Theorem meadjun 38294

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 11714	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		meadjun.m	. . . . . . . . 9 |- ( ph -> M e. Meas )
3		meadjun.x	. . . . . . . . 9 |- S = dom M
4	2, 3	meaf 38285	. . . . . . . 8 |- ( ph -> M : S --> ( 0 [,] +oo ) )
5		meadjun.b	. . . . . . . 8 |- ( ph -> B e. S )
6	4, 5	ffvelrnd 6021	. . . . . . 7 |- ( ph -> ( M ` B ) e. ( 0 [,] +oo ) )
7	1, 6	sseldi 3429	. . . . . 6 |- ( ph -> ( M ` B ) e. RR* )
8		xaddid2 11530	. . . . . 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 2456	. . . 4 |- ( ph -> ( M ` B ) = ( 0 +e ( M ` B ) ) )
11	10	adantr 467	. . 3 |- ( ( ph /\ A = (/) ) -> ( M ` B ) = ( 0 +e ( M ` B ) ) )
12		uneq1 3580	. . . . . 6 |- ( A = (/) -> ( A u. B ) = ( (/) u. B ) )
13		0un 37380	. . . . . . 7 |- ( (/) u. B ) = B
14	13	a1i 11	. . . . . 6 |- ( A = (/) -> ( (/) u. B ) = B )
15	12, 14	eqtrd 2484	. . . . 5 |- ( A = (/) -> ( A u. B ) = B )
16	15	fveq2d 5867	. . . 4 |- ( A = (/) -> ( M ` ( A u. B ) ) = ( M ` B ) )
17	16	adantl 468	. . 3 |- ( ( ph /\ A = (/) ) -> ( M ` ( A u. B ) ) = ( M ` B ) )
18		fveq2 5863	. . . . . 6 |- ( A = (/) -> ( M ` A ) = ( M ` (/) ) )
19	18	adantl 468	. . . . 5 |- ( ( ph /\ A = (/) ) -> ( M ` A ) = ( M ` (/) ) )
20	2	mea0 38286	. . . . . 6 |- ( ph -> ( M ` (/) ) = 0 )
21	20	adantr 467	. . . . 5 |- ( ( ph /\ A = (/) ) -> ( M ` (/) ) = 0 )
22	19, 21	eqtrd 2484	. . . 4 |- ( ( ph /\ A = (/) ) -> ( M ` A ) = 0 )
23	22	oveq1d 6303	. . 3 |- ( ( ph /\ A = (/) ) -> ( ( M ` A ) +e ( M ` B ) ) = ( 0 +e ( M ` B ) ) )
24	11, 17, 23	3eqtr4d 2494	. 2 |- ( ( ph /\ A = (/) ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
25		simpl 459	. . 3 |- ( ( ph /\ -. A = (/) ) -> ph )
26		meadjun.dj	. . . . . 6 |- ( ph -> ( A i^i B ) = (/) )
27	26	ad2antrr 731	. . . . 5 |- ( ( ( ph /\ -. A = (/) ) /\ A = B ) -> ( A i^i B ) = (/) )
28		inidm 3640	. . . . . . . . . . 11 |- ( A i^i A ) = A
29	28	eqcomi 2459	. . . . . . . . . 10 |- A = ( A i^i A )
30		ineq2 3627	. . . . . . . . . 10 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
31	29, 30	syl5req 2497	. . . . . . . . 9 |- ( A = B -> ( A i^i B ) = A )
32	31	adantl 468	. . . . . . . 8 |- ( ( -. A = (/) /\ A = B ) -> ( A i^i B ) = A )
33		neqne 37368	. . . . . . . . 9 |- ( -. A = (/) -> A =/= (/) )
34	33	adantr 467	. . . . . . . 8 |- ( ( -. A = (/) /\ A = B ) -> A =/= (/) )
35	32, 34	eqnetrd 2690	. . . . . . 7 |- ( ( -. A = (/) /\ A = B ) -> ( A i^i B ) =/= (/) )
36	35	neneqd 2628	. . . . . 6 |- ( ( -. A = (/) /\ A = B ) -> -. ( A i^i B ) = (/) )
37	36	adantll 719	. . . . 5 |- ( ( ( ph /\ -. A = (/) ) /\ A = B ) -> -. ( A i^i B ) = (/) )
38	27, 37	pm2.65da 579	. . . 4 |- ( ( ph /\ -. A = (/) ) -> -. A = B )
39	38	neqned 2630	. . 3 |- ( ( ph /\ -. A = (/) ) -> A =/= B )
40		meadjun.a	. . . . . . . 8 |- ( ph -> A e. S )
41		uniprg 4211	. . . . . . . 8 |- ( ( A e. S /\ B e. S ) -> U. { A , B } = ( A u. B ) )
42	40, 5, 41	syl2anc 666	. . . . . . 7 |- ( ph -> U. { A , B } = ( A u. B ) )
43	42	eqcomd 2456	. . . . . 6 |- ( ph -> ( A u. B ) = U. { A , B } )
44	43	fveq2d 5867	. . . . 5 |- ( ph -> ( M ` ( A u. B ) ) = ( M ` U. { A , B } ) )
45	44	adantr 467	. . . 4 |- ( ( ph /\ A =/= B ) -> ( M ` ( A u. B ) ) = ( M ` U. { A , B } ) )
46	40, 5	prssd 37378	. . . . . 6 |- ( ph -> { A , B } C_ S )
47		prfi 7843	. . . . . . . 8 |- { A , B } e. Fin
48		isfinite 8154	. . . . . . . . . 10 |- ( { A , B } e. Fin <-> { A , B } ~< om )
49	48	biimpi 198	. . . . . . . . 9 |- ( { A , B } e. Fin -> { A , B } ~< om )
50		sdomdom 7594	. . . . . . . . 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 4395	. . . . . . . . . 10 |- Disj x e. { B } x
55	54	a1i 11	. . . . . . . . 9 |- ( A = B -> Disj x e. { B } x )
56		preq1 4050	. . . . . . . . . . 11 |- ( A = B -> { A , B } = { B , B } )
57		dfsn2 3980	. . . . . . . . . . . . 13 |- { B } = { B , B }
58	57	eqcomi 2459	. . . . . . . . . . . 12 |- { B , B } = { B }
59	58	a1i 11	. . . . . . . . . . 11 |- ( A = B -> { B , B } = { B } )
60	56, 59	eqtrd 2484	. . . . . . . . . 10 |- ( A = B -> { A , B } = { B } )
61	60	disjeq1d 4380	. . . . . . . . 9 |- ( A = B -> (Disj x e. { A , B } x <-> Disj x e. { B } x ) )
62	55, 61	mpbird 236	. . . . . . . 8 |- ( A = B -> Disj x e. { A , B } x )
63	62	adantl 468	. . . . . . 7 |- ( ( ph /\ A = B ) -> Disj x e. { A , B } x )
64		simpl 459	. . . . . . . 8 |- ( ( ph /\ -. A = B ) -> ph )
65		neqne 37368	. . . . . . . . 9 |- ( -. A = B -> A =/= B )
66	65	adantl 468	. . . . . . . 8 |- ( ( ph /\ -. A = B ) -> A =/= B )
67	26	adantr 467	. . . . . . . . 9 |- ( ( ph /\ A =/= B ) -> ( A i^i B ) = (/) )
68	40	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ A =/= B ) -> A e. S )
69	5	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ A =/= B ) -> B e. S )
70		simpr 463	. . . . . . . . . 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 4397	. . . . . . . . . 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 1267	. . . . . . . . 9 |- ( ( ph /\ A =/= B ) -> (Disj x e. { A , B } x <-> ( A i^i B ) = (/) ) )
75	67, 74	mpbird 236	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> Disj x e. { A , B } x )
76	64, 66, 75	syl2anc 666	. . . . . . 7 |- ( ( ph /\ -. A = B ) -> Disj x e. { A , B } x )
77	63, 76	pm2.61dan 799	. . . . . 6 |- ( ph -> Disj x e. { A , B } x )
78	2, 3, 46, 53, 77	meadjuni 38289	. . . . 5 |- ( ph -> ( M ` U. { A , B } ) = (Σ^ ` ( M |` { A , B } ) ) )
79	78	adantr 467	. . . 4 |- ( ( ph /\ A =/= B ) -> ( M ` U. { A , B } ) = (Σ^ ` ( M |` { A , B } ) ) )
80	4, 40	ffvelrnd 6021	. . . . . . 7 |- ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )
81	80	adantr 467	. . . . . 6 |- ( ( ph /\ A =/= B ) -> ( M ` A ) e. ( 0 [,] +oo ) )
82	6	adantr 467	. . . . . 6 |- ( ( ph /\ A =/= B ) -> ( M ` B ) e. ( 0 [,] +oo ) )
83		fveq2 5863	. . . . . 6 |- ( x = A -> ( M ` x ) = ( M ` A ) )
84		fveq2 5863	. . . . . 6 |- ( x = B -> ( M ` x ) = ( M ` B ) )
85	68, 69, 81, 82, 83, 84, 70	sge0pr 38230	. . . . 5 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) = ( ( M ` A ) +e ( M ` B ) ) )
86	4, 46	fssresd 5748	. . . . . . . . 9 |- ( ph -> ( M |` { A , B } ) : { A , B } --> ( 0 [,] +oo ) )
87	86	feqmptd 5916	. . . . . . . 8 |- ( ph -> ( M |` { A , B } ) = ( x e. { A , B } |-> ( ( M |` { A , B } ) ` x ) ) )
88		fvres 5877	. . . . . . . . . 10 |- ( x e. { A , B } -> ( ( M |` { A , B } ) ` x ) = ( M ` x ) )
89	88	mpteq2ia 4484	. . . . . . . . 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 2484	. . . . . . 7 |- ( ph -> ( M |` { A , B } ) = ( x e. { A , B } |-> ( M ` x ) ) )
92	91	fveq2d 5867	. . . . . 6 |- ( ph -> (Σ^ ` ( M |` { A , B } ) ) = (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) )
93	92	adantr 467	. . . . 5 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( M |` { A , B } ) ) = (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) )
94		eqidd 2451	. . . . 5 |- ( ( ph /\ A =/= B ) -> ( ( M ` A ) +e ( M ` B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
95	85, 93, 94	3eqtr4d 2494	. . . 4 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( M |` { A , B } ) ) = ( ( M ` A ) +e ( M ` B ) ) )
96	45, 79, 95	3eqtrd 2488	. . 3 |- ( ( ph /\ A =/= B ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
97	25, 39, 96	syl2anc 666	. 2 |- ( ( ph /\ -. A = (/) ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
98	24, 97	pm2.61dan 799	1 |- ( ph -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   u. cun 3401   i^i cin 3402   (/)c0 3730   {csn 3967   {cpr 3969   U.cuni 4197  Disj wdisj 4372   class class class wbr 4401   |-> cmpt 4460   dom cdm 4833   |` cres 4835   ` cfv 5581  (class class class)co 6288   omcom 6689   ~<_ cdom 7564   ~< csdm 7565   Fincfn 7566   0cc0 9536   +oocpnf 9669   RR*cxr 9671   +ecxad 11404   [,]cicc 11635  Σ^{^}csumge0 38198  Meascmea 38281

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

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-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-oi 8022  df-card 8370  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-rp 11300  df-xadd 11407  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  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-sumge0 38199  df-mea 38282

This theorem is referenced by:  meassle  38295  meaunle  38296