ax-addf 10015

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-ofr 6898 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-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 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-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 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-sum 14417 df-rest 16083 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-top 20699 df-topon 20716 df-bases 20750 df-cmp 21190 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391

This theorem is referenced by: iblsplitf 40186

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