Theorem vitalilem1 23376

Description: Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof shortened by AV, 1-May-2021.)

Hypothesis

Ref	Expression
vitali.1	|- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }

Assertion

Ref	Expression
vitalilem1	|- .~ Er ( 0 [,] 1 )

Distinct variable group:   x, y, .~

Proof of Theorem vitalilem1

Dummy variables v w u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vitali.1	. . 3 |- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
2	1	relopabi 5245	. 2 |- Rel .~
3		simplr 792	. . . 4 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> v e. ( 0 [,] 1 ) )
4		simpll 790	. . . 4 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> u e. ( 0 [,] 1 ) )
5		unitssre 12319	. . . . . . . . 9 |- ( 0 [,] 1 ) C_ RR
6	5	sseli 3599	. . . . . . . 8 |- ( u e. ( 0 [,] 1 ) -> u e. RR )
7	6	recnd 10068	. . . . . . 7 |- ( u e. ( 0 [,] 1 ) -> u e. CC )
8	7	ad2antrr 762	. . . . . 6 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> u e. CC )
9	5	sseli 3599	. . . . . . . 8 |- ( v e. ( 0 [,] 1 ) -> v e. RR )
10	9	recnd 10068	. . . . . . 7 |- ( v e. ( 0 [,] 1 ) -> v e. CC )
11	10	ad2antlr 763	. . . . . 6 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> v e. CC )
12	8, 11	negsubdi2d 10408	. . . . 5 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> -u ( u - v ) = ( v - u ) )
13		qnegcl 11805	. . . . . 6 |- ( ( u - v ) e. QQ -> -u ( u - v ) e. QQ )
14	13	adantl 482	. . . . 5 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> -u ( u - v ) e. QQ )
15	12, 14	eqeltrrd 2702	. . . 4 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> ( v - u ) e. QQ )
16	3, 4, 15	jca31 557	. . 3 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> ( ( v e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( v - u ) e. QQ ) )
17		oveq12 6659	. . . . 5 |- ( ( x = u /\ y = v ) -> ( x - y ) = ( u - v ) )
18	17	eleq1d 2686	. . . 4 |- ( ( x = u /\ y = v ) -> ( ( x - y ) e. QQ <-> ( u - v ) e. QQ ) )
19	18, 1	brab2a 5194	. . 3 |- ( u .~ v <-> ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) )
20		oveq12 6659	. . . . 5 |- ( ( x = v /\ y = u ) -> ( x - y ) = ( v - u ) )
21	20	eleq1d 2686	. . . 4 |- ( ( x = v /\ y = u ) -> ( ( x - y ) e. QQ <-> ( v - u ) e. QQ ) )
22	21, 1	brab2a 5194	. . 3 |- ( v .~ u <-> ( ( v e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( v - u ) e. QQ ) )
23	16, 19, 22	3imtr4i 281	. 2 |- ( u .~ v -> v .~ u )
24		simpl 473	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> u .~ v )
25	24, 19	sylib 208	. . . . . 6 |- ( ( u .~ v /\ v .~ w ) -> ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) )
26	25	simpld 475	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) )
27	26	simpld 475	. . . 4 |- ( ( u .~ v /\ v .~ w ) -> u e. ( 0 [,] 1 ) )
28		simpr 477	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> v .~ w )
29		oveq12 6659	. . . . . . . . 9 |- ( ( x = v /\ y = w ) -> ( x - y ) = ( v - w ) )
30	29	eleq1d 2686	. . . . . . . 8 |- ( ( x = v /\ y = w ) -> ( ( x - y ) e. QQ <-> ( v - w ) e. QQ ) )
31	30, 1	brab2a 5194	. . . . . . 7 |- ( v .~ w <-> ( ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( v - w ) e. QQ ) )
32	28, 31	sylib 208	. . . . . 6 |- ( ( u .~ v /\ v .~ w ) -> ( ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( v - w ) e. QQ ) )
33	32	simpld 475	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) )
34	33	simprd 479	. . . 4 |- ( ( u .~ v /\ v .~ w ) -> w e. ( 0 [,] 1 ) )
35	27, 34	jca 554	. . 3 |- ( ( u .~ v /\ v .~ w ) -> ( u e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) )
36	27, 7	syl 17	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> u e. CC )
37	25, 11	syl 17	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> v e. CC )
38	5, 34	sseldi 3601	. . . . . 6 |- ( ( u .~ v /\ v .~ w ) -> w e. RR )
39	38	recnd 10068	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> w e. CC )
40	36, 37, 39	npncand 10416	. . . 4 |- ( ( u .~ v /\ v .~ w ) -> ( ( u - v ) + ( v - w ) ) = ( u - w ) )
41	25	simprd 479	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> ( u - v ) e. QQ )
42	32	simprd 479	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> ( v - w ) e. QQ )
43		qaddcl 11804	. . . . 5 |- ( ( ( u - v ) e. QQ /\ ( v - w ) e. QQ ) -> ( ( u - v ) + ( v - w ) ) e. QQ )
44	41, 42, 43	syl2anc 693	. . . 4 |- ( ( u .~ v /\ v .~ w ) -> ( ( u - v ) + ( v - w ) ) e. QQ )
45	40, 44	eqeltrrd 2702	. . 3 |- ( ( u .~ v /\ v .~ w ) -> ( u - w ) e. QQ )
46		oveq12 6659	. . . . 5 |- ( ( x = u /\ y = w ) -> ( x - y ) = ( u - w ) )
47	46	eleq1d 2686	. . . 4 |- ( ( x = u /\ y = w ) -> ( ( x - y ) e. QQ <-> ( u - w ) e. QQ ) )
48	47, 1	brab2a 5194	. . 3 |- ( u .~ w <-> ( ( u e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( u - w ) e. QQ ) )
49	35, 45, 48	sylanbrc 698	. 2 |- ( ( u .~ v /\ v .~ w ) -> u .~ w )
50	7	subidd 10380	. . . . . 6 |- ( u e. ( 0 [,] 1 ) -> ( u - u ) = 0 )
51		0z 11388	. . . . . . 7 |- 0 e. ZZ
52		zq 11794	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
53	51, 52	ax-mp 5	. . . . . 6 |- 0 e. QQ
54	50, 53	syl6eqel 2709	. . . . 5 |- ( u e. ( 0 [,] 1 ) -> ( u - u ) e. QQ )
55	54	adantr 481	. . . 4 |- ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) -> ( u - u ) e. QQ )
56	55	pm4.71i 664	. . 3 |- ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) <-> ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( u - u ) e. QQ ) )
57		pm4.24 675	. . 3 |- ( u e. ( 0 [,] 1 ) <-> ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) )
58		oveq12 6659	. . . . 5 |- ( ( x = u /\ y = u ) -> ( x - y ) = ( u - u ) )
59	58	eleq1d 2686	. . . 4 |- ( ( x = u /\ y = u ) -> ( ( x - y ) e. QQ <-> ( u - u ) e. QQ ) )
60	59, 1	brab2a 5194	. . 3 |- ( u .~ u <-> ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( u - u ) e. QQ ) )
61	56, 57, 60	3bitr4i 292	. 2 |- ( u e. ( 0 [,] 1 ) <-> u .~ u )
62	2, 23, 49, 61	iseri 7769	1 |- .~ Er ( 0 [,] 1 )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   {copab 4712  (class class class)co 6650   Er wer 7739   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   -ucneg 10267   ZZcz 11377   QQcq 11788   [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-q 11789  df-icc 12182

This theorem is referenced by:  vitalilem2  23378  vitalilem3  23379  vitalilem5  23381  vitali  23382