Theorem vitalilem1OLD 23377

Description: Obsolete proof of vitalilem1 23376 as of 1-May-2021. Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
vitalilem1OLD	|- .~ Er ( 0 [,] 1 )

Distinct variable group:   x, y, .~

Proof of Theorem vitalilem1OLD

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

Step	Hyp	Ref	Expression
1		vitali.1	. . . . 5 |- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
2	1	relopabi 5245	. . . 4 |- Rel .~
3	2	a1i 11	. . 3 |- ( T. -> Rel .~ )
4		simplr 792	. . . . . 6 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> v e. ( 0 [,] 1 ) )
5		simpll 790	. . . . . 6 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> u e. ( 0 [,] 1 ) )
6		unitssre 12319	. . . . . . . . . . 11 |- ( 0 [,] 1 ) C_ RR
7	6	sseli 3599	. . . . . . . . . 10 |- ( u e. ( 0 [,] 1 ) -> u e. RR )
8	7	recnd 10068	. . . . . . . . 9 |- ( u e. ( 0 [,] 1 ) -> u e. CC )
9	8	ad2antrr 762	. . . . . . . 8 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> u e. CC )
10	6	sseli 3599	. . . . . . . . . 10 |- ( v e. ( 0 [,] 1 ) -> v e. RR )
11	10	recnd 10068	. . . . . . . . 9 |- ( v e. ( 0 [,] 1 ) -> v e. CC )
12	11	ad2antlr 763	. . . . . . . 8 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> v e. CC )
13	9, 12	negsubdi2d 10408	. . . . . . 7 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> -u ( u - v ) = ( v - u ) )
14		qnegcl 11805	. . . . . . . 8 |- ( ( u - v ) e. QQ -> -u ( u - v ) e. QQ )
15	14	adantl 482	. . . . . . 7 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> -u ( u - v ) e. QQ )
16	13, 15	eqeltrrd 2702	. . . . . 6 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> ( v - u ) e. QQ )
17	4, 5, 16	jca31 557	. . . . 5 |- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> ( ( v e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( v - u ) e. QQ ) )
18		oveq12 6659	. . . . . . 7 |- ( ( x = u /\ y = v ) -> ( x - y ) = ( u - v ) )
19	18	eleq1d 2686	. . . . . 6 |- ( ( x = u /\ y = v ) -> ( ( x - y ) e. QQ <-> ( u - v ) e. QQ ) )
20	19, 1	brab2a 5194	. . . . 5 |- ( u .~ v <-> ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) )
21		oveq12 6659	. . . . . . 7 |- ( ( x = v /\ y = u ) -> ( x - y ) = ( v - u ) )
22	21	eleq1d 2686	. . . . . 6 |- ( ( x = v /\ y = u ) -> ( ( x - y ) e. QQ <-> ( v - u ) e. QQ ) )
23	22, 1	brab2a 5194	. . . . 5 |- ( v .~ u <-> ( ( v e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( v - u ) e. QQ ) )
24	17, 20, 23	3imtr4i 281	. . . 4 |- ( u .~ v -> v .~ u )
25	24	adantl 482	. . 3 |- ( ( T. /\ u .~ v ) -> v .~ u )
26		simpl 473	. . . . . . . . 9 |- ( ( u .~ v /\ v .~ w ) -> u .~ v )
27	26, 20	sylib 208	. . . . . . . 8 |- ( ( u .~ v /\ v .~ w ) -> ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) )
28	27	simpld 475	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) )
29	28	simpld 475	. . . . . 6 |- ( ( u .~ v /\ v .~ w ) -> u e. ( 0 [,] 1 ) )
30		simpr 477	. . . . . . . . 9 |- ( ( u .~ v /\ v .~ w ) -> v .~ w )
31		oveq12 6659	. . . . . . . . . . 11 |- ( ( x = v /\ y = w ) -> ( x - y ) = ( v - w ) )
32	31	eleq1d 2686	. . . . . . . . . 10 |- ( ( x = v /\ y = w ) -> ( ( x - y ) e. QQ <-> ( v - w ) e. QQ ) )
33	32, 1	brab2a 5194	. . . . . . . . 9 |- ( v .~ w <-> ( ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( v - w ) e. QQ ) )
34	30, 33	sylib 208	. . . . . . . 8 |- ( ( u .~ v /\ v .~ w ) -> ( ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( v - w ) e. QQ ) )
35	34	simpld 475	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) )
36	35	simprd 479	. . . . . 6 |- ( ( u .~ v /\ v .~ w ) -> w e. ( 0 [,] 1 ) )
37	29, 36	jca 554	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> ( u e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) )
38	29, 8	syl 17	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> u e. CC )
39	27, 12	syl 17	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> v e. CC )
40	6, 36	sseldi 3601	. . . . . . . 8 |- ( ( u .~ v /\ v .~ w ) -> w e. RR )
41	40	recnd 10068	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> w e. CC )
42	38, 39, 41	npncand 10416	. . . . . 6 |- ( ( u .~ v /\ v .~ w ) -> ( ( u - v ) + ( v - w ) ) = ( u - w ) )
43	27	simprd 479	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> ( u - v ) e. QQ )
44	34	simprd 479	. . . . . . 7 |- ( ( u .~ v /\ v .~ w ) -> ( v - w ) e. QQ )
45		qaddcl 11804	. . . . . . 7 |- ( ( ( u - v ) e. QQ /\ ( v - w ) e. QQ ) -> ( ( u - v ) + ( v - w ) ) e. QQ )
46	43, 44, 45	syl2anc 693	. . . . . 6 |- ( ( u .~ v /\ v .~ w ) -> ( ( u - v ) + ( v - w ) ) e. QQ )
47	42, 46	eqeltrrd 2702	. . . . 5 |- ( ( u .~ v /\ v .~ w ) -> ( u - w ) e. QQ )
48		oveq12 6659	. . . . . . 7 |- ( ( x = u /\ y = w ) -> ( x - y ) = ( u - w ) )
49	48	eleq1d 2686	. . . . . 6 |- ( ( x = u /\ y = w ) -> ( ( x - y ) e. QQ <-> ( u - w ) e. QQ ) )
50	49, 1	brab2a 5194	. . . . 5 |- ( u .~ w <-> ( ( u e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( u - w ) e. QQ ) )
51	37, 47, 50	sylanbrc 698	. . . 4 |- ( ( u .~ v /\ v .~ w ) -> u .~ w )
52	51	adantl 482	. . 3 |- ( ( T. /\ ( u .~ v /\ v .~ w ) ) -> u .~ w )
53	8	subidd 10380	. . . . . . . 8 |- ( u e. ( 0 [,] 1 ) -> ( u - u ) = 0 )
54		0z 11388	. . . . . . . . 9 |- 0 e. ZZ
55		zq 11794	. . . . . . . . 9 |- ( 0 e. ZZ -> 0 e. QQ )
56	54, 55	ax-mp 5	. . . . . . . 8 |- 0 e. QQ
57	53, 56	syl6eqel 2709	. . . . . . 7 |- ( u e. ( 0 [,] 1 ) -> ( u - u ) e. QQ )
58	57	adantr 481	. . . . . 6 |- ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) -> ( u - u ) e. QQ )
59	58	pm4.71i 664	. . . . 5 |- ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) <-> ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( u - u ) e. QQ ) )
60		pm4.24 675	. . . . 5 |- ( u e. ( 0 [,] 1 ) <-> ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) )
61		oveq12 6659	. . . . . . 7 |- ( ( x = u /\ y = u ) -> ( x - y ) = ( u - u ) )
62	61	eleq1d 2686	. . . . . 6 |- ( ( x = u /\ y = u ) -> ( ( x - y ) e. QQ <-> ( u - u ) e. QQ ) )
63	62, 1	brab2a 5194	. . . . 5 |- ( u .~ u <-> ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( u - u ) e. QQ ) )
64	59, 60, 63	3bitr4i 292	. . . 4 |- ( u e. ( 0 [,] 1 ) <-> u .~ u )
65	64	a1i 11	. . 3 |- ( T. -> ( u e. ( 0 [,] 1 ) <-> u .~ u ) )
66	3, 25, 52, 65	iserd 7768	. 2 |- ( T. -> .~ Er ( 0 [,] 1 ) )
67	66	trud 1493	1 |- .~ Er ( 0 [,] 1 )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   class class class wbr 4653   {copab 4712   Rel wrel 5119  (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: (None)