Theorem vitali 23382

Description: If the reals can be well-ordered, then there are non-measurable sets. The proof uses "Vitali sets", named for Giuseppe Vitali (1905). (Contributed by Mario Carneiro, 16-Jun-2014.)

Assertion

Ref	Expression
vitali	|- ( .< We RR -> dom vol C. ~P RR )

Proof of Theorem vitali

Dummy variables a b c f g m n s t w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 10027	. . . 4 |- RR e. _V
2	1	pwex 4848	. . 3 |- ~P RR e. _V
3		weinxp 5186	. . . . 5 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We RR )
4		unipw 4918	. . . . . 6 |- U. ~P RR = RR
5		weeq2 5103	. . . . . 6 |- ( U. ~P RR = RR -> ( ( .< i^i ( RR X. RR ) ) We U. ~P RR <-> ( .< i^i ( RR X. RR ) ) We RR ) )
6	4, 5	ax-mp 5	. . . . 5 |- ( ( .< i^i ( RR X. RR ) ) We U. ~P RR <-> ( .< i^i ( RR X. RR ) ) We RR )
7	3, 6	bitr4i 267	. . . 4 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We U. ~P RR )
8	1, 1	xpex 6962	. . . . . 6 |- ( RR X. RR ) e. _V
9	8	inex2 4800	. . . . 5 |- ( .< i^i ( RR X. RR ) ) e. _V
10		weeq1 5102	. . . . 5 |- ( x = ( .< i^i ( RR X. RR ) ) -> ( x We U. ~P RR <-> ( .< i^i ( RR X. RR ) ) We U. ~P RR ) )
11	9, 10	spcev 3300	. . . 4 |- ( ( .< i^i ( RR X. RR ) ) We U. ~P RR -> E. x x We U. ~P RR )
12	7, 11	sylbi 207	. . 3 |- ( .< We RR -> E. x x We U. ~P RR )
13		dfac8c 8856	. . 3 |- ( ~P RR e. _V -> ( E. x x We U. ~P RR -> E. f A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) )
14	2, 12, 13	mpsyl 68	. 2 |- ( .< We RR -> E. f A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) )
15		qex 11800	. . . . . . 7 |- QQ e. _V
16	15	inex1 4799	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) e. _V
17		nnrecq 11811	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. QQ )
18		nnrecre 11057	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) e. RR )
19		neg1rr 11125	. . . . . . . . . . 11 |- -u 1 e. RR
20	19	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 e. RR )
21		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
22	21	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> 0 e. RR )
23		neg1lt0 11127	. . . . . . . . . . . 12 |- -u 1 < 0
24	19, 21, 23	ltleii 10160	. . . . . . . . . . 11 |- -u 1 <_ 0
25	24	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 <_ 0 )
26		nnrp 11842	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR+ )
27	26	rpreccld 11882	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 / x ) e. RR+ )
28	27	rpge0d 11876	. . . . . . . . . 10 |- ( x e. NN -> 0 <_ ( 1 / x ) )
29	20, 22, 18, 25, 28	letrd 10194	. . . . . . . . 9 |- ( x e. NN -> -u 1 <_ ( 1 / x ) )
30		nnge1 11046	. . . . . . . . . . 11 |- ( x e. NN -> 1 <_ x )
31		nnre 11027	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR )
32		nngt0 11049	. . . . . . . . . . . 12 |- ( x e. NN -> 0 < x )
33		1re 10039	. . . . . . . . . . . . 13 |- 1 e. RR
34		0lt1 10550	. . . . . . . . . . . . 13 |- 0 < 1
35		lerec 10906	. . . . . . . . . . . . 13 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( x e. RR /\ 0 < x ) ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
36	33, 34, 35	mpanl12 718	. . . . . . . . . . . 12 |- ( ( x e. RR /\ 0 < x ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
37	31, 32, 36	syl2anc 693	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
38	30, 37	mpbid 222	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / x ) <_ ( 1 / 1 ) )
39		1div1e1 10717	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	38, 39	syl6breq 4694	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) <_ 1 )
41	19, 33	elicc2i 12239	. . . . . . . . 9 |- ( ( 1 / x ) e. ( -u 1 [,] 1 ) <-> ( ( 1 / x ) e. RR /\ -u 1 <_ ( 1 / x ) /\ ( 1 / x ) <_ 1 ) )
42	18, 29, 40, 41	syl3anbrc 1246	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. ( -u 1 [,] 1 ) )
43	17, 42	elind 3798	. . . . . . 7 |- ( x e. NN -> ( 1 / x ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
44		oveq2 6658	. . . . . . . . 9 |- ( ( 1 / x ) = ( 1 / y ) -> ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) )
45		nncn 11028	. . . . . . . . . . 11 |- ( x e. NN -> x e. CC )
46		nnne0 11053	. . . . . . . . . . 11 |- ( x e. NN -> x =/= 0 )
47	45, 46	recrecd 10798	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / ( 1 / x ) ) = x )
48		nncn 11028	. . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
49		nnne0 11053	. . . . . . . . . . 11 |- ( y e. NN -> y =/= 0 )
50	48, 49	recrecd 10798	. . . . . . . . . 10 |- ( y e. NN -> ( 1 / ( 1 / y ) ) = y )
51	47, 50	eqeqan12d 2638	. . . . . . . . 9 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) <-> x = y ) )
52	44, 51	syl5ib 234	. . . . . . . 8 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) -> x = y ) )
53		oveq2 6658	. . . . . . . 8 |- ( x = y -> ( 1 / x ) = ( 1 / y ) )
54	52, 53	impbid1 215	. . . . . . 7 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) <-> x = y ) )
55	43, 54	dom2 7998	. . . . . 6 |- ( ( QQ i^i ( -u 1 [,] 1 ) ) e. _V -> NN ~<_ ( QQ i^i ( -u 1 [,] 1 ) ) )
56	16, 55	ax-mp 5	. . . . 5 |- NN ~<_ ( QQ i^i ( -u 1 [,] 1 ) )
57		inss1 3833	. . . . . . 7 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ
58		ssdomg 8001	. . . . . . 7 |- ( QQ e. _V -> ( ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ -> ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ ) )
59	15, 57, 58	mp2 9	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ
60		qnnen 14942	. . . . . 6 |- QQ ~~ NN
61		domentr 8015	. . . . . 6 |- ( ( ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ /\ QQ ~~ NN ) -> ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN )
62	59, 60, 61	mp2an 708	. . . . 5 |- ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN
63		sbth 8080	. . . . 5 |- ( ( NN ~<_ ( QQ i^i ( -u 1 [,] 1 ) ) /\ ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN ) -> NN ~~ ( QQ i^i ( -u 1 [,] 1 ) ) )
64	56, 62, 63	mp2an 708	. . . 4 |- NN ~~ ( QQ i^i ( -u 1 [,] 1 ) )
65		bren 7964	. . . 4 |- ( NN ~~ ( QQ i^i ( -u 1 [,] 1 ) ) <-> E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
66	64, 65	mpbi 220	. . 3 |- E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) )
67		eleq1 2689	. . . . . . . . . . . . 13 |- ( a = x -> ( a e. ( 0 [,] 1 ) <-> x e. ( 0 [,] 1 ) ) )
68		eleq1 2689	. . . . . . . . . . . . 13 |- ( b = y -> ( b e. ( 0 [,] 1 ) <-> y e. ( 0 [,] 1 ) ) )
69	67, 68	bi2anan9 917	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) <-> ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) )
70		oveq12 6659	. . . . . . . . . . . . 13 |- ( ( a = x /\ b = y ) -> ( a - b ) = ( x - y ) )
71	70	eleq1d 2686	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a - b ) e. QQ <-> ( x - y ) e. QQ ) )
72	69, 71	anbi12d 747	. . . . . . . . . . 11 |- ( ( a = x /\ b = y ) -> ( ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) <-> ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) ) )
73	72	cbvopabv 4722	. . . . . . . . . 10 |- { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
74		eqid 2622	. . . . . . . . . 10 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) = ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } )
75		fvex 6201	. . . . . . . . . . . 12 |- ( f ` c ) e. _V
76		eqid 2622	. . . . . . . . . . . 12 |- ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) = ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) )
77	75, 76	fnmpti 6022	. . . . . . . . . . 11 |- ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) Fn ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } )
78	77	a1i 11	. . . . . . . . . 10 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ ( g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) ) ) -> ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) Fn ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) )
79		neeq1 2856	. . . . . . . . . . . . . . 15 |- ( z = w -> ( z =/= (/) <-> w =/= (/) ) )
80		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( f ` z ) = ( f ` w ) )
81		id 22	. . . . . . . . . . . . . . . 16 |- ( z = w -> z = w )
82	80, 81	eleq12d 2695	. . . . . . . . . . . . . . 15 |- ( z = w -> ( ( f ` z ) e. z <-> ( f ` w ) e. w ) )
83	79, 82	imbi12d 334	. . . . . . . . . . . . . 14 |- ( z = w -> ( ( z =/= (/) -> ( f ` z ) e. z ) <-> ( w =/= (/) -> ( f ` w ) e. w ) ) )
84	83	cbvralv 3171	. . . . . . . . . . . . 13 |- ( A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) <-> A. w e. ~P RR ( w =/= (/) -> ( f ` w ) e. w ) )
85	73	vitalilem1 23376	. . . . . . . . . . . . . . . . . 18 |- { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } Er ( 0 [,] 1 )
86	85	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( T. -> { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } Er ( 0 [,] 1 ) )
87	86	qsss 7808	. . . . . . . . . . . . . . . 16 |- ( T. -> ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P ( 0 [,] 1 ) )
88	87	trud 1493	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P ( 0 [,] 1 )
89		unitssre 12319	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) C_ RR
90		sspwb 4917	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,] 1 ) C_ RR <-> ~P ( 0 [,] 1 ) C_ ~P RR )
91	89, 90	mpbi 220	. . . . . . . . . . . . . . 15 |- ~P ( 0 [,] 1 ) C_ ~P RR
92	88, 91	sstri 3612	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P RR
93		ssralv 3666	. . . . . . . . . . . . . 14 |- ( ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P RR -> ( A. w e. ~P RR ( w =/= (/) -> ( f ` w ) e. w ) -> A. w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) ( w =/= (/) -> ( f ` w ) e. w ) ) )
94	92, 93	ax-mp 5	. . . . . . . . . . . . 13 |- ( A. w e. ~P RR ( w =/= (/) -> ( f ` w ) e. w ) -> A. w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) ( w =/= (/) -> ( f ` w ) e. w ) )
95	84, 94	sylbi 207	. . . . . . . . . . . 12 |- ( A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) -> A. w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) ( w =/= (/) -> ( f ` w ) e. w ) )
96		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( c = w -> ( f ` c ) = ( f ` w ) )
97		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( f ` w ) e. _V
98	96, 76, 97	fvmpt 6282	. . . . . . . . . . . . . . 15 |- ( w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) -> ( ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ` w ) = ( f ` w ) )
99	98	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) -> ( ( ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ` w ) e. w <-> ( f ` w ) e. w ) )
100	99	imbi2d 330	. . . . . . . . . . . . 13 |- ( w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) -> ( ( w =/= (/) -> ( ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ` w ) e. w ) <-> ( w =/= (/) -> ( f ` w ) e. w ) ) )
101	100	ralbiia 2979	. . . . . . . . . . . 12 |- ( A. w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) ( w =/= (/) -> ( ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ` w ) e. w ) <-> A. w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) ( w =/= (/) -> ( f ` w ) e. w ) )
102	95, 101	sylibr 224	. . . . . . . . . . 11 |- ( A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) -> A. w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) ( w =/= (/) -> ( ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ` w ) e. w ) )
103	102	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ ( g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) ) ) -> A. w e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) ( w =/= (/) -> ( ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ` w ) e. w ) )
104		simprl 794	. . . . . . . . . 10 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ ( g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) ) ) -> g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
105		oveq1 6657	. . . . . . . . . . . . . 14 |- ( t = s -> ( t - ( g ` m ) ) = ( s - ( g ` m ) ) )
106	105	eleq1d 2686	. . . . . . . . . . . . 13 |- ( t = s -> ( ( t - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) <-> ( s - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ) )
107	106	cbvrabv 3199	. . . . . . . . . . . 12 |- { t e. RR | ( t - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) } = { s e. RR | ( s - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) }
108		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( m = n -> ( g ` m ) = ( g ` n ) )
109	108	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( m = n -> ( s - ( g ` m ) ) = ( s - ( g ` n ) ) )
110	109	eleq1d 2686	. . . . . . . . . . . . 13 |- ( m = n -> ( ( s - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) <-> ( s - ( g ` n ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) ) )
111	110	rabbidv 3189	. . . . . . . . . . . 12 |- ( m = n -> { s e. RR | ( s - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) } = { s e. RR | ( s - ( g ` n ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) } )
112	107, 111	syl5eq 2668	. . . . . . . . . . 11 |- ( m = n -> { t e. RR | ( t - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) } = { s e. RR | ( s - ( g ` n ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) } )
113	112	cbvmptv 4750	. . . . . . . . . 10 |- ( m e. NN |-> { t e. RR | ( t - ( g ` m ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) } ) = ( n e. NN |-> { s e. RR | ( s - ( g ` n ) ) e. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) } )
114		simprr 796	. . . . . . . . . 10 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ ( g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) ) ) -> -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) )
115	73, 74, 78, 103, 104, 113, 114	vitalilem5 23381	. . . . . . . . 9 |- -. ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ ( g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) ) )
116	115	pm2.21i 116	. . . . . . . 8 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ ( g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) ) ) -> ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) )
117	116	expr 643	. . . . . . 7 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) ) -> ( -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) -> ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) ) )
118	117	pm2.18d 124	. . . . . 6 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) ) -> ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) )
119		eldif 3584	. . . . . . 7 |- ( ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) <-> ( ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ~P RR /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. dom vol ) )
120		mblss 23299	. . . . . . . . . 10 |- ( x e. dom vol -> x C_ RR )
121		selpw 4165	. . . . . . . . . 10 |- ( x e. ~P RR <-> x C_ RR )
122	120, 121	sylibr 224	. . . . . . . . 9 |- ( x e. dom vol -> x e. ~P RR )
123	122	ssriv 3607	. . . . . . . 8 |- dom vol C_ ~P RR
124		ssnelpss 3718	. . . . . . . 8 |- ( dom vol C_ ~P RR -> ( ( ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ~P RR /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. dom vol ) -> dom vol C. ~P RR ) )
125	123, 124	ax-mp 5	. . . . . . 7 |- ( ( ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ~P RR /\ -. ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. dom vol ) -> dom vol C. ~P RR )
126	119, 125	sylbi 207	. . . . . 6 |- ( ran ( c e. ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) |-> ( f ` c ) ) e. ( ~P RR \ dom vol ) -> dom vol C. ~P RR )
127	118, 126	syl 17	. . . . 5 |- ( ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) /\ g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) ) -> dom vol C. ~P RR )
128	127	ex 450	. . . 4 |- ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) -> ( g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) -> dom vol C. ~P RR ) )
129	128	exlimdv 1861	. . 3 |- ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) -> ( E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) -> dom vol C. ~P RR ) )
130	66, 129	mpi 20	. 2 |- ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) -> dom vol C. ~P RR )
131	14, 130	exlimddv 1863	1 |- ( .< We RR -> dom vol C. ~P RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   E.wex 1704   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   C. wpss 3575   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   {copab 4712   |-> cmpt 4729   We wwe 5072   X. cxp 5112   dom cdm 5114   ran crn 5115   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Er wer 7739   /.cqs 7741   ~~ cen 7952   ~<_ cdom 7953   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   QQcq 11788   [,]cicc 12178   volcvol 23232

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-cc 9257  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-cda 8990  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-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  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-rlim 14220  df-sum 14417  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234

This theorem is referenced by:  vitali2  40908