iblsplit - Mathbox for Glauco Siliprandi

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Theorem iblsplit 37837

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 2450	. . . 4
3	1, 2	fmptd 6044	. . 3
4		ssun1 3596	. . . . . 6
5		iblsplit.2	. . . . . 6
6	4, 5	syl5sseqr 3480	. . . . 5
7	6	resmptd 5155	. . . 4
8		iblsplit.4	. . . . . 6
9		eqidd 2451	. . . . . . 7 $/\$ $/\$
10		eqidd 2451	. . . . . . 7 $/\$
11	6	sseld 3430	. . . . . . . . 9
12	11	imdistani 695	. . . . . . . 8 $/\$ $/\$
13	12, 1	syl 17	. . . . . . 7 $/\$
14	9, 10, 13	isibl2 22717	. . . . . 6 MblFn $/\$ $/\$
15	8, 14	mpbid 214	. . . . 5 MblFn $/\$ $/\$
16	15	simpld 461	. . . 4 MblFn
17	7, 16	eqeltrd 2528	. . 3 MblFn
18		ssun2 3597	. . . . . 6
19	18, 5	syl5sseqr 3480	. . . . 5
20	19	resmptd 5155	. . . 4
21		iblsplit.5	. . . . . 6
22		eqidd 2451	. . . . . . 7 $/\$ $/\$
23		eqidd 2451	. . . . . . 7 $/\$
24	19	sseld 3430	. . . . . . . . 9
25	24	imdistani 695	. . . . . . . 8 $/\$ $/\$
26	25, 1	syl 17	. . . . . . 7 $/\$
27	22, 23, 26	isibl2 22717	. . . . . 6 MblFn $/\$ $/\$
28	21, 27	mpbid 214	. . . . 5 MblFn $/\$ $/\$
29	28	simpld 461	. . . 4 MblFn
30	20, 29	eqeltrd 2528	. . 3 MblFn
31	5	eqcomd 2456	. . 3
32	3, 17, 30, 31	mbfres2cn 37829	. 2 MblFn
33	16, 13	mbfdm2 22587	. . . . . 6
34	33	adantr 467	. . . . 5 $/\$
35	29, 26	mbfdm2 22587	. . . . . 6
36	35	adantr 467	. . . . 5 $/\$
37		iblsplit.1	. . . . . 6
38	37	adantr 467	. . . . 5 $/\$
39	5	adantr 467	. . . . 5 $/\$
40	1	adantlr 720	. . . . . . . . . . 11 $/\$ $/\$
41		ax-icn 9595	. . . . . . . . . . . . . 14
42	41	a1i 11	. . . . . . . . . . . . 13
43		elfznn0 11884	. . . . . . . . . . . . 13
44	42, 43	expcld 12413	. . . . . . . . . . . 12
45	44	ad2antlr 732	. . . . . . . . . . 11 $/\$ $/\$
46	41	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$
47		ine0 10051	. . . . . . . . . . . . 13
48	47	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$
49		elfzelz 11797	. . . . . . . . . . . . 13
50	49	ad2antlr 732	. . . . . . . . . . . 12 $/\$ $/\$
51	46, 48, 50	expne0d 12419	. . . . . . . . . . 11 $/\$ $/\$
52	40, 45, 51	divcld 10380	. . . . . . . . . 10 $/\$ $/\$
53	52	recld 13250	. . . . . . . . 9 $/\$ $/\$
54	53	rexrd 9687	. . . . . . . 8 $/\$ $/\$
55	54	adantr 467	. . . . . . 7 $/\$ $/\$ $/\$
56		simpr 463	. . . . . . 7 $/\$ $/\$ $/\$
57		pnfge 11429	. . . . . . . 8
58	55, 57	syl 17	. . . . . . 7 $/\$ $/\$ $/\$
59		0xr 9684	. . . . . . . 8
60		pnfxr 11409	. . . . . . . 8
61		elicc1 11677	. . . . . . . 8 $/\$ $/\$ $/\$
62	59, 60, 61	mp2an 677	. . . . . . 7 $/\$ $/\$
63	55, 56, 58, 62	syl3anbrc 1191	. . . . . 6 $/\$ $/\$ $/\$
64		0e0iccpnf 11740	. . . . . . 7
65	64	a1i 11	. . . . . 6 $/\$ $/\$ $/\$
66	63, 65	ifclda 3912	. . . . 5 $/\$ $/\$
67		eqid 2450	. . . . 5
68		eqid 2450	. . . . 5
69		ifan 3926	. . . . . 6 $/\$
70	69	mpteq2i 4485	. . . . 5 $/\$
71		ifan 3926	. . . . . . . . . 10 $/\$
72	71	eqcomi 2459	. . . . . . . . 9 $/\$
73	72	mpteq2i 4485	. . . . . . . 8 $/\$
74	73	a1i 11	. . . . . . 7 $/\$ $/\$
75	74	fveq2d 5867	. . . . . 6 $/\$ $/\$
76		eqidd 2451	. . . . . . . . . 10 $/\$ $/\$
77		eqidd 2451	. . . . . . . . . 10 $/\$
78	76, 77, 13	isibl2 22717	. . . . . . . . 9 MblFn $/\$ $/\$
79	8, 78	mpbid 214	. . . . . . . 8 MblFn $/\$ $/\$
80	79	simprd 465	. . . . . . 7 $/\$
81	80	r19.21bi 2756	. . . . . 6 $/\$ $/\$
82	75, 81	eqeltrd 2528	. . . . 5 $/\$
83		ifan 3926	. . . . . . . . 9 $/\$
84	83	eqcomi 2459	. . . . . . . 8 $/\$
85	84	mpteq2i 4485	. . . . . . 7 $/\$
86	85	fveq2i 5866	. . . . . 6 $/\$
87		eqidd 2451	. . . . . . . . . 10 $/\$ $/\$
88		eqidd 2451	. . . . . . . . . 10 $/\$
89	87, 88, 26	isibl2 22717	. . . . . . . . 9 MblFn $/\$ $/\$
90	21, 89	mpbid 214	. . . . . . . 8 MblFn $/\$ $/\$
91	90	simprd 465	. . . . . . 7 $/\$
92	91	r19.21bi 2756	. . . . . 6 $/\$ $/\$
93	86, 92	syl5eqel 2532	. . . . 5 $/\$
94	34, 36, 38, 39, 66, 67, 68, 70, 82, 93	itg2split 22700	. . . 4 $/\$ $/\$
95	82, 93	readdcld 9667	. . . 4 $/\$
96	94, 95	eqeltrd 2528	. . 3 $/\$ $/\$
97	96	ralrimiva 2801	. 2 $/\$
98		eqidd 2451	. . 3 $/\$ $/\$
99		eqidd 2451	. . 3 $/\$
100	98, 99, 1	isibl2 22717	. 2 MblFn $/\$ $/\$
101	32, 97, 100	mpbir2and 932	1

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 188 $/\$ wa 371 $/\$ w3a 984

wceq 1443

wcel 1886

wne 2621

wral 2736

cun 3401

cin 3402

cif 3880 class class class wbr 4401

cmpt 4460

cdm 4833

cres 4835

cfv 5581 (class class class)co 6288

cc 9534

cr 9535

cc0 9536

ci 9538

caddc 9539

cpnf 9669

cxr 9671

cle 9673

cdiv 10266

c3 10657

cz 10934

cicc 11635

cfz 11781

cexp 12269

cre 13153

covol 22406

cvol 22408 MblFncmbf 22565

citg2 22567

cibl 22568

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-inf2 8143 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 ax-pre-sup 9614 ax-addf 9615

This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-fal 1449 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-disj 4373 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-se 4793 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-isom 5590 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-of 6528 df-ofr 6529 df-om 6690 df-1st 6790 df-2nd 6791 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-1o 7179 df-2o 7180 df-oadd 7183 df-er 7360 df-map 7471 df-pm 7472 df-en 7567 df-dom 7568 df-sdom 7569 df-fin 7570 df-fi 7922 df-sup 7953 df-inf 7954 df-oi 8022 df-card 8370 df-cda 8595 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-div 10267 df-nn 10607 df-2 10665 df-3 10666 df-n0 10867 df-z 10935 df-uz 11157 df-q 11262 df-rp 11300 df-xneg 11406 df-xadd 11407 df-xmul 11408 df-ioo 11636 df-ico 11638 df-icc 11639 df-fz 11782 df-fzo 11913 df-fl 12025 df-seq 12211 df-exp 12270 df-hash 12513 df-cj 13155 df-re 13156 df-im 13157 df-sqrt 13291 df-abs 13292 df-clim 13545 df-sum 13746 df-rest 15314 df-topgen 15335 df-psmet 18955 df-xmet 18956 df-met 18957 df-bl 18958 df-mopn 18959 df-top 19914 df-bases 19915 df-topon 19916 df-cmp 20395 df-ovol 22409 df-vol 22411 df-mbf 22570 df-itg1 22571 df-itg2 22572 df-ibl 22573

This theorem is referenced by: iblsplitf 37841

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