Theorem vitali 19458

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 9037	. . . 4 |- RR e. _V
2	1	pwex 4342	. . 3 |- ~P RR e. _V
3		weinxp 4904	. . . . 5 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We RR )
4		unipw 4374	. . . . . 6 |- U. ~P RR = RR
5		weeq2 4531	. . . . . 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 8	. . . . 5 |- ( ( .< i^i ( RR X. RR ) ) We U. ~P RR <-> ( .< i^i ( RR X. RR ) ) We RR )
7	3, 6	bitr4i 244	. . . 4 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We U. ~P RR )
8	1, 1	xpex 4949	. . . . . 6 |- ( RR X. RR ) e. _V
9	8	inex2 4305	. . . . 5 |- ( .< i^i ( RR X. RR ) ) e. _V
10		weeq1 4530	. . . . 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 3003	. . . 4 |- ( ( .< i^i ( RR X. RR ) ) We U. ~P RR -> E. x x We U. ~P RR )
12	7, 11	sylbi 188	. . 3 |- ( .< We RR -> E. x x We U. ~P RR )
13		dfac8c 7870	. . 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 61	. 2 |- ( .< We RR -> E. f A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) )
15		qex 10542	. . . . . . 7 |- QQ e. _V
16	15	inex1 4304	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) e. _V
17		nnrecq 10553	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. QQ )
18		nnrecre 9992	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) e. RR )
19		1re 9046	. . . . . . . . . . . 12 |- 1 e. RR
20	19	renegcli 9318	. . . . . . . . . . 11 |- -u 1 e. RR
21	20	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 e. RR )
22		0re 9047	. . . . . . . . . . 11 |- 0 e. RR
23	22	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> 0 e. RR )
24		0lt1 9506	. . . . . . . . . . . . 13 |- 0 < 1
25		lt0neg2 9491	. . . . . . . . . . . . . 14 |- ( 1 e. RR -> ( 0 < 1 <-> -u 1 < 0 ) )
26	19, 25	ax-mp 8	. . . . . . . . . . . . 13 |- ( 0 < 1 <-> -u 1 < 0 )
27	24, 26	mpbi 200	. . . . . . . . . . . 12 |- -u 1 < 0
28	20, 22, 27	ltleii 9152	. . . . . . . . . . 11 |- -u 1 <_ 0
29	28	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 <_ 0 )
30		nnrp 10577	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR+ )
31	30	rpreccld 10614	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 / x ) e. RR+ )
32	31	rpge0d 10608	. . . . . . . . . 10 |- ( x e. NN -> 0 <_ ( 1 / x ) )
33	21, 23, 18, 29, 32	letrd 9183	. . . . . . . . 9 |- ( x e. NN -> -u 1 <_ ( 1 / x ) )
34		nnge1 9982	. . . . . . . . . . 11 |- ( x e. NN -> 1 <_ x )
35		nnre 9963	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR )
36		nngt0 9985	. . . . . . . . . . . 12 |- ( x e. NN -> 0 < x )
37		lerec 9848	. . . . . . . . . . . . 13 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( x e. RR /\ 0 < x ) ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
38	19, 24, 37	mpanl12 664	. . . . . . . . . . . 12 |- ( ( x e. RR /\ 0 < x ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
39	35, 36, 38	syl2anc 643	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
40	34, 39	mpbid 202	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / x ) <_ ( 1 / 1 ) )
41		ax-1cn 9004	. . . . . . . . . . 11 |- 1 e. CC
42	41	div1i 9698	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
43	40, 42	syl6breq 4211	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) <_ 1 )
44	20, 19	elicc2i 10932	. . . . . . . . 9 |- ( ( 1 / x ) e. ( -u 1 [,] 1 ) <-> ( ( 1 / x ) e. RR /\ -u 1 <_ ( 1 / x ) /\ ( 1 / x ) <_ 1 ) )
45	18, 33, 43, 44	syl3anbrc 1138	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. ( -u 1 [,] 1 ) )
46		elin 3490	. . . . . . . 8 |- ( ( 1 / x ) e. ( QQ i^i ( -u 1 [,] 1 ) ) <-> ( ( 1 / x ) e. QQ /\ ( 1 / x ) e. ( -u 1 [,] 1 ) ) )
47	17, 45, 46	sylanbrc 646	. . . . . . 7 |- ( x e. NN -> ( 1 / x ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
48		oveq2 6048	. . . . . . . . 9 |- ( ( 1 / x ) = ( 1 / y ) -> ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) )
49		nncn 9964	. . . . . . . . . . 11 |- ( x e. NN -> x e. CC )
50		nnne0 9988	. . . . . . . . . . 11 |- ( x e. NN -> x =/= 0 )
51	49, 50	recrecd 9743	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / ( 1 / x ) ) = x )
52		nncn 9964	. . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
53		nnne0 9988	. . . . . . . . . . 11 |- ( y e. NN -> y =/= 0 )
54	52, 53	recrecd 9743	. . . . . . . . . 10 |- ( y e. NN -> ( 1 / ( 1 / y ) ) = y )
55	51, 54	eqeqan12d 2419	. . . . . . . . 9 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) <-> x = y ) )
56	48, 55	syl5ib 211	. . . . . . . 8 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) -> x = y ) )
57		oveq2 6048	. . . . . . . 8 |- ( x = y -> ( 1 / x ) = ( 1 / y ) )
58	56, 57	impbid1 195	. . . . . . 7 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) <-> x = y ) )
59	47, 58	dom2 7109	. . . . . 6 |- ( ( QQ i^i ( -u 1 [,] 1 ) ) e. _V -> NN ~<_ ( QQ i^i ( -u 1 [,] 1 ) ) )
60	16, 59	ax-mp 8	. . . . 5 |- NN ~<_ ( QQ i^i ( -u 1 [,] 1 ) )
61		inss1 3521	. . . . . . 7 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ
62		ssdomg 7112	. . . . . . 7 |- ( QQ e. _V -> ( ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ -> ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ ) )
63	15, 61, 62	mp2 9	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ
64		qnnen 12768	. . . . . 6 |- QQ ~~ NN
65		domentr 7125	. . . . . 6 |- ( ( ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ /\ QQ ~~ NN ) -> ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN )
66	63, 64, 65	mp2an 654	. . . . 5 |- ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN
67		sbth 7186	. . . . 5 |- ( ( NN ~<_ ( QQ i^i ( -u 1 [,] 1 ) ) /\ ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN ) -> NN ~~ ( QQ i^i ( -u 1 [,] 1 ) ) )
68	60, 66, 67	mp2an 654	. . . 4 |- NN ~~ ( QQ i^i ( -u 1 [,] 1 ) )
69		bren 7076	. . . 4 |- ( NN ~~ ( QQ i^i ( -u 1 [,] 1 ) ) <-> E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
70	68, 69	mpbi 200	. . 3 |- E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) )
71		eleq1 2464	. . . . . . . . . . . . 13 |- ( a = x -> ( a e. ( 0 [,] 1 ) <-> x e. ( 0 [,] 1 ) ) )
72		eleq1 2464	. . . . . . . . . . . . 13 |- ( b = y -> ( b e. ( 0 [,] 1 ) <-> y e. ( 0 [,] 1 ) ) )
73	71, 72	bi2anan9 844	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) <-> ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) )
74		oveq12 6049	. . . . . . . . . . . . 13 |- ( ( a = x /\ b = y ) -> ( a - b ) = ( x - y ) )
75	74	eleq1d 2470	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a - b ) e. QQ <-> ( x - y ) e. QQ ) )
76	73, 75	anbi12d 692	. . . . . . . . . . 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 ) ) )
77	76	cbvopabv 4237	. . . . . . . . . 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 ) }
78		eqid 2404	. . . . . . . . . 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 ) } )
79		fvex 5701	. . . . . . . . . . . 12 |- ( f ` c ) e. _V
80		eqid 2404	. . . . . . . . . . . 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 ) )
81	79, 80	fnmpti 5532	. . . . . . . . . . 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 ) } )
82	81	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 ) } ) )
83		neeq1 2575	. . . . . . . . . . . . . . 15 |- ( z = w -> ( z =/= (/) <-> w =/= (/) ) )
84		fveq2 5687	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( f ` z ) = ( f ` w ) )
85		id 20	. . . . . . . . . . . . . . . 16 |- ( z = w -> z = w )
86	84, 85	eleq12d 2472	. . . . . . . . . . . . . . 15 |- ( z = w -> ( ( f ` z ) e. z <-> ( f ` w ) e. w ) )
87	83, 86	imbi12d 312	. . . . . . . . . . . . . 14 |- ( z = w -> ( ( z =/= (/) -> ( f ` z ) e. z ) <-> ( w =/= (/) -> ( f ` w ) e. w ) ) )
88	87	cbvralv 2892	. . . . . . . . . . . . 13 |- ( A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) <-> A. w e. ~P RR ( w =/= (/) -> ( f ` w ) e. w ) )
89	77	vitalilem1 19453	. . . . . . . . . . . . . . . . . 18 |- { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } Er ( 0 [,] 1 )
90	89	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( T. -> { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } Er ( 0 [,] 1 ) )
91	90	qsss 6924	. . . . . . . . . . . . . . . 16 |- ( T. -> ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P ( 0 [,] 1 ) )
92	91	trud 1329	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P ( 0 [,] 1 )
93		unitssre 10998	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) C_ RR
94		sspwb 4373	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,] 1 ) C_ RR <-> ~P ( 0 [,] 1 ) C_ ~P RR )
95	93, 94	mpbi 200	. . . . . . . . . . . . . . 15 |- ~P ( 0 [,] 1 ) C_ ~P RR
96	92, 95	sstri 3317	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P RR
97		ssralv 3367	. . . . . . . . . . . . . 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 ) ) )
98	96, 97	ax-mp 8	. . . . . . . . . . . . 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 ) )
99	88, 98	sylbi 188	. . . . . . . . . . . 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 ) )
100		fveq2 5687	. . . . . . . . . . . . . . . 16 |- ( c = w -> ( f ` c ) = ( f ` w ) )
101		fvex 5701	. . . . . . . . . . . . . . . 16 |- ( f ` w ) e. _V
102	100, 80, 101	fvmpt 5765	. . . . . . . . . . . . . . 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 ) )
103	102	eleq1d 2470	. . . . . . . . . . . . . 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 ) )
104	103	imbi2d 308	. . . . . . . . . . . . 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 ) ) )
105	104	ralbiia 2698	. . . . . . . . . . . 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 ) )
106	99, 105	sylibr 204	. . . . . . . . . . 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 ) )
107	106	ad2antlr 708	. . . . . . . . . 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 ) )
108		simprl 733	. . . . . . . . . 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 ) ) )
109		oveq1 6047	. . . . . . . . . . . . . 14 |- ( t = s -> ( t - ( g ` m ) ) = ( s - ( g ` m ) ) )
110	109	eleq1d 2470	. . . . . . . . . . . . 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 ) ) ) )
111	110	cbvrabv 2915	. . . . . . . . . . . 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 ) ) }
112		fveq2 5687	. . . . . . . . . . . . . . 15 |- ( m = n -> ( g ` m ) = ( g ` n ) )
113	112	oveq2d 6056	. . . . . . . . . . . . . 14 |- ( m = n -> ( s - ( g ` m ) ) = ( s - ( g ` n ) ) )
114	113	eleq1d 2470	. . . . . . . . . . . . 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 ) ) ) )
115	114	rabbidv 2908	. . . . . . . . . . . 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 ) ) } )
116	111, 115	syl5eq 2448	. . . . . . . . . . 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 ) ) } )
117	116	cbvmptv 4260	. . . . . . . . . 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 ) ) } )
118		simprr 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 ) ) ) -> -. 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	77, 78, 82, 107, 108, 117, 118	vitalilem5 19457	. . . . . . . . 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 ) ) )
120	119	pm2.21i 125	. . . . . . . 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 ) )
121	120	expr 599	. . . . . . 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 ) ) )
122	121	pm2.18d 105	. . . . . 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 ) )
123		eldif 3290	. . . . . . 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 ) )
124		mblss 19380	. . . . . . . . . 10 |- ( x e. dom vol -> x C_ RR )
125		vex 2919	. . . . . . . . . . 11 |- x e. _V
126	125	elpw 3765	. . . . . . . . . 10 |- ( x e. ~P RR <-> x C_ RR )
127	124, 126	sylibr 204	. . . . . . . . 9 |- ( x e. dom vol -> x e. ~P RR )
128	127	ssriv 3312	. . . . . . . 8 |- dom vol C_ ~P RR
129		ssnelpss 3651	. . . . . . . 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 ) )
130	128, 129	ax-mp 8	. . . . . . 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 )
131	123, 130	sylbi 188	. . . . . 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 )
132	122, 131	syl 16	. . . . 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 )
133	132	ex 424	. . . 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 ) )
134	133	exlimdv 1643	. . 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 ) )
135	70, 134	mpi 17	. 2 |- ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) -> dom vol C. ~P RR )
136	14, 135	exlimddv 1645	1 |- ( .< We RR -> dom vol C. ~P RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   /\ wa 359   T. wtru 1322   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   {crab 2670   _Vcvv 2916   \ cdif 3277   i^i cin 3279   C_ wss 3280   C. wpss 3281   (/)c0 3588   ~Pcpw 3759   U.cuni 3975   class class class wbr 4172   {copab 4225   e. cmpt 4226   We wwe 4500   X. cxp 4835   dom cdm 4837   ran crn 4838   Fn wfn 5408   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Er wer 6861   /.cqs 6863   ~~ cen 7065   ~<_ cdom 7066   RRcr 8945   0cc0 8946   1c1 8947   < clt 9076   <_ cle 9077   - cmin 9247   -ucneg 9248   / cdiv 9633   NNcn 9956   QQcq 10530   [,]cicc 10875   volcvol 19313

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-rest 13605  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cmp 17404  df-ovol 19314  df-vol 19315