Theorem meadjun 38416

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 11742	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		meadjun.m	. . . . . . . . 9 |- ( ph -> M e. Meas )
3		meadjun.x	. . . . . . . . 9 |- S = dom M
4	2, 3	meaf 38407	. . . . . . . 8 |- ( ph -> M : S --> ( 0 [,] +oo ) )
5		meadjun.b	. . . . . . . 8 |- ( ph -> B e. S )
6	4, 5	ffvelrnd 6038	. . . . . . 7 |- ( ph -> ( M ` B ) e. ( 0 [,] +oo ) )
7	1, 6	sseldi 3416	. . . . . 6 |- ( ph -> ( M ` B ) e. RR* )
8		xaddid2 11557	. . . . . 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 2477	. . . 4 |- ( ph -> ( M ` B ) = ( 0 +e ( M ` B ) ) )
11	10	adantr 472	. . 3 |- ( ( ph /\ A = (/) ) -> ( M ` B ) = ( 0 +e ( M ` B ) ) )
12		uneq1 3572	. . . . . 6 |- ( A = (/) -> ( A u. B ) = ( (/) u. B ) )
13		0un 37445	. . . . . . 7 |- ( (/) u. B ) = B
14	13	a1i 11	. . . . . 6 |- ( A = (/) -> ( (/) u. B ) = B )
15	12, 14	eqtrd 2505	. . . . 5 |- ( A = (/) -> ( A u. B ) = B )
16	15	fveq2d 5883	. . . 4 |- ( A = (/) -> ( M ` ( A u. B ) ) = ( M ` B ) )
17	16	adantl 473	. . 3 |- ( ( ph /\ A = (/) ) -> ( M ` ( A u. B ) ) = ( M ` B ) )
18		fveq2 5879	. . . . . 6 |- ( A = (/) -> ( M ` A ) = ( M ` (/) ) )
19	18	adantl 473	. . . . 5 |- ( ( ph /\ A = (/) ) -> ( M ` A ) = ( M ` (/) ) )
20	2	mea0 38408	. . . . . 6 |- ( ph -> ( M ` (/) ) = 0 )
21	20	adantr 472	. . . . 5 |- ( ( ph /\ A = (/) ) -> ( M ` (/) ) = 0 )
22	19, 21	eqtrd 2505	. . . 4 |- ( ( ph /\ A = (/) ) -> ( M ` A ) = 0 )
23	22	oveq1d 6323	. . 3 |- ( ( ph /\ A = (/) ) -> ( ( M ` A ) +e ( M ` B ) ) = ( 0 +e ( M ` B ) ) )
24	11, 17, 23	3eqtr4d 2515	. 2 |- ( ( ph /\ A = (/) ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
25		simpl 464	. . 3 |- ( ( ph /\ -. A = (/) ) -> ph )
26		meadjun.dj	. . . . . 6 |- ( ph -> ( A i^i B ) = (/) )
27	26	ad2antrr 740	. . . . 5 |- ( ( ( ph /\ -. A = (/) ) /\ A = B ) -> ( A i^i B ) = (/) )
28		inidm 3632	. . . . . . . . . . 11 |- ( A i^i A ) = A
29	28	eqcomi 2480	. . . . . . . . . 10 |- A = ( A i^i A )
30		ineq2 3619	. . . . . . . . . 10 |- ( A = B -> ( A i^i A ) = ( A i^i B ) )
31	29, 30	syl5req 2518	. . . . . . . . 9 |- ( A = B -> ( A i^i B ) = A )
32	31	adantl 473	. . . . . . . 8 |- ( ( -. A = (/) /\ A = B ) -> ( A i^i B ) = A )
33		neqne 2651	. . . . . . . . 9 |- ( -. A = (/) -> A =/= (/) )
34	33	adantr 472	. . . . . . . 8 |- ( ( -. A = (/) /\ A = B ) -> A =/= (/) )
35	32, 34	eqnetrd 2710	. . . . . . 7 |- ( ( -. A = (/) /\ A = B ) -> ( A i^i B ) =/= (/) )
36	35	neneqd 2648	. . . . . 6 |- ( ( -. A = (/) /\ A = B ) -> -. ( A i^i B ) = (/) )
37	36	adantll 728	. . . . 5 |- ( ( ( ph /\ -. A = (/) ) /\ A = B ) -> -. ( A i^i B ) = (/) )
38	27, 37	pm2.65da 586	. . . 4 |- ( ( ph /\ -. A = (/) ) -> -. A = B )
39	38	neqned 2650	. . 3 |- ( ( ph /\ -. A = (/) ) -> A =/= B )
40		meadjun.a	. . . . . . . 8 |- ( ph -> A e. S )
41		uniprg 4204	. . . . . . . 8 |- ( ( A e. S /\ B e. S ) -> U. { A , B } = ( A u. B ) )
42	40, 5, 41	syl2anc 673	. . . . . . 7 |- ( ph -> U. { A , B } = ( A u. B ) )
43	42	eqcomd 2477	. . . . . 6 |- ( ph -> ( A u. B ) = U. { A , B } )
44	43	fveq2d 5883	. . . . 5 |- ( ph -> ( M ` ( A u. B ) ) = ( M ` U. { A , B } ) )
45	44	adantr 472	. . . 4 |- ( ( ph /\ A =/= B ) -> ( M ` ( A u. B ) ) = ( M ` U. { A , B } ) )
46	40, 5	prssd 4120	. . . . . 6 |- ( ph -> { A , B } C_ S )
47		prfi 7864	. . . . . . . 8 |- { A , B } e. Fin
48		isfinite 8175	. . . . . . . . . 10 |- ( { A , B } e. Fin <-> { A , B } ~< om )
49	48	biimpi 199	. . . . . . . . 9 |- ( { A , B } e. Fin -> { A , B } ~< om )
50		sdomdom 7615	. . . . . . . . 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 4389	. . . . . . . . . 10 |- Disj x e. { B } x
55	54	a1i 11	. . . . . . . . 9 |- ( A = B -> Disj x e. { B } x )
56		preq1 4042	. . . . . . . . . . 11 |- ( A = B -> { A , B } = { B , B } )
57		dfsn2 3972	. . . . . . . . . . . . 13 |- { B } = { B , B }
58	57	eqcomi 2480	. . . . . . . . . . . 12 |- { B , B } = { B }
59	58	a1i 11	. . . . . . . . . . 11 |- ( A = B -> { B , B } = { B } )
60	56, 59	eqtrd 2505	. . . . . . . . . 10 |- ( A = B -> { A , B } = { B } )
61	60	disjeq1d 4374	. . . . . . . . 9 |- ( A = B -> (Disj x e. { A , B } x <-> Disj x e. { B } x ) )
62	55, 61	mpbird 240	. . . . . . . 8 |- ( A = B -> Disj x e. { A , B } x )
63	62	adantl 473	. . . . . . 7 |- ( ( ph /\ A = B ) -> Disj x e. { A , B } x )
64		simpl 464	. . . . . . . 8 |- ( ( ph /\ -. A = B ) -> ph )
65		neqne 2651	. . . . . . . . 9 |- ( -. A = B -> A =/= B )
66	65	adantl 473	. . . . . . . 8 |- ( ( ph /\ -. A = B ) -> A =/= B )
67	26	adantr 472	. . . . . . . . 9 |- ( ( ph /\ A =/= B ) -> ( A i^i B ) = (/) )
68	40	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ A =/= B ) -> A e. S )
69	5	adantr 472	. . . . . . . . . 10 |- ( ( ph /\ A =/= B ) -> B e. S )
70		simpr 468	. . . . . . . . . 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 4391	. . . . . . . . . 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 1292	. . . . . . . . 9 |- ( ( ph /\ A =/= B ) -> (Disj x e. { A , B } x <-> ( A i^i B ) = (/) ) )
75	67, 74	mpbird 240	. . . . . . . 8 |- ( ( ph /\ A =/= B ) -> Disj x e. { A , B } x )
76	64, 66, 75	syl2anc 673	. . . . . . 7 |- ( ( ph /\ -. A = B ) -> Disj x e. { A , B } x )
77	63, 76	pm2.61dan 808	. . . . . 6 |- ( ph -> Disj x e. { A , B } x )
78	2, 3, 46, 53, 77	meadjuni 38411	. . . . 5 |- ( ph -> ( M ` U. { A , B } ) = (Σ^ ` ( M |` { A , B } ) ) )
79	78	adantr 472	. . . 4 |- ( ( ph /\ A =/= B ) -> ( M ` U. { A , B } ) = (Σ^ ` ( M |` { A , B } ) ) )
80	4, 40	ffvelrnd 6038	. . . . . . 7 |- ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )
81	80	adantr 472	. . . . . 6 |- ( ( ph /\ A =/= B ) -> ( M ` A ) e. ( 0 [,] +oo ) )
82	6	adantr 472	. . . . . 6 |- ( ( ph /\ A =/= B ) -> ( M ` B ) e. ( 0 [,] +oo ) )
83		fveq2 5879	. . . . . 6 |- ( x = A -> ( M ` x ) = ( M ` A ) )
84		fveq2 5879	. . . . . 6 |- ( x = B -> ( M ` x ) = ( M ` B ) )
85	68, 69, 81, 82, 83, 84, 70	sge0pr 38350	. . . . 5 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) = ( ( M ` A ) +e ( M ` B ) ) )
86	4, 46	fssresd 5762	. . . . . . . . 9 |- ( ph -> ( M |` { A , B } ) : { A , B } --> ( 0 [,] +oo ) )
87	86	feqmptd 5932	. . . . . . . 8 |- ( ph -> ( M |` { A , B } ) = ( x e. { A , B } |-> ( ( M |` { A , B } ) ` x ) ) )
88		fvres 5893	. . . . . . . . . 10 |- ( x e. { A , B } -> ( ( M |` { A , B } ) ` x ) = ( M ` x ) )
89	88	mpteq2ia 4478	. . . . . . . . 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 2505	. . . . . . 7 |- ( ph -> ( M |` { A , B } ) = ( x e. { A , B } |-> ( M ` x ) ) )
92	91	fveq2d 5883	. . . . . 6 |- ( ph -> (Σ^ ` ( M |` { A , B } ) ) = (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) )
93	92	adantr 472	. . . . 5 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( M |` { A , B } ) ) = (Σ^ ` ( x e. { A , B } |-> ( M ` x ) ) ) )
94		eqidd 2472	. . . . 5 |- ( ( ph /\ A =/= B ) -> ( ( M ` A ) +e ( M ` B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
95	85, 93, 94	3eqtr4d 2515	. . . 4 |- ( ( ph /\ A =/= B ) -> (Σ^ ` ( M |` { A , B } ) ) = ( ( M ` A ) +e ( M ` B ) ) )
96	45, 79, 95	3eqtrd 2509	. . 3 |- ( ( ph /\ A =/= B ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
97	25, 39, 96	syl2anc 673	. 2 |- ( ( ph /\ -. A = (/) ) -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )
98	24, 97	pm2.61dan 808	1 |- ( ph -> ( M ` ( A u. B ) ) = ( ( M ` A ) +e ( M ` B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   u. cun 3388   i^i cin 3389   (/)c0 3722   {csn 3959   {cpr 3961   U.cuni 4190  Disj wdisj 4366   class class class wbr 4395   |-> cmpt 4454   dom cdm 4839   |` cres 4841   ` cfv 5589  (class class class)co 6308   omcom 6711   ~<_ cdom 7585   ~< csdm 7586   Fincfn 7587   0cc0 9557   +oocpnf 9690   RR*cxr 9692   +ecxad 11430   [,]cicc 11663  Σ^{^}csumge0 38318  Meascmea 38403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-xadd 11433  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-sumge0 38319  df-mea 38404

This theorem is referenced by:  meassle  38417  meaunle  38418