Theorem meadjiunlem 40682

Description: The sum of nonnegative extended reals, restricted to the range of another function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
meadjiunlem.f	|- ( ph -> M e. Meas )
meadjiunlem.3	|- S = dom M
meadjiunlem.x	|- ( ph -> X e. V )
meadjiunlem.g	|- ( ph -> G : X --> S )
meadjiunlem.y	|- Y = { i e. X | ( G ` i ) =/= (/) }
meadjiunlem.dj	|- ( ph -> Disj i e. X ( G ` i ) )

Assertion

Ref	Expression
meadjiunlem	|- ( ph -> (Σ^ ` ( M |` ran G ) ) = (Σ^ ` ( M o. G ) ) )

Distinct variable groups:   i, G   i, X   i, Y   ph, i

Allowed substitution hints:   S( i)   M( i)   V( i)

Proof of Theorem meadjiunlem

Dummy variables x k j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ k ph
2		meadjiunlem.g	. . . . . 6 |- ( ph -> G : X --> S )
3		meadjiunlem.x	. . . . . 6 |- ( ph -> X e. V )
4	2, 3	jca 554	. . . . 5 |- ( ph -> ( G : X --> S /\ X e. V ) )
5		fex 6490	. . . . 5 |- ( ( G : X --> S /\ X e. V ) -> G e. _V )
6		rnexg 7098	. . . . 5 |- ( G e. _V -> ran G e. _V )
7	4, 5, 6	3syl 18	. . . 4 |- ( ph -> ran G e. _V )
8		difssd 3738	. . . 4 |- ( ph -> ( ran G \ { (/) } ) C_ ran G )
9		meadjiunlem.f	. . . . . . 7 |- ( ph -> M e. Meas )
10		meadjiunlem.3	. . . . . . 7 |- S = dom M
11	9, 10	meaf 40670	. . . . . 6 |- ( ph -> M : S --> ( 0 [,] +oo ) )
12	11	adantr 481	. . . . 5 |- ( ( ph /\ k e. ( ran G \ { (/) } ) ) -> M : S --> ( 0 [,] +oo ) )
13		frn 6053	. . . . . . . 8 |- ( G : X --> S -> ran G C_ S )
14	2, 13	syl 17	. . . . . . 7 |- ( ph -> ran G C_ S )
15	14	adantr 481	. . . . . 6 |- ( ( ph /\ k e. ( ran G \ { (/) } ) ) -> ran G C_ S )
16	8	sselda 3603	. . . . . 6 |- ( ( ph /\ k e. ( ran G \ { (/) } ) ) -> k e. ran G )
17	15, 16	sseldd 3604	. . . . 5 |- ( ( ph /\ k e. ( ran G \ { (/) } ) ) -> k e. S )
18	12, 17	ffvelrnd 6360	. . . 4 |- ( ( ph /\ k e. ( ran G \ { (/) } ) ) -> ( M ` k ) e. ( 0 [,] +oo ) )
19		simpl 473	. . . . 5 |- ( ( ph /\ k e. ( ran G \ ( ran G \ { (/) } ) ) ) -> ph )
20		id 22	. . . . . . . 8 |- ( k e. ( ran G \ ( ran G \ { (/) } ) ) -> k e. ( ran G \ ( ran G \ { (/) } ) ) )
21		dfin4 3867	. . . . . . . . 9 |- ( ran G i^i { (/) } ) = ( ran G \ ( ran G \ { (/) } ) )
22	21	eqcomi 2631	. . . . . . . 8 |- ( ran G \ ( ran G \ { (/) } ) ) = ( ran G i^i { (/) } )
23	20, 22	syl6eleq 2711	. . . . . . 7 |- ( k e. ( ran G \ ( ran G \ { (/) } ) ) -> k e. ( ran G i^i { (/) } ) )
24		elinel2 3800	. . . . . . . 8 |- ( k e. ( ran G i^i { (/) } ) -> k e. { (/) } )
25		elsni 4194	. . . . . . . 8 |- ( k e. { (/) } -> k = (/) )
26	24, 25	syl 17	. . . . . . 7 |- ( k e. ( ran G i^i { (/) } ) -> k = (/) )
27	23, 26	syl 17	. . . . . 6 |- ( k e. ( ran G \ ( ran G \ { (/) } ) ) -> k = (/) )
28	27	adantl 482	. . . . 5 |- ( ( ph /\ k e. ( ran G \ ( ran G \ { (/) } ) ) ) -> k = (/) )
29		simpr 477	. . . . . . 7 |- ( ( ph /\ k = (/) ) -> k = (/) )
30	29	fveq2d 6195	. . . . . 6 |- ( ( ph /\ k = (/) ) -> ( M ` k ) = ( M ` (/) ) )
31	9	mea0 40671	. . . . . . 7 |- ( ph -> ( M ` (/) ) = 0 )
32	31	adantr 481	. . . . . 6 |- ( ( ph /\ k = (/) ) -> ( M ` (/) ) = 0 )
33	30, 32	eqtrd 2656	. . . . 5 |- ( ( ph /\ k = (/) ) -> ( M ` k ) = 0 )
34	19, 28, 33	syl2anc 693	. . . 4 |- ( ( ph /\ k e. ( ran G \ ( ran G \ { (/) } ) ) ) -> ( M ` k ) = 0 )
35	1, 7, 8, 18, 34	sge0ss 40629	. . 3 |- ( ph -> (Σ^ ` ( k e. ( ran G \ { (/) } ) |-> ( M ` k ) ) ) = (Σ^ ` ( k e. ran G |-> ( M ` k ) ) ) )
36	35	eqcomd 2628	. 2 |- ( ph -> (Σ^ ` ( k e. ran G |-> ( M ` k ) ) ) = (Σ^ ` ( k e. ( ran G \ { (/) } ) |-> ( M ` k ) ) ) )
37	11, 14	feqresmpt 6250	. . 3 |- ( ph -> ( M |` ran G ) = ( k e. ran G |-> ( M ` k ) ) )
38	37	fveq2d 6195	. 2 |- ( ph -> (Σ^ ` ( M |` ran G ) ) = (Σ^ ` ( k e. ran G |-> ( M ` k ) ) ) )
39	2	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ j e. X ) -> ( G ` j ) e. S )
40	2	feqmptd 6249	. . . . 5 |- ( ph -> G = ( j e. X |-> ( G ` j ) ) )
41	11	feqmptd 6249	. . . . 5 |- ( ph -> M = ( k e. S |-> ( M ` k ) ) )
42		fveq2 6191	. . . . 5 |- ( k = ( G ` j ) -> ( M ` k ) = ( M ` ( G ` j ) ) )
43	39, 40, 41, 42	fmptco 6396	. . . 4 |- ( ph -> ( M o. G ) = ( j e. X |-> ( M ` ( G ` j ) ) ) )
44	43	fveq2d 6195	. . 3 |- ( ph -> (Σ^ ` ( M o. G ) ) = (Σ^ ` ( j e. X |-> ( M ` ( G ` j ) ) ) ) )
45		nfv 1843	. . . . 5 |- F/ j ph
46		meadjiunlem.y	. . . . . 6 |- Y = { i e. X | ( G ` i ) =/= (/) }
47		ssrab2 3687	. . . . . . 7 |- { i e. X | ( G ` i ) =/= (/) } C_ X
48	47	a1i 11	. . . . . 6 |- ( ph -> { i e. X | ( G ` i ) =/= (/) } C_ X )
49	46, 48	syl5eqss 3649	. . . . 5 |- ( ph -> Y C_ X )
50	11	adantr 481	. . . . . 6 |- ( ( ph /\ j e. Y ) -> M : S --> ( 0 [,] +oo ) )
51	2	adantr 481	. . . . . . 7 |- ( ( ph /\ j e. Y ) -> G : X --> S )
52	49	sselda 3603	. . . . . . 7 |- ( ( ph /\ j e. Y ) -> j e. X )
53	51, 52	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ j e. Y ) -> ( G ` j ) e. S )
54	50, 53	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ j e. Y ) -> ( M ` ( G ` j ) ) e. ( 0 [,] +oo ) )
55		eldifi 3732	. . . . . . . . . . 11 |- ( j e. ( X \ Y ) -> j e. X )
56	55	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) -> j e. X )
57		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( ( G ` j ) = (/) -> ( M ` ( G ` j ) ) = ( M ` (/) ) )
58	57	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( G ` j ) = (/) ) -> ( M ` ( G ` j ) ) = ( M ` (/) ) )
59	9	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( G ` j ) = (/) ) -> M e. Meas )
60	59	mea0 40671	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( G ` j ) = (/) ) -> ( M ` (/) ) = 0 )
61	58, 60	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ph /\ ( G ` j ) = (/) ) -> ( M ` ( G ` j ) ) = 0 )
62	61	ad4ant14 1293	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) /\ ( G ` j ) = (/) ) -> ( M ` ( G ` j ) ) = 0 )
63		neneq 2800	. . . . . . . . . . . . 13 |- ( ( M ` ( G ` j ) ) =/= 0 -> -. ( M ` ( G ` j ) ) = 0 )
64	63	ad2antlr 763	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) /\ ( G ` j ) = (/) ) -> -. ( M ` ( G ` j ) ) = 0 )
65	62, 64	pm2.65da 600	. . . . . . . . . . 11 |- ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) -> -. ( G ` j ) = (/) )
66	65	neqned 2801	. . . . . . . . . 10 |- ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) -> ( G ` j ) =/= (/) )
67	56, 66	jca 554	. . . . . . . . 9 |- ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) -> ( j e. X /\ ( G ` j ) =/= (/) ) )
68		fveq2 6191	. . . . . . . . . . 11 |- ( i = j -> ( G ` i ) = ( G ` j ) )
69	68	neeq1d 2853	. . . . . . . . . 10 |- ( i = j -> ( ( G ` i ) =/= (/) <-> ( G ` j ) =/= (/) ) )
70	69	elrab 3363	. . . . . . . . 9 |- ( j e. { i e. X | ( G ` i ) =/= (/) } <-> ( j e. X /\ ( G ` j ) =/= (/) ) )
71	67, 70	sylibr 224	. . . . . . . 8 |- ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) -> j e. { i e. X | ( G ` i ) =/= (/) } )
72	71, 46	syl6eleqr 2712	. . . . . . 7 |- ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) -> j e. Y )
73		eldifn 3733	. . . . . . . 8 |- ( j e. ( X \ Y ) -> -. j e. Y )
74	73	ad2antlr 763	. . . . . . 7 |- ( ( ( ph /\ j e. ( X \ Y ) ) /\ ( M ` ( G ` j ) ) =/= 0 ) -> -. j e. Y )
75	72, 74	pm2.65da 600	. . . . . 6 |- ( ( ph /\ j e. ( X \ Y ) ) -> -. ( M ` ( G ` j ) ) =/= 0 )
76		nne 2798	. . . . . 6 |- ( -. ( M ` ( G ` j ) ) =/= 0 <-> ( M ` ( G ` j ) ) = 0 )
77	75, 76	sylib 208	. . . . 5 |- ( ( ph /\ j e. ( X \ Y ) ) -> ( M ` ( G ` j ) ) = 0 )
78	45, 3, 49, 54, 77	sge0ss 40629	. . . 4 |- ( ph -> (Σ^ ` ( j e. Y |-> ( M ` ( G ` j ) ) ) ) = (Σ^ ` ( j e. X |-> ( M ` ( G ` j ) ) ) ) )
79	78	eqcomd 2628	. . 3 |- ( ph -> (Σ^ ` ( j e. X |-> ( M ` ( G ` j ) ) ) ) = (Σ^ ` ( j e. Y |-> ( M ` ( G ` j ) ) ) ) )
80	3, 49	ssexd 4805	. . . . 5 |- ( ph -> Y e. _V )
81		nfv 1843	. . . . . . . . 9 |- F/ i ph
82		eqid 2622	. . . . . . . . 9 |- ( i e. Y |-> ( G ` i ) ) = ( i e. Y |-> ( G ` i ) )
83	2	ffnd 6046	. . . . . . . . . . . . 13 |- ( ph -> G Fn X )
84		dffn3 6054	. . . . . . . . . . . . 13 |- ( G Fn X <-> G : X --> ran G )
85	83, 84	sylib 208	. . . . . . . . . . . 12 |- ( ph -> G : X --> ran G )
86	85	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ i e. Y ) -> G : X --> ran G )
87	49	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ i e. Y ) -> i e. X )
88	86, 87	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ i e. Y ) -> ( G ` i ) e. ran G )
89	46	eleq2i 2693	. . . . . . . . . . . . . . 15 |- ( i e. Y <-> i e. { i e. X | ( G ` i ) =/= (/) } )
90		rabid 3116	. . . . . . . . . . . . . . 15 |- ( i e. { i e. X | ( G ` i ) =/= (/) } <-> ( i e. X /\ ( G ` i ) =/= (/) ) )
91	89, 90	bitri 264	. . . . . . . . . . . . . 14 |- ( i e. Y <-> ( i e. X /\ ( G ` i ) =/= (/) ) )
92	91	biimpi 206	. . . . . . . . . . . . 13 |- ( i e. Y -> ( i e. X /\ ( G ` i ) =/= (/) ) )
93	92	simprd 479	. . . . . . . . . . . 12 |- ( i e. Y -> ( G ` i ) =/= (/) )
94	93	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ i e. Y ) -> ( G ` i ) =/= (/) )
95		nelsn 4212	. . . . . . . . . . 11 |- ( ( G ` i ) =/= (/) -> -. ( G ` i ) e. { (/) } )
96	94, 95	syl 17	. . . . . . . . . 10 |- ( ( ph /\ i e. Y ) -> -. ( G ` i ) e. { (/) } )
97	88, 96	eldifd 3585	. . . . . . . . 9 |- ( ( ph /\ i e. Y ) -> ( G ` i ) e. ( ran G \ { (/) } ) )
98		meadjiunlem.dj	. . . . . . . . . 10 |- ( ph -> Disj i e. X ( G ` i ) )
99		disjss1 4626	. . . . . . . . . 10 |- ( Y C_ X -> (Disj i e. X ( G ` i ) -> Disj i e. Y ( G ` i ) ) )
100	49, 98, 99	sylc 65	. . . . . . . . 9 |- ( ph -> Disj i e. Y ( G ` i ) )
101	81, 82, 97, 94, 100	disjf1 39369	. . . . . . . 8 |- ( ph -> ( i e. Y |-> ( G ` i ) ) : Y -1-1-> ( ran G \ { (/) } ) )
102	2, 49	feqresmpt 6250	. . . . . . . . 9 |- ( ph -> ( G |` Y ) = ( i e. Y |-> ( G ` i ) ) )
103		f1eq1 6096	. . . . . . . . 9 |- ( ( G |` Y ) = ( i e. Y |-> ( G ` i ) ) -> ( ( G |` Y ) : Y -1-1-> ( ran G \ { (/) } ) <-> ( i e. Y |-> ( G ` i ) ) : Y -1-1-> ( ran G \ { (/) } ) ) )
104	102, 103	syl 17	. . . . . . . 8 |- ( ph -> ( ( G |` Y ) : Y -1-1-> ( ran G \ { (/) } ) <-> ( i e. Y |-> ( G ` i ) ) : Y -1-1-> ( ran G \ { (/) } ) ) )
105	101, 104	mpbird 247	. . . . . . 7 |- ( ph -> ( G |` Y ) : Y -1-1-> ( ran G \ { (/) } ) )
106	102	rneqd 5353	. . . . . . . . 9 |- ( ph -> ran ( G |` Y ) = ran ( i e. Y |-> ( G ` i ) ) )
107	97	ralrimiva 2966	. . . . . . . . . 10 |- ( ph -> A. i e. Y ( G ` i ) e. ( ran G \ { (/) } ) )
108	82	rnmptss 6392	. . . . . . . . . 10 |- ( A. i e. Y ( G ` i ) e. ( ran G \ { (/) } ) -> ran ( i e. Y |-> ( G ` i ) ) C_ ( ran G \ { (/) } ) )
109	107, 108	syl 17	. . . . . . . . 9 |- ( ph -> ran ( i e. Y |-> ( G ` i ) ) C_ ( ran G \ { (/) } ) )
110	106, 109	eqsstrd 3639	. . . . . . . 8 |- ( ph -> ran ( G |` Y ) C_ ( ran G \ { (/) } ) )
111		simpl 473	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ran G \ { (/) } ) ) -> ph )
112		eldifi 3732	. . . . . . . . . . . 12 |- ( x e. ( ran G \ { (/) } ) -> x e. ran G )
113	112	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ran G \ { (/) } ) ) -> x e. ran G )
114		eldifsni 4320	. . . . . . . . . . . 12 |- ( x e. ( ran G \ { (/) } ) -> x =/= (/) )
115	114	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( ran G \ { (/) } ) ) -> x =/= (/) )
116		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ran G ) -> x e. ran G )
117		fvelrnb 6243	. . . . . . . . . . . . . . . 16 |- ( G Fn X -> ( x e. ran G <-> E. i e. X ( G ` i ) = x ) )
118	83, 117	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( x e. ran G <-> E. i e. X ( G ` i ) = x ) )
119	118	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ran G ) -> ( x e. ran G <-> E. i e. X ( G ` i ) = x ) )
120	116, 119	mpbid 222	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ran G ) -> E. i e. X ( G ` i ) = x )
121	120	3adant3 1081	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ran G /\ x =/= (/) ) -> E. i e. X ( G ` i ) = x )
122		id 22	. . . . . . . . . . . . . . . . . 18 |- ( ( G ` i ) = x -> ( G ` i ) = x )
123	122	eqcomd 2628	. . . . . . . . . . . . . . . . 17 |- ( ( G ` i ) = x -> x = ( G ` i ) )
124	123	3ad2ant3 1084	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x =/= (/) ) /\ i e. X /\ ( G ` i ) = x ) -> x = ( G ` i ) )
125		simp1l 1085	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x =/= (/) ) /\ i e. X /\ ( G ` i ) = x ) -> ph )
126		simp2 1062	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x =/= (/) ) /\ i e. X /\ ( G ` i ) = x ) -> i e. X )
127		simpr 477	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x =/= (/) /\ ( G ` i ) = x ) -> ( G ` i ) = x )
128		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( x =/= (/) /\ ( G ` i ) = x ) -> x =/= (/) )
129	127, 128	eqnetrd 2861	. . . . . . . . . . . . . . . . . . 19 |- ( ( x =/= (/) /\ ( G ` i ) = x ) -> ( G ` i ) =/= (/) )
130	129	adantll 750	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ x =/= (/) ) /\ ( G ` i ) = x ) -> ( G ` i ) =/= (/) )
131	130	3adant2 1080	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x =/= (/) ) /\ i e. X /\ ( G ` i ) = x ) -> ( G ` i ) =/= (/) )
132	91	biimpri 218	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. X /\ ( G ` i ) =/= (/) ) -> i e. Y )
133		fvexd 6203	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. X /\ ( G ` i ) =/= (/) ) -> ( G ` i ) e. _V )
134	82	elrnmpt1 5374	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. Y /\ ( G ` i ) e. _V ) -> ( G ` i ) e. ran ( i e. Y |-> ( G ` i ) ) )
135	132, 133, 134	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( i e. X /\ ( G ` i ) =/= (/) ) -> ( G ` i ) e. ran ( i e. Y |-> ( G ` i ) ) )
136	135	3adant1 1079	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. X /\ ( G ` i ) =/= (/) ) -> ( G ` i ) e. ran ( i e. Y |-> ( G ` i ) ) )
137	106	eqcomd 2628	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ran ( i e. Y |-> ( G ` i ) ) = ran ( G |` Y ) )
138	137	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ i e. X /\ ( G ` i ) =/= (/) ) -> ran ( i e. Y |-> ( G ` i ) ) = ran ( G |` Y ) )
139	136, 138	eleqtrd 2703	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ i e. X /\ ( G ` i ) =/= (/) ) -> ( G ` i ) e. ran ( G |` Y ) )
140	125, 126, 131, 139	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x =/= (/) ) /\ i e. X /\ ( G ` i ) = x ) -> ( G ` i ) e. ran ( G |` Y ) )
141	124, 140	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x =/= (/) ) /\ i e. X /\ ( G ` i ) = x ) -> x e. ran ( G |` Y ) )
142	141	3exp 1264	. . . . . . . . . . . . . 14 |- ( ( ph /\ x =/= (/) ) -> ( i e. X -> ( ( G ` i ) = x -> x e. ran ( G |` Y ) ) ) )
143	142	rexlimdv 3030	. . . . . . . . . . . . 13 |- ( ( ph /\ x =/= (/) ) -> ( E. i e. X ( G ` i ) = x -> x e. ran ( G |` Y ) ) )
144	143	3adant2 1080	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ran G /\ x =/= (/) ) -> ( E. i e. X ( G ` i ) = x -> x e. ran ( G |` Y ) ) )
145	121, 144	mpd 15	. . . . . . . . . . 11 |- ( ( ph /\ x e. ran G /\ x =/= (/) ) -> x e. ran ( G |` Y ) )
146	111, 113, 115, 145	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ x e. ( ran G \ { (/) } ) ) -> x e. ran ( G |` Y ) )
147	146	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. x e. ( ran G \ { (/) } ) x e. ran ( G |` Y ) )
148		dfss3 3592	. . . . . . . . 9 |- ( ( ran G \ { (/) } ) C_ ran ( G |` Y ) <-> A. x e. ( ran G \ { (/) } ) x e. ran ( G |` Y ) )
149	147, 148	sylibr 224	. . . . . . . 8 |- ( ph -> ( ran G \ { (/) } ) C_ ran ( G |` Y ) )
150	110, 149	eqssd 3620	. . . . . . 7 |- ( ph -> ran ( G |` Y ) = ( ran G \ { (/) } ) )
151	105, 150	jca 554	. . . . . 6 |- ( ph -> ( ( G |` Y ) : Y -1-1-> ( ran G \ { (/) } ) /\ ran ( G |` Y ) = ( ran G \ { (/) } ) ) )
152		dff1o5 6146	. . . . . 6 |- ( ( G |` Y ) : Y -1-1-onto-> ( ran G \ { (/) } ) <-> ( ( G |` Y ) : Y -1-1-> ( ran G \ { (/) } ) /\ ran ( G |` Y ) = ( ran G \ { (/) } ) ) )
153	151, 152	sylibr 224	. . . . 5 |- ( ph -> ( G |` Y ) : Y -1-1-onto-> ( ran G \ { (/) } ) )
154		fvres 6207	. . . . . 6 |- ( j e. Y -> ( ( G |` Y ) ` j ) = ( G ` j ) )
155	154	adantl 482	. . . . 5 |- ( ( ph /\ j e. Y ) -> ( ( G |` Y ) ` j ) = ( G ` j ) )
156	1, 45, 42, 80, 153, 155, 18	sge0f1o 40599	. . . 4 |- ( ph -> (Σ^ ` ( k e. ( ran G \ { (/) } ) |-> ( M ` k ) ) ) = (Σ^ ` ( j e. Y |-> ( M ` ( G ` j ) ) ) ) )
157	156	eqcomd 2628	. . 3 |- ( ph -> (Σ^ ` ( j e. Y |-> ( M ` ( G ` j ) ) ) ) = (Σ^ ` ( k e. ( ran G \ { (/) } ) |-> ( M ` k ) ) ) )
158	44, 79, 157	3eqtrd 2660	. 2 |- ( ph -> (Σ^ ` ( M o. G ) ) = (Σ^ ` ( k e. ( ran G \ { (/) } ) |-> ( M ` k ) ) ) )
159	36, 38, 158	3eqtr4d 2666	1 |- ( ph -> (Σ^ ` ( M |` ran G ) ) = (Σ^ ` ( M o. G ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177  Disj wdisj 4620   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   [,]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:  meadjiun  40683