Theorem vitali 22650

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 9648	. . . 4 |- RR e. _V
2	1	pwex 4584	. . 3 |- ~P RR e. _V
3		weinxp 4907	. . . . 5 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We RR )
4		unipw 4650	. . . . . 6 |- U. ~P RR = RR
5		weeq2 4828	. . . . . 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 260	. . . 4 |- ( .< We RR <-> ( .< i^i ( RR X. RR ) ) We U. ~P RR )
8	1, 1	xpex 6614	. . . . . 6 |- ( RR X. RR ) e. _V
9	8	inex2 4538	. . . . 5 |- ( .< i^i ( RR X. RR ) ) e. _V
10		weeq1 4827	. . . . 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 3127	. . . 4 |- ( ( .< i^i ( RR X. RR ) ) We U. ~P RR -> E. x x We U. ~P RR )
12	7, 11	sylbi 200	. . 3 |- ( .< We RR -> E. x x We U. ~P RR )
13		dfac8c 8482	. . 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 64	. 2 |- ( .< We RR -> E. f A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) )
15		qex 11299	. . . . . . 7 |- QQ e. _V
16	15	inex1 4537	. . . . . 6 |- ( QQ i^i ( -u 1 [,] 1 ) ) e. _V
17		nnrecq 11310	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. QQ )
18		nnrecre 10668	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) e. RR )
19		neg1rr 10736	. . . . . . . . . . 11 |- -u 1 e. RR
20	19	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 e. RR )
21		0re 9661	. . . . . . . . . . 11 |- 0 e. RR
22	21	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> 0 e. RR )
23		neg1lt0 10738	. . . . . . . . . . . 12 |- -u 1 < 0
24	19, 21, 23	ltleii 9775	. . . . . . . . . . 11 |- -u 1 <_ 0
25	24	a1i 11	. . . . . . . . . 10 |- ( x e. NN -> -u 1 <_ 0 )
26		nnrp 11334	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR+ )
27	26	rpreccld 11374	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 / x ) e. RR+ )
28	27	rpge0d 11368	. . . . . . . . . 10 |- ( x e. NN -> 0 <_ ( 1 / x ) )
29	20, 22, 18, 25, 28	letrd 9809	. . . . . . . . 9 |- ( x e. NN -> -u 1 <_ ( 1 / x ) )
30		nnge1 10657	. . . . . . . . . . 11 |- ( x e. NN -> 1 <_ x )
31		nnre 10638	. . . . . . . . . . . 12 |- ( x e. NN -> x e. RR )
32		nngt0 10660	. . . . . . . . . . . 12 |- ( x e. NN -> 0 < x )
33		1re 9660	. . . . . . . . . . . . 13 |- 1 e. RR
34		0lt1 10157	. . . . . . . . . . . . 13 |- 0 < 1
35		lerec 10511	. . . . . . . . . . . . 13 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( x e. RR /\ 0 < x ) ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
36	33, 34, 35	mpanl12 696	. . . . . . . . . . . 12 |- ( ( x e. RR /\ 0 < x ) -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
37	31, 32, 36	syl2anc 673	. . . . . . . . . . 11 |- ( x e. NN -> ( 1 <_ x <-> ( 1 / x ) <_ ( 1 / 1 ) ) )
38	30, 37	mpbid 215	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / x ) <_ ( 1 / 1 ) )
39		1div1e1 10322	. . . . . . . . . 10 |- ( 1 / 1 ) = 1
40	38, 39	syl6breq 4435	. . . . . . . . 9 |- ( x e. NN -> ( 1 / x ) <_ 1 )
41	19, 33	elicc2i 11725	. . . . . . . . 9 |- ( ( 1 / x ) e. ( -u 1 [,] 1 ) <-> ( ( 1 / x ) e. RR /\ -u 1 <_ ( 1 / x ) /\ ( 1 / x ) <_ 1 ) )
42	18, 29, 40, 41	syl3anbrc 1214	. . . . . . . 8 |- ( x e. NN -> ( 1 / x ) e. ( -u 1 [,] 1 ) )
43	17, 42	elind 3609	. . . . . . 7 |- ( x e. NN -> ( 1 / x ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
44		oveq2 6316	. . . . . . . . 9 |- ( ( 1 / x ) = ( 1 / y ) -> ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) )
45		nncn 10639	. . . . . . . . . . 11 |- ( x e. NN -> x e. CC )
46		nnne0 10664	. . . . . . . . . . 11 |- ( x e. NN -> x =/= 0 )
47	45, 46	recrecd 10402	. . . . . . . . . 10 |- ( x e. NN -> ( 1 / ( 1 / x ) ) = x )
48		nncn 10639	. . . . . . . . . . 11 |- ( y e. NN -> y e. CC )
49		nnne0 10664	. . . . . . . . . . 11 |- ( y e. NN -> y =/= 0 )
50	48, 49	recrecd 10402	. . . . . . . . . 10 |- ( y e. NN -> ( 1 / ( 1 / y ) ) = y )
51	47, 50	eqeqan12d 2487	. . . . . . . . 9 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / ( 1 / x ) ) = ( 1 / ( 1 / y ) ) <-> x = y ) )
52	44, 51	syl5ib 227	. . . . . . . 8 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) -> x = y ) )
53		oveq2 6316	. . . . . . . 8 |- ( x = y -> ( 1 / x ) = ( 1 / y ) )
54	52, 53	impbid1 208	. . . . . . 7 |- ( ( x e. NN /\ y e. NN ) -> ( ( 1 / x ) = ( 1 / y ) <-> x = y ) )
55	43, 54	dom2 7630	. . . . . 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 3643	. . . . . . 7 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ QQ
58		ssdomg 7633	. . . . . . 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 14343	. . . . . 6 |- QQ ~~ NN
61		domentr 7646	. . . . . 6 |- ( ( ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ QQ /\ QQ ~~ NN ) -> ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN )
62	59, 60, 61	mp2an 686	. . . . 5 |- ( QQ i^i ( -u 1 [,] 1 ) ) ~<_ NN
63		sbth 7710	. . . . 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 686	. . . 4 |- NN ~~ ( QQ i^i ( -u 1 [,] 1 ) )
65		bren 7596	. . . 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 213	. . 3 |- E. g g : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) )
67		eleq1 2537	. . . . . . . . . . . . 13 |- ( a = x -> ( a e. ( 0 [,] 1 ) <-> x e. ( 0 [,] 1 ) ) )
68		eleq1 2537	. . . . . . . . . . . . 13 |- ( b = y -> ( b e. ( 0 [,] 1 ) <-> y e. ( 0 [,] 1 ) ) )
69	67, 68	bi2anan9 890	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) <-> ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) ) )
70		oveq12 6317	. . . . . . . . . . . . 13 |- ( ( a = x /\ b = y ) -> ( a - b ) = ( x - y ) )
71	70	eleq1d 2533	. . . . . . . . . . . 12 |- ( ( a = x /\ b = y ) -> ( ( a - b ) e. QQ <-> ( x - y ) e. QQ ) )
72	69, 71	anbi12d 725	. . . . . . . . . . 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 4465	. . . . . . . . . 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 2471	. . . . . . . . . 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 5889	. . . . . . . . . . . 12 |- ( f ` c ) e. _V
76		eqid 2471	. . . . . . . . . . . 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 5716	. . . . . . . . . . 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 2705	. . . . . . . . . . . . . . 15 |- ( z = w -> ( z =/= (/) <-> w =/= (/) ) )
80		fveq2 5879	. . . . . . . . . . . . . . . 16 |- ( z = w -> ( f ` z ) = ( f ` w ) )
81		id 22	. . . . . . . . . . . . . . . 16 |- ( z = w -> z = w )
82	80, 81	eleq12d 2543	. . . . . . . . . . . . . . 15 |- ( z = w -> ( ( f ` z ) e. z <-> ( f ` w ) e. w ) )
83	79, 82	imbi12d 327	. . . . . . . . . . . . . 14 |- ( z = w -> ( ( z =/= (/) -> ( f ` z ) e. z ) <-> ( w =/= (/) -> ( f ` w ) e. w ) ) )
84	83	cbvralv 3005	. . . . . . . . . . . . 13 |- ( A. z e. ~P RR ( z =/= (/) -> ( f ` z ) e. z ) <-> A. w e. ~P RR ( w =/= (/) -> ( f ` w ) e. w ) )
85	73	vitalilem1 22645	. . . . . . . . . . . . . . . . . 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 7442	. . . . . . . . . . . . . . . 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 1461	. . . . . . . . . . . . . . 15 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P ( 0 [,] 1 )
89		unitssre 11805	. . . . . . . . . . . . . . . 16 |- ( 0 [,] 1 ) C_ RR
90		sspwb 4649	. . . . . . . . . . . . . . . 16 |- ( ( 0 [,] 1 ) C_ RR <-> ~P ( 0 [,] 1 ) C_ ~P RR )
91	89, 90	mpbi 213	. . . . . . . . . . . . . . 15 |- ~P ( 0 [,] 1 ) C_ ~P RR
92	88, 91	sstri 3427	. . . . . . . . . . . . . 14 |- ( ( 0 [,] 1 ) /. { <. a , b >. | ( ( a e. ( 0 [,] 1 ) /\ b e. ( 0 [,] 1 ) ) /\ ( a - b ) e. QQ ) } ) C_ ~P RR
93		ssralv 3479	. . . . . . . . . . . . . 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 200	. . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . . 16 |- ( c = w -> ( f ` c ) = ( f ` w ) )
97		fvex 5889	. . . . . . . . . . . . . . . 16 |- ( f ` w ) e. _V
98	96, 76, 97	fvmpt 5963	. . . . . . . . . . . . . . 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 2533	. . . . . . . . . . . . . 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 323	. . . . . . . . . . . . 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 2822	. . . . . . . . . . . 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 217	. . . . . . . . . . 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 741	. . . . . . . . . 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 772	. . . . . . . . . 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 6315	. . . . . . . . . . . . . 14 |- ( t = s -> ( t - ( g ` m ) ) = ( s - ( g ` m ) ) )
106	105	eleq1d 2533	. . . . . . . . . . . . 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 3030	. . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . 15 |- ( m = n -> ( g ` m ) = ( g ` n ) )
109	108	oveq2d 6324	. . . . . . . . . . . . . 14 |- ( m = n -> ( s - ( g ` m ) ) = ( s - ( g ` n ) ) )
110	109	eleq1d 2533	. . . . . . . . . . . . 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 3022	. . . . . . . . . . . 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 2517	. . . . . . . . . . 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 4488	. . . . . . . . . 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 774	. . . . . . . . . 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 22649	. . . . . . . . 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 136	. . . . . . . 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 626	. . . . . . 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 3400	. . . . . . 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 22563	. . . . . . . . . 10 |- ( x e. dom vol -> x C_ RR )
121		selpw 3949	. . . . . . . . . 10 |- ( x e. ~P RR <-> x C_ RR )
122	120, 121	sylibr 217	. . . . . . . . 9 |- ( x e. dom vol -> x e. ~P RR )
123	122	ssriv 3422	. . . . . . . 8 |- dom vol C_ ~P RR
124		ssnelpss 3530	. . . . . . . 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 200	. . . . . 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 441	. . . 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 1787	. . 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 1789	1 |- ( .< We RR -> dom vol C. ~P RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   T. wtru 1453   E.wex 1671   e. wcel 1904   =/= wne 2641   A.wral 2756   {crab 2760   _Vcvv 3031   \ cdif 3387   i^i cin 3389   C_ wss 3390   C. wpss 3391   (/)c0 3722   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395   {copab 4453   |-> cmpt 4454   We wwe 4797   X. cxp 4837   dom cdm 4839   ran crn 4840   Fn wfn 5584   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   Er wer 7378   /.cqs 7380   ~~ cen 7584   ~<_ cdom 7585   RRcr 9556   0cc0 9557   1c1 9558   < clt 9693   <_ cle 9694   - cmin 9880   -ucneg 9881   / cdiv 10291   NNcn 10631   QQcq 11287   [,]cicc 11663   volcvol 22493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-disj 4367  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-acn 8394  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-rest 15399  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cmp 20479  df-ovol 22494  df-vol 22496

This theorem is referenced by: (None)