Theorem ovolsplit 40205

Description: The Lebesgue outer measure function is finitely sub-additive: application to a set split in two parts, using addition for extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypothesis

Ref	Expression
ovolsplit.1	|- ( ph -> A C_ RR )

Assertion

Ref	Expression
ovolsplit	|- ( ph -> ( vol* ` A ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )

Proof of Theorem ovolsplit

Step	Hyp	Ref	Expression
1		inundif 4046	. . . . 5 |- ( ( A i^i B ) u. ( A \ B ) ) = A
2	1	eqcomi 2631	. . . 4 |- A = ( ( A i^i B ) u. ( A \ B ) )
3	2	a1i 11	. . 3 |- ( ph -> A = ( ( A i^i B ) u. ( A \ B ) ) )
4	3	fveq2d 6195	. 2 |- ( ph -> ( vol* ` A ) = ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) )
5		ovolsplit.1	. . . . . . . . 9 |- ( ph -> A C_ RR )
6	5	ssinss1d 39214	. . . . . . . 8 |- ( ph -> ( A i^i B ) C_ RR )
7	5	ssdifssd 3748	. . . . . . . 8 |- ( ph -> ( A \ B ) C_ RR )
8	6, 7	unssd 3789	. . . . . . 7 |- ( ph -> ( ( A i^i B ) u. ( A \ B ) ) C_ RR )
9		ovolcl 23246	. . . . . . 7 |- ( ( ( A i^i B ) u. ( A \ B ) ) C_ RR -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) e. RR* )
10	8, 9	syl 17	. . . . . 6 |- ( ph -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) e. RR* )
11		pnfge 11964	. . . . . 6 |- ( ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) e. RR* -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ +oo )
12	10, 11	syl 17	. . . . 5 |- ( ph -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ +oo )
13	12	adantr 481	. . . 4 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ +oo )
14		oveq1 6657	. . . . . 6 |- ( ( vol* ` ( A i^i B ) ) = +oo -> ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) = ( +oo +e ( vol* ` ( A \ B ) ) ) )
15	14	adantl 482	. . . . 5 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) = +oo ) -> ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) = ( +oo +e ( vol* ` ( A \ B ) ) ) )
16		ovolcl 23246	. . . . . . . 8 |- ( ( A \ B ) C_ RR -> ( vol* ` ( A \ B ) ) e. RR* )
17	7, 16	syl 17	. . . . . . 7 |- ( ph -> ( vol* ` ( A \ B ) ) e. RR* )
18	17	adantr 481	. . . . . 6 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( A \ B ) ) e. RR* )
19		reex 10027	. . . . . . . . . . . . . 14 |- RR e. _V
20	19	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> RR e. _V )
21	20, 5	ssexd 4805	. . . . . . . . . . . 12 |- ( ph -> A e. _V )
22		difexg 4808	. . . . . . . . . . . 12 |- ( A e. _V -> ( A \ B ) e. _V )
23	21, 22	syl 17	. . . . . . . . . . 11 |- ( ph -> ( A \ B ) e. _V )
24		elpwg 4166	. . . . . . . . . . 11 |- ( ( A \ B ) e. _V -> ( ( A \ B ) e. ~P RR <-> ( A \ B ) C_ RR ) )
25	23, 24	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( A \ B ) e. ~P RR <-> ( A \ B ) C_ RR ) )
26	7, 25	mpbird 247	. . . . . . . . 9 |- ( ph -> ( A \ B ) e. ~P RR )
27		ovolf 23250	. . . . . . . . . 10 |- vol* : ~P RR --> ( 0 [,] +oo )
28	27	ffvelrni 6358	. . . . . . . . 9 |- ( ( A \ B ) e. ~P RR -> ( vol* ` ( A \ B ) ) e. ( 0 [,] +oo ) )
29	26, 28	syl 17	. . . . . . . 8 |- ( ph -> ( vol* ` ( A \ B ) ) e. ( 0 [,] +oo ) )
30	29	xrge0nemnfd 39548	. . . . . . 7 |- ( ph -> ( vol* ` ( A \ B ) ) =/= -oo )
31	30	adantr 481	. . . . . 6 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( A \ B ) ) =/= -oo )
32		xaddpnf2 12058	. . . . . 6 |- ( ( ( vol* ` ( A \ B ) ) e. RR* /\ ( vol* ` ( A \ B ) ) =/= -oo ) -> ( +oo +e ( vol* ` ( A \ B ) ) ) = +oo )
33	18, 31, 32	syl2anc 693	. . . . 5 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) = +oo ) -> ( +oo +e ( vol* ` ( A \ B ) ) ) = +oo )
34	15, 33	eqtr2d 2657	. . . 4 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) = +oo ) -> +oo = ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
35	13, 34	breqtrd 4679	. . 3 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
36		simpl 473	. . . 4 |- ( ( ph /\ -. ( vol* ` ( A i^i B ) ) = +oo ) -> ph )
37	20, 6	sselpwd 4807	. . . . . . 7 |- ( ph -> ( A i^i B ) e. ~P RR )
38	27	ffvelrni 6358	. . . . . . 7 |- ( ( A i^i B ) e. ~P RR -> ( vol* ` ( A i^i B ) ) e. ( 0 [,] +oo ) )
39	37, 38	syl 17	. . . . . 6 |- ( ph -> ( vol* ` ( A i^i B ) ) e. ( 0 [,] +oo ) )
40	39	adantr 481	. . . . 5 |- ( ( ph /\ -. ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( A i^i B ) ) e. ( 0 [,] +oo ) )
41		neqne 2802	. . . . . 6 |- ( -. ( vol* ` ( A i^i B ) ) = +oo -> ( vol* ` ( A i^i B ) ) =/= +oo )
42	41	adantl 482	. . . . 5 |- ( ( ph /\ -. ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( A i^i B ) ) =/= +oo )
43		ge0xrre 39758	. . . . 5 |- ( ( ( vol* ` ( A i^i B ) ) e. ( 0 [,] +oo ) /\ ( vol* ` ( A i^i B ) ) =/= +oo ) -> ( vol* ` ( A i^i B ) ) e. RR )
44	40, 42, 43	syl2anc 693	. . . 4 |- ( ( ph /\ -. ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( A i^i B ) ) e. RR )
45	12	adantr 481	. . . . . . 7 |- ( ( ph /\ ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ +oo )
46		oveq2 6658	. . . . . . . . 9 |- ( ( vol* ` ( A \ B ) ) = +oo -> ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) = ( ( vol* ` ( A i^i B ) ) +e +oo ) )
47	46	adantl 482	. . . . . . . 8 |- ( ( ph /\ ( vol* ` ( A \ B ) ) = +oo ) -> ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) = ( ( vol* ` ( A i^i B ) ) +e +oo ) )
48		ovolcl 23246	. . . . . . . . . . 11 |- ( ( A i^i B ) C_ RR -> ( vol* ` ( A i^i B ) ) e. RR* )
49	6, 48	syl 17	. . . . . . . . . 10 |- ( ph -> ( vol* ` ( A i^i B ) ) e. RR* )
50	39	xrge0nemnfd 39548	. . . . . . . . . 10 |- ( ph -> ( vol* ` ( A i^i B ) ) =/= -oo )
51		xaddpnf1 12057	. . . . . . . . . 10 |- ( ( ( vol* ` ( A i^i B ) ) e. RR* /\ ( vol* ` ( A i^i B ) ) =/= -oo ) -> ( ( vol* ` ( A i^i B ) ) +e +oo ) = +oo )
52	49, 50, 51	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( vol* ` ( A i^i B ) ) +e +oo ) = +oo )
53	52	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( vol* ` ( A \ B ) ) = +oo ) -> ( ( vol* ` ( A i^i B ) ) +e +oo ) = +oo )
54	47, 53	eqtr2d 2657	. . . . . . 7 |- ( ( ph /\ ( vol* ` ( A \ B ) ) = +oo ) -> +oo = ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
55	45, 54	breqtrd 4679	. . . . . 6 |- ( ( ph /\ ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
56	55	adantlr 751	. . . . 5 |- ( ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR ) /\ ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
57		simpll 790	. . . . . 6 |- ( ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR ) /\ -. ( vol* ` ( A \ B ) ) = +oo ) -> ph )
58		simplr 792	. . . . . 6 |- ( ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR ) /\ -. ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( A i^i B ) ) e. RR )
59	29	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( A \ B ) ) e. ( 0 [,] +oo ) )
60		neqne 2802	. . . . . . . . 9 |- ( -. ( vol* ` ( A \ B ) ) = +oo -> ( vol* ` ( A \ B ) ) =/= +oo )
61	60	adantl 482	. . . . . . . 8 |- ( ( ph /\ -. ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( A \ B ) ) =/= +oo )
62		ge0xrre 39758	. . . . . . . 8 |- ( ( ( vol* ` ( A \ B ) ) e. ( 0 [,] +oo ) /\ ( vol* ` ( A \ B ) ) =/= +oo ) -> ( vol* ` ( A \ B ) ) e. RR )
63	59, 61, 62	syl2anc 693	. . . . . . 7 |- ( ( ph /\ -. ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( A \ B ) ) e. RR )
64	63	adantlr 751	. . . . . 6 |- ( ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR ) /\ -. ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( A \ B ) ) e. RR )
65	6	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( A i^i B ) C_ RR )
66		simp2 1062	. . . . . . . 8 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( vol* ` ( A i^i B ) ) e. RR )
67	7	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( A \ B ) C_ RR )
68		simp3 1063	. . . . . . . 8 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( vol* ` ( A \ B ) ) e. RR )
69		ovolun 23267	. . . . . . . 8 |- ( ( ( ( A i^i B ) C_ RR /\ ( vol* ` ( A i^i B ) ) e. RR ) /\ ( ( A \ B ) C_ RR /\ ( vol* ` ( A \ B ) ) e. RR ) ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) + ( vol* ` ( A \ B ) ) ) )
70	65, 66, 67, 68, 69	syl22anc 1327	. . . . . . 7 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) + ( vol* ` ( A \ B ) ) ) )
71		rexadd 12063	. . . . . . . . 9 |- ( ( ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) = ( ( vol* ` ( A i^i B ) ) + ( vol* ` ( A \ B ) ) ) )
72	71	eqcomd 2628	. . . . . . . 8 |- ( ( ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( ( vol* ` ( A i^i B ) ) + ( vol* ` ( A \ B ) ) ) = ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
73	72	3adant1 1079	. . . . . . 7 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( ( vol* ` ( A i^i B ) ) + ( vol* ` ( A \ B ) ) ) = ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
74	70, 73	breqtrd 4679	. . . . . 6 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR /\ ( vol* ` ( A \ B ) ) e. RR ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
75	57, 58, 64, 74	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR ) /\ -. ( vol* ` ( A \ B ) ) = +oo ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
76	56, 75	pm2.61dan 832	. . . 4 |- ( ( ph /\ ( vol* ` ( A i^i B ) ) e. RR ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
77	36, 44, 76	syl2anc 693	. . 3 |- ( ( ph /\ -. ( vol* ` ( A i^i B ) ) = +oo ) -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
78	35, 77	pm2.61dan 832	. 2 |- ( ph -> ( vol* ` ( ( A i^i B ) u. ( A \ B ) ) ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )
79	4, 78	eqbrtrd 4675	1 |- ( ph -> ( vol* ` A ) <_ ( ( vol* ` ( A i^i B ) ) +e ( vol* ` ( A \ B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   <_ cle 10075   +ecxad 11944   [,]cicc 12178   vol*covol 23231

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-ico 12181  df-icc 12182  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-ovol 23233

This theorem is referenced by:  ismbl4  40210