Theorem meadjun 40679

Description: The measure of the union of two disjoint sets is the sum of the measures, Property 112C (a) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
meadjun.m	|- ( ph -> M e. Meas )
meadjun.x	|- S = dom M
meadjun.a	|- ( ph -> A e. S )
meadjun.b	|- ( ph -> B e. S )
meadjun.dj	|- ( ph -> ( A i^i B ) = (/) )

Assertion

Ref	Expression
meadjun	|- ( ph -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )

Proof of Theorem meadjun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 12256	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		meadjun.m	. . . . . . . . 9 |- ( ph -> M e. Meas )
3		meadjun.x	. . . . . . . . 9 |- S = dom M
4	2, 3	meaf 40670	. . . . . . . 8 |- ( ph -> M : S --> ( 0 [,] +oo ) )
5		meadjun.b	. . . . . . . 8 |- ( ph -> B e. S )
6	4, 5	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( M ` B ) e. ( 0 [,] +oo ) )
7	1, 6	sseldi 3601	. . . . . 6 |- ( ph -> ( M ` B ) e. RR* )
8		xaddid2 12073	. . . . . 6 |- ( ( M ` B ) e. RR* -> ( 0 +e ( M ` B ) ) = ( M ` B ) )
9	7, 8	syl 17	. . . . 5 |- ( ph -> ( 0 +e ( M ` B ) ) = ( M ` B ) )
10	9	eqcomd 2628	. . . 4 |- ( ph -> ( M ` B ) = ( 0 +e ( M ` B ) ) )
11	10	adantr 481	. . 3 |- ( ( ph /\ A = (/) ) -> ( M ` B ) = ( 0 +e ( M ` B ) ) )
12		uneq1 3760	. . . . . 6 |- ( A = (/) -> ( A u. B ) = ( (/) u. B ) )
13		0un 39215	. . . . . . 7 |- ( (/) u. B ) = B
14	13	a1i 11	. . . . . 6 |- ( A = (/) -> ( (/) u. B ) = B )
15	12, 14	eqtrd 2656	. . . . 5 |- ( A = (/) -> ( A u. B ) = B )
16	15	fveq2d 6195	. . . 4 |- ( A = (/) -> ( M ` ( A u. B ) ) = ( M ` B ) )
17	16	adantl 482	. . 3 |- ( ( ph /\ A = (/) ) -> ( M ` ( A u. B ) ) = ( M ` B ) )
18		fveq2 6191	. . . . . 6 |- ( A = (/) -> ( M ` A ) = ( M ` (/) ) )
19	18	adantl 482	. . . . 5 |- ( ( ph /\ A = (/) ) -> ( M ` A ) = ( M ` (/) ) )
20	2	mea0 40671	. . . . . 6 |- ( ph -> ( M ` (/) ) = 0 )
21	20	adantr 481	. . . . 5 |- ( ( ph /\ A = (/) ) -> ( M ` (/) ) = 0 )
22	19, 21	eqtrd 2656	. . . 4 |- ( ( ph /\ A = (/) ) -> ( M ` A ) = 0 )
23	22	oveq1d 6665	. . 3 |- ( ( ph /\ A = (/) ) -> ( ( M ` A ) +e ( M ` B ) ) = ( 0 +e ( M ` B ) ) )
24	11, 17, 23	3eqtr4d 2666	. 2 |- ( ( ph /\ A = (/) ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
25		simpl 473	. . 3 |- ( ( ph /\ -. A = (/) ) -> ph )
26		meadjun.dj	. . . . . 6 |- ( ph -> ( A i^i B ) = (/) )
27	26	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ -. A = (/) ) /\ A = B ) -> ( A i^i B ) = (/) )
28		inidm 3822	. . . . . . . . . . 11 |- ( A i^i A ) = A
29	28	eqcomi 2631	. . . . . . . . . 10 |- A = ( A i^i A )
30		ineq2 3808	. . . . . . . . . 10 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
31	29, 30	syl5req 2669	. . . . . . . . 9 |- ( A = B -> ( A i^i B ) = A )
32	31	adantl 482	. . . . . . . 8 |- ( ( -. A = (/) /\ A = B ) -> ( A i^i B ) = A )
33		neqne 2802	. . . . . . . . 9 |- ( -. A = (/) -> A =/= (/) )
34	33	adantr 481	. . . . . . . 8 |- ( ( -. A = (/) /\ A = B ) -> A =/= (/) )
35	32, 34	eqnetrd 2861	. . . . . . 7 |- ( ( -. A = (/) /\ A = B ) -> ( A i^i B ) =/= (/) )
36	35	neneqd 2799	. . . . . 6 |- ( ( -. A = (/) /\ A = B ) -> -. ( A i^i B ) = (/) )
37	36	adantll 750	. . . . 5 |- ( ( ( ph /\ -. A = (/) ) /\ A = B ) -> -. ( A i^i B ) = (/) )
38	27, 37	pm2.65da 600	. . . 4 |- ( ( ph /\ -. A = (/) ) -> -. A = B )
39	38	neqned 2801	. . 3 |- ( ( ph /\ -. A = (/) ) -> A =/= B )
40		meadjun.a	. . . . . . . 8 |- ( ph -> A e. S )
41		uniprg 4450	. . . . . . . 8 |- ( ( A e. S /\ B e. S ) -> U. { A , B } = ( A u. B ) )
42	40, 5, 41	syl2anc 693	. . . . . . 7 |- ( ph -> U. { A , B } = ( A u. B ) )
43	42	eqcomd 2628	. . . . . 6 |- ( ph -> ( A u. B ) = U. { A , B } )
44	43	fveq2d 6195	. . . . 5 |- ( ph -> ( M ` ( A u. B ) ) = ( M ` U. { A , B } ) )
45	44	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> ( M ` ( A u. B ) ) = ( M ` U. { A , B } ) )
46	40, 5	prssd 4354	. . . . . 6 |- ( ph -> { A , B } C_ S )
47		prfi 8235	. . . . . . . 8 |- { A , B } e. Fin
48		isfinite 8549	. . . . . . . . . 10 |- ( { A , B } e. Fin <-> { A , B } ~< om )
49	48	biimpi 206	. . . . . . . . 9 |- ( { A , B } e. Fin -> { A , B } ~< om )
50		sdomdom 7983	. . . . . . . . 9 |- ( { A , B } ~< om -> { A , B } ~<_ om )
51	49, 50	syl 17	. . . . . . . 8 |- ( { A , B } e. Fin -> { A , B } ~<_ om )
52	47, 51	ax-mp 5	. . . . . . 7 |- { A , B } ~<_ om
53	52	a1i 11	. . . . . 6 |- ( ph -> { A , B } ~<_ om )
54		disjxsn 4646	. . . . . . . . . 10 |- Disj x e. { B } x
55	54	a1i 11	. . . . . . . . 9 |- ( A = B -> Disj x e. { B } x )
56		preq1 4268	. . . . . . . . . . 11 |- ( A = B -> { A , B } = { B , B } )
57		dfsn2 4190	. . . . . . . . . . . . 13 |- { B } = { B , B }
58	57	eqcomi 2631	. . . . . . . . . . . 12 |- { B , B } = { B }
59	58	a1i 11	. . . . . . . . . . 11 |- ( A = B -> { B , B } = { B } )
60	56, 59	eqtrd 2656	. . . . . . . . . 10 |- ( A = B -> { A , B } = { B } )
61	60	disjeq1d 4628	. . . . . . . . 9 |- ( A = B -> (Disj x e. { A , B } x <-> Disj x e. { B } x ) )
62	55, 61	mpbird 247	. . . . . . . 8 |- ( A = B -> Disj x e. { A , B } x )
63	62	adantl 482	. . . . . . 7 |- ( ( ph /\ A = B ) -> Disj x e. { A , B } x )
64		simpl 473	. . . . . . . 8 |- ( ( ph /\ -. A = B ) -> ph )
65		neqne 2802	. . . . . . . . 9 |- ( -. A = B -> A =/= B )
66	65	adantl 482	. . . . . . . 8 |- ( ( ph /\ -. A = B ) -> A =/= B )
67	26	adantr 481	. . . . . . . . 9 |- ( ( ph /\ A =/= B ) -> ( A i^i B ) = (/) )
68	40	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ A =/= B ) -> A e. S )
69	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ A =/= B ) -> B e. S )
70		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ A =/= B ) -> A =/= B )
71		id 22	. . . . . . . . . . 11 |- ( x = A -> x = A )
72		id 22	. . . . . . . . . . 11 |- ( x = B -> x = B )
73	71, 72	disjprg 4648	. . . . . . . . . 10 |- ( ( A e. S /\ B e. S /\ A =/= B ) -> (Disj x e. { A , B } x <-> ( A i^i B ) = (/) ) )
74	68, 69, 70, 73	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ A =/= B ) -> (Disj x e. { A , B } x <-> ( A i^i B ) = (/) ) )
75	67, 74	mpbird 247	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> Disj x e. { A , B } x )
76	64, 66, 75	syl2anc 693	. . . . . . 7 |- ( ( ph /\ -. A = B ) -> Disj x e. { A , B } x )
77	63, 76	pm2.61dan 832	. . . . . 6 |- ( ph -> Disj x e. { A , B } x )
78	2, 3, 46, 53, 77	meadjuni 40674	. . . . 5 |- ( ph -> ( M ` U. { A , B } ) = (Σ^ ` ( M |` { A , B } ) ) )
79	78	adantr 481	. . . 4 |- ( ( ph /\ A =/= B ) -> ( M ` U. { A , B } ) = (Σ^ ` ( M |` { A , B } ) ) )
80	4, 40	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )
81	80	adantr 481	. . . . . 6 |- ( ( ph /\ A =/= B ) -> ( M ` A ) e. ( 0 [,] +oo ) )
82	6	adantr 481	. . . . . 6 |- ( ( ph /\ A =/= B ) -> ( M ` B ) e. ( 0 [,] +oo ) )
83		fveq2 6191	. . . . . 6 |- ( x = A -> ( M ` x ) = ( M ` A ) )
84		fveq2 6191	. . . . . 6 |- ( x = B -> ( M ` x ) = ( M ` B ) )
85	68, 69, 81, 82, 83, 84, 70	sge0pr 40611	. . . . 5 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) = ( ( M ` A ) +e ( M ` B ) ) )
86	4, 46	fssresd 6071	. . . . . . . . 9 |- ( ph -> ( M |` { A , B } ) : { A , B } --> ( 0 [,] +oo ) )
87	86	feqmptd 6249	. . . . . . . 8 |- ( ph -> ( M |` { A , B } ) = ( x e. { A , B } |-> ( ( M |` { A , B } ) ` x ) ) )
88		fvres 6207	. . . . . . . . . 10 |- ( x e. { A , B } -> ( ( M |` { A , B } ) ` x ) = ( M ` x ) )
89	88	mpteq2ia 4740	. . . . . . . . 9 |- ( x e. { A , B } |-> ( ( M |` { A , B } ) ` x ) ) = ( x e. { A , B } |-> ( M ` x ) )
90	89	a1i 11	. . . . . . . 8 |- ( ph -> ( x e. { A , B } |-> ( ( M |` { A , B } ) ` x ) ) = ( x e. { A , B } |-> ( M ` x ) ) )
91	87, 90	eqtrd 2656	. . . . . . 7 |- ( ph -> ( M |` { A , B } ) = ( x e. { A , B } |-> ( M ` x ) ) )
92	91	fveq2d 6195	. . . . . 6 |- ( ph -> (Σ^ ` ( M |` { A , B } ) ) = (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) )
93	92	adantr 481	. . . . 5 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( M |` { A , B } ) ) = (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) )
94		eqidd 2623	. . . . 5 |- ( ( ph /\ A =/= B ) -> ( ( M ` A ) +e ( M ` B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
95	85, 93, 94	3eqtr4d 2666	. . . 4 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( M |` { A , B } ) ) = ( ( M ` A ) +e ( M ` B ) ) )
96	45, 79, 95	3eqtrd 2660	. . 3 |- ( ( ph /\ A =/= B ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
97	25, 39, 96	syl2anc 693	. 2 |- ( ( ph /\ -. A = (/) ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
98	24, 97	pm2.61dan 832	1 |- ( ph -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   {cpr 4179   U.cuni 4436  Disj wdisj 4620   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   0cc0 9936   +oocpnf 10071   RR*cxr 10073   +ecxad 11944   [,]cicc 12178  Σ^{^}csumge0 40579  Meascmea 40666

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

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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  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-rp 11833  df-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  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-sumge0 40580  df-mea 40667

This theorem is referenced by:  meassle  40680  meaunle  40681  meadjunre  40693