Theorem vitali 22148

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 9600	. . . 4 |- RR e. _V
2	1	pwex 4639	. . 3 |- ~P RR e. _V
3		weinxp 5076	. . . . 5 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We RR )
4		unipw 4706	. . . . . 6 |- U. ~P RR = RR
5		weeq2 4877	. . . . . 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 252	. . . 4 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We U. ~P RR )
8	1, 1	xpex 6603	. . . . . 6 |- ( RR X. RR ) e. _V
9	8	inex2 4598	. . . . 5 |- ( .< i^i ( RR X. RR ) ) e. _V
10		weeq1 4876	. . . . 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 3201	. . . 4 |- ( ( .< i^i ( RR X. RR ) ) We U. ~P RR -> E. x x We U. ~P RR )
12	7, 11	sylbi 195	. . 3 |- ( .< We RR -> E. x x We U. ~P RR )
13		dfac8c 8431	. . 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 63	. 2 |- ( .< We RR -> E. f A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) )
15		qex 11219	. . . . . . 7 |- QQ e. _V
16	15	inex1 4597	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) e. _V
17		nnrecq 11230	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. QQ )
18		nnrecre 10593	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) e. RR )
19		neg1rr 10661	. . . . . . . . . . 11 |- -u 1 e. RR
20	19	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 e. RR )
21		0re 9613	. . . . . . . . . . 11 |- 0 e. RR
22	21	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> 0 e. RR )
23		neg1lt0 10663	. . . . . . . . . . . 12 |- -u 1 < 0
24	19, 21, 23	ltleii 9724	. . . . . . . . . . 11 |- -u 1 <_ 0
25	24	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 <_ 0 )
26		nnrp 11254	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR+ )
27	26	rpreccld 11291	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 / x ) e. RR+ )
28	27	rpge0d 11285	. . . . . . . . . 10 |- ( x e. NN -> 0 <_ ( 1 / x ) )
29	20, 22, 18, 25, 28	letrd 9756	. . . . . . . . 9 |- ( x e. NN -> -u 1 <_ ( 1 / x ) )
30		nnge1 10582	. . . . . . . . . . 11 |- ( x e. NN -> 1 <_ x )
31		nnre 10563	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR )
32		nngt0 10585	. . . . . . . . . . . 12 |- ( x e. NN -> 0 < x )
33		1re 9612	. . . . . . . . . . . . 13 |- 1 e. RR
34		0lt1 10096	. . . . . . . . . . . . 13 |- 0 < 1
35		lerec 10447	. . . . . . . . . . . . 13 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( x e. RR /\ 0 < x ) ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
36	33, 34, 35	mpanl12 682	. . . . . . . . . . . 12 |- ( ( x e. RR /\ 0 < x ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
37	31, 32, 36	syl2anc 661	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
38	30, 37	mpbid 210	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / x ) <_ ( 1 / 1 ) )
39		1div1e1 10258	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	38, 39	syl6breq 4495	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) <_ 1 )
41	19, 33	elicc2i 11615	. . . . . . . . 9 |- ( ( 1 / x ) e. ( -u 1 [,] 1 ) <-> ( ( 1 / x ) e. RR /\ -u 1 <_ ( 1 / x ) /\ ( 1 / x ) <_ 1 ) )
42	18, 29, 40, 41	syl3anbrc 1180	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. ( -u 1 [,] 1 ) )
43	17, 42	elind 3684	. . . . . . 7 |- ( x e. NN -> ( 1 / x ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
44		oveq2 6304	. . . . . . . . 9 |- ( ( 1 / x ) = ( 1 / y ) -> ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) )
45		nncn 10564	. . . . . . . . . . 11 |- ( x e. NN -> x e. CC )
46		nnne0 10589	. . . . . . . . . . 11 |- ( x e. NN -> x =/= 0 )
47	45, 46	recrecd 10338	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / ( 1 / x ) ) = x )
48		nncn 10564	. . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
49		nnne0 10589	. . . . . . . . . . 11 |- ( y e. NN -> y =/= 0 )
50	48, 49	recrecd 10338	. . . . . . . . . 10 |- ( y e. NN -> ( 1 / ( 1 / y ) ) = y )
51	47, 50	eqeqan12d 2480	. . . . . . . . 9 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) <-> x = y ) )
52	44, 51	syl5ib 219	. . . . . . . 8 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) -> x = y ) )
53		oveq2 6304	. . . . . . . 8 |- ( x = y -> ( 1 / x ) = ( 1 / y ) )
54	52, 53	impbid1 203	. . . . . . 7 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) <-> x = y ) )
55	43, 54	dom2 7577	. . . . . 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 3714	. . . . . . 7 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ
58		ssdomg 7580	. . . . . . 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 13959	. . . . . 6 |- QQ ~~ NN
61		domentr 7593	. . . . . 6 |- ( ( ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ /\ QQ ~~ NN ) -> ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN )
62	59, 60, 61	mp2an 672	. . . . 5 |- ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN
63		sbth 7656	. . . . 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 672	. . . 4 |- NN ~~ ( QQ i^i ( -u 1 [,] 1 ) )
65		bren 7544	. . . 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 208	. . 3 |- E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) )
67		eleq1 2529	. . . . . . . . . . . . 13 |- ( a = x -> ( a e. ( 0 [,] 1 ) <-> x e. ( 0 [,] 1 ) ) )
68		eleq1 2529	. . . . . . . . . . . . 13 |- ( b = y -> ( b e. ( 0 [,] 1 ) <-> y e. ( 0 [,] 1 ) ) )
69	67, 68	bi2anan9 873	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) <-> ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) )
70		oveq12 6305	. . . . . . . . . . . . 13 |- ( ( a = x /\ b = y ) -> ( a - b ) = ( x - y ) )
71	70	eleq1d 2526	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a - b ) e. QQ <-> ( x - y ) e. QQ ) )
72	69, 71	anbi12d 710	. . . . . . . . . . 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 4526	. . . . . . . . . 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 2457	. . . . . . . . . 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 5882	. . . . . . . . . . . 12 |- ( f ` c ) e. _V
76		eqid 2457	. . . . . . . . . . . 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 5715	. . . . . . . . . . 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 2738	. . . . . . . . . . . . . . 15 |- ( z = w -> ( z =/= (/) <-> w =/= (/) ) )
80		fveq2 5872	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( f ` z ) = ( f ` w ) )
81		id 22	. . . . . . . . . . . . . . . 16 |- ( z = w -> z = w )
82	80, 81	eleq12d 2539	. . . . . . . . . . . . . . 15 |- ( z = w -> ( ( f ` z ) e. z <-> ( f ` w ) e. w ) )
83	79, 82	imbi12d 320	. . . . . . . . . . . . . 14 |- ( z = w -> ( ( z =/= (/) -> ( f ` z ) e. z ) <-> ( w =/= (/) -> ( f ` w ) e. w ) ) )
84	83	cbvralv 3084	. . . . . . . . . . . . 13 |- ( A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) <-> A. w e. ~P RR ( w =/= (/) -> ( f ` w ) e. w ) )
85	73	vitalilem1 22143	. . . . . . . . . . . . . . . . . 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 7390	. . . . . . . . . . . . . . . 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 1404	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P ( 0 [,] 1 )
89		unitssre 11692	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) C_ RR
90		sspwb 4705	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,] 1 ) C_ RR <-> ~P ( 0 [,] 1 ) C_ ~P RR )
91	89, 90	mpbi 208	. . . . . . . . . . . . . . 15 |- ~P ( 0 [,] 1 ) C_ ~P RR
92	88, 91	sstri 3508	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P RR
93		ssralv 3560	. . . . . . . . . . . . . 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 195	. . . . . . . . . . . 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 5872	. . . . . . . . . . . . . . . 16 |- ( c = w -> ( f ` c ) = ( f ` w ) )
97		fvex 5882	. . . . . . . . . . . . . . . 16 |- ( f ` w ) e. _V
98	96, 76, 97	fvmpt 5956	. . . . . . . . . . . . . . 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 2526	. . . . . . . . . . . . . 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 316	. . . . . . . . . . . . 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 2887	. . . . . . . . . . . 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 212	. . . . . . . . . . 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 726	. . . . . . . . . 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 756	. . . . . . . . . 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 6303	. . . . . . . . . . . . . 14 |- ( t = s -> ( t - ( g ` m ) ) = ( s - ( g ` m ) ) )
106	105	eleq1d 2526	. . . . . . . . . . . . 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 3108	. . . . . . . . . . . 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 5872	. . . . . . . . . . . . . . 15 |- ( m = n -> ( g ` m ) = ( g ` n ) )
109	108	oveq2d 6312	. . . . . . . . . . . . . 14 |- ( m = n -> ( s - ( g ` m ) ) = ( s - ( g ` n ) ) )
110	109	eleq1d 2526	. . . . . . . . . . . . 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 3101	. . . . . . . . . . . 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 2510	. . . . . . . . . . 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 4548	. . . . . . . . . 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 757	. . . . . . . . . 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 22147	. . . . . . . . 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 131	. . . . . . . 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 615	. . . . . . 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 111	. . . . . 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 3481	. . . . . . 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 22068	. . . . . . . . . 10 |- ( x e. dom vol -> x C_ RR )
121		selpw 4022	. . . . . . . . . 10 |- ( x e. ~P RR <-> x C_ RR )
122	120, 121	sylibr 212	. . . . . . . . 9 |- ( x e. dom vol -> x e. ~P RR )
123	122	ssriv 3503	. . . . . . . 8 |- dom vol C_ ~P RR
124		ssnelpss 3893	. . . . . . . 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 195	. . . . . 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 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 )
128	127	ex 434	. . . 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 1725	. . 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 17	. 2 |- ( ( .< We RR /\ A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) ) -> dom vol C. ~P RR )
131	14, 130	exlimddv 1727	1 |- ( .< We RR -> dom vol C. ~P RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   T. wtru 1396   E.wex 1613   e. wcel 1819   =/= wne 2652   A.wral 2807   {crab 2811   _Vcvv 3109   \ cdif 3468   i^i cin 3470   C_ wss 3471   C. wpss 3472   (/)c0 3793   ~Pcpw 4015   U.cuni 4251   class class class wbr 4456   {copab 4514   |-> cmpt 4515   We wwe 4846   X. cxp 5006   dom cdm 5008   ran crn 5009   Fn wfn 5589   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   Er wer 7326   /.cqs 7328   ~~ cen 7532   ~<_ cdom 7533   RRcr 9508   0cc0 9509   1c1 9510   < clt 9645   <_ cle 9646   - cmin 9824   -ucneg 9825   / cdiv 10227   NNcn 10556   QQcq 11207   [,]cicc 11557   volcvol 22001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-disj 4428  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-er 7329  df-ec 7331  df-qs 7335  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-acn 8340  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-rest 14840  df-topgen 14861  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-top 19526  df-bases 19528  df-topon 19529  df-cmp 20014  df-ovol 22002  df-vol 22003

This theorem is referenced by: (None)