Theorem vitali 22583

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 9635	. . . 4 |- RR e. _V
2	1	pwex 4589	. . 3 |- ~P RR e. _V
3		weinxp 4905	. . . . 5 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We RR )
4		unipw 4653	. . . . . 6 |- U. ~P RR = RR
5		weeq2 4826	. . . . . 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 256	. . . 4 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We U. ~P RR )
8	1, 1	xpex 6600	. . . . . 6 |- ( RR X. RR ) e. _V
9	8	inex2 4548	. . . . 5 |- ( .< i^i ( RR X. RR ) ) e. _V
10		weeq1 4825	. . . . 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 3143	. . . 4 |- ( ( .< i^i ( RR X. RR ) ) We U. ~P RR -> E. x x We U. ~P RR )
12	7, 11	sylbi 199	. . 3 |- ( .< We RR -> E. x x We U. ~P RR )
13		dfac8c 8469	. . 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 65	. 2 |- ( .< We RR -> E. f A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) )
15		qex 11283	. . . . . . 7 |- QQ e. _V
16	15	inex1 4547	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) e. _V
17		nnrecq 11294	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. QQ )
18		nnrecre 10653	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) e. RR )
19		neg1rr 10721	. . . . . . . . . . 11 |- -u 1 e. RR
20	19	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 e. RR )
21		0re 9648	. . . . . . . . . . 11 |- 0 e. RR
22	21	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> 0 e. RR )
23		neg1lt0 10723	. . . . . . . . . . . 12 |- -u 1 < 0
24	19, 21, 23	ltleii 9762	. . . . . . . . . . 11 |- -u 1 <_ 0
25	24	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 <_ 0 )
26		nnrp 11318	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR+ )
27	26	rpreccld 11358	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 / x ) e. RR+ )
28	27	rpge0d 11352	. . . . . . . . . 10 |- ( x e. NN -> 0 <_ ( 1 / x ) )
29	20, 22, 18, 25, 28	letrd 9797	. . . . . . . . 9 |- ( x e. NN -> -u 1 <_ ( 1 / x ) )
30		nnge1 10642	. . . . . . . . . . 11 |- ( x e. NN -> 1 <_ x )
31		nnre 10623	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR )
32		nngt0 10645	. . . . . . . . . . . 12 |- ( x e. NN -> 0 < x )
33		1re 9647	. . . . . . . . . . . . 13 |- 1 e. RR
34		0lt1 10143	. . . . . . . . . . . . 13 |- 0 < 1
35		lerec 10496	. . . . . . . . . . . . 13 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( x e. RR /\ 0 < x ) ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
36	33, 34, 35	mpanl12 689	. . . . . . . . . . . 12 |- ( ( x e. RR /\ 0 < x ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
37	31, 32, 36	syl2anc 667	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
38	30, 37	mpbid 214	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / x ) <_ ( 1 / 1 ) )
39		1div1e1 10307	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	38, 39	syl6breq 4445	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) <_ 1 )
41	19, 33	elicc2i 11707	. . . . . . . . 9 |- ( ( 1 / x ) e. ( -u 1 [,] 1 ) <-> ( ( 1 / x ) e. RR /\ -u 1 <_ ( 1 / x ) /\ ( 1 / x ) <_ 1 ) )
42	18, 29, 40, 41	syl3anbrc 1193	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. ( -u 1 [,] 1 ) )
43	17, 42	elind 3620	. . . . . . 7 |- ( x e. NN -> ( 1 / x ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
44		oveq2 6303	. . . . . . . . 9 |- ( ( 1 / x ) = ( 1 / y ) -> ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) )
45		nncn 10624	. . . . . . . . . . 11 |- ( x e. NN -> x e. CC )
46		nnne0 10649	. . . . . . . . . . 11 |- ( x e. NN -> x =/= 0 )
47	45, 46	recrecd 10387	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / ( 1 / x ) ) = x )
48		nncn 10624	. . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
49		nnne0 10649	. . . . . . . . . . 11 |- ( y e. NN -> y =/= 0 )
50	48, 49	recrecd 10387	. . . . . . . . . 10 |- ( y e. NN -> ( 1 / ( 1 / y ) ) = y )
51	47, 50	eqeqan12d 2469	. . . . . . . . 9 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) <-> x = y ) )
52	44, 51	syl5ib 223	. . . . . . . 8 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) -> x = y ) )
53		oveq2 6303	. . . . . . . 8 |- ( x = y -> ( 1 / x ) = ( 1 / y ) )
54	52, 53	impbid1 207	. . . . . . 7 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) <-> x = y ) )
55	43, 54	dom2 7617	. . . . . 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 3654	. . . . . . 7 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ
58		ssdomg 7620	. . . . . . 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 14278	. . . . . 6 |- QQ ~~ NN
61		domentr 7633	. . . . . 6 |- ( ( ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ /\ QQ ~~ NN ) -> ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN )
62	59, 60, 61	mp2an 679	. . . . 5 |- ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN
63		sbth 7697	. . . . 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 679	. . . 4 |- NN ~~ ( QQ i^i ( -u 1 [,] 1 ) )
65		bren 7583	. . . 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 212	. . 3 |- E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) )
67		eleq1 2519	. . . . . . . . . . . . 13 |- ( a = x -> ( a e. ( 0 [,] 1 ) <-> x e. ( 0 [,] 1 ) ) )
68		eleq1 2519	. . . . . . . . . . . . 13 |- ( b = y -> ( b e. ( 0 [,] 1 ) <-> y e. ( 0 [,] 1 ) ) )
69	67, 68	bi2anan9 885	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) <-> ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) )
70		oveq12 6304	. . . . . . . . . . . . 13 |- ( ( a = x /\ b = y ) -> ( a - b ) = ( x - y ) )
71	70	eleq1d 2515	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a - b ) e. QQ <-> ( x - y ) e. QQ ) )
72	69, 71	anbi12d 718	. . . . . . . . . . 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 4475	. . . . . . . . . 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 2453	. . . . . . . . . 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 5880	. . . . . . . . . . . 12 |- ( f ` c ) e. _V
76		eqid 2453	. . . . . . . . . . . 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 5711	. . . . . . . . . . 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 2688	. . . . . . . . . . . . . . 15 |- ( z = w -> ( z =/= (/) <-> w =/= (/) ) )
80		fveq2 5870	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( f ` z ) = ( f ` w ) )
81		id 22	. . . . . . . . . . . . . . . 16 |- ( z = w -> z = w )
82	80, 81	eleq12d 2525	. . . . . . . . . . . . . . 15 |- ( z = w -> ( ( f ` z ) e. z <-> ( f ` w ) e. w ) )
83	79, 82	imbi12d 322	. . . . . . . . . . . . . 14 |- ( z = w -> ( ( z =/= (/) -> ( f ` z ) e. z ) <-> ( w =/= (/) -> ( f ` w ) e. w ) ) )
84	83	cbvralv 3021	. . . . . . . . . . . . 13 |- ( A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) <-> A. w e. ~P RR ( w =/= (/) -> ( f ` w ) e. w ) )
85	73	vitalilem1 22578	. . . . . . . . . . . . . . . . . 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 7429	. . . . . . . . . . . . . . . 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 1455	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P ( 0 [,] 1 )
89		unitssre 11786	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) C_ RR
90		sspwb 4652	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,] 1 ) C_ RR <-> ~P ( 0 [,] 1 ) C_ ~P RR )
91	89, 90	mpbi 212	. . . . . . . . . . . . . . 15 |- ~P ( 0 [,] 1 ) C_ ~P RR
92	88, 91	sstri 3443	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P RR
93		ssralv 3495	. . . . . . . . . . . . . 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 199	. . . . . . . . . . . 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 5870	. . . . . . . . . . . . . . . 16 |- ( c = w -> ( f ` c ) = ( f ` w ) )
97		fvex 5880	. . . . . . . . . . . . . . . 16 |- ( f ` w ) e. _V
98	96, 76, 97	fvmpt 5953	. . . . . . . . . . . . . . 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 2515	. . . . . . . . . . . . . 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 318	. . . . . . . . . . . . 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 2820	. . . . . . . . . . . 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 216	. . . . . . . . . . 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 734	. . . . . . . . . 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 765	. . . . . . . . . 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 6302	. . . . . . . . . . . . . 14 |- ( t = s -> ( t - ( g ` m ) ) = ( s - ( g ` m ) ) )
106	105	eleq1d 2515	. . . . . . . . . . . . 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 3046	. . . . . . . . . . . 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 5870	. . . . . . . . . . . . . . 15 |- ( m = n -> ( g ` m ) = ( g ` n ) )
109	108	oveq2d 6311	. . . . . . . . . . . . . 14 |- ( m = n -> ( s - ( g ` m ) ) = ( s - ( g ` n ) ) )
110	109	eleq1d 2515	. . . . . . . . . . . . 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 3038	. . . . . . . . . . . 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 2499	. . . . . . . . . . 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 4498	. . . . . . . . . 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 767	. . . . . . . . . 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 22582	. . . . . . . . 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 135	. . . . . . . 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 620	. . . . . . 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 115	. . . . . 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 3416	. . . . . . 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 22497	. . . . . . . . . 10 |- ( x e. dom vol -> x C_ RR )
121		selpw 3960	. . . . . . . . . 10 |- ( x e. ~P RR <-> x C_ RR )
122	120, 121	sylibr 216	. . . . . . . . 9 |- ( x e. dom vol -> x e. ~P RR )
123	122	ssriv 3438	. . . . . . . 8 |- dom vol C_ ~P RR
124		ssnelpss 3831	. . . . . . . 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 199	. . . . . 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 436	. . . 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 1781	. . 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 1783	1 |- ( .< We RR -> dom vol C. ~P RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   T. wtru 1447   E.wex 1665   e. wcel 1889   =/= wne 2624   A.wral 2739   {crab 2743   _Vcvv 3047   \ cdif 3403   i^i cin 3405   C_ wss 3406   C. wpss 3407   (/)c0 3733   ~Pcpw 3953   U.cuni 4201   class class class wbr 4405   {copab 4463   |-> cmpt 4464   We wwe 4795   X. cxp 4835   dom cdm 4837   ran crn 4838   Fn wfn 5580   -1-1-onto->wf1o 5584   ` cfv 5585  (class class class)co 6295   Er wer 7365   /.cqs 7367   ~~ cen 7571   ~<_ cdom 7572   RRcr 9543   0cc0 9544   1c1 9545   < clt 9680   <_ cle 9681   - cmin 9865   -ucneg 9866   / cdiv 10276   NNcn 10616   QQcq 11271   [,]cicc 11645   volcvol 22427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cc 8870  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-disj 4377  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-omul 7192  df-er 7368  df-ec 7370  df-qs 7374  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-acn 8381  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-rlim 13565  df-sum 13765  df-rest 15333  df-topgen 15354  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-top 19933  df-bases 19934  df-topon 19935  df-cmp 20414  df-ovol 22428  df-vol 22430

This theorem is referenced by: (None)