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Theorem vitalilem1 21883
Description: Lemma for vitali 21888. (Contributed by Mario Carneiro, 16-Jun-2014.)
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.
StepHypRef Expression
1 vitali.1 . . . . 5  |-  .~  =  { <. x ,  y
>.  |  ( (
x  e.  ( 0 [,] 1 )  /\  y  e.  ( 0 [,] 1 ) )  /\  ( x  -  y )  e.  QQ ) }
21relopabi 5114 . . . 4  |-  Rel  .~
32a1i 11 . . 3  |-  ( T. 
->  Rel  .~  )
4 simplr 754 . . . . . 6  |-  ( ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ )  ->  v  e.  ( 0 [,] 1
) )
5 simpll 753 . . . . . 6  |-  ( ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ )  ->  u  e.  ( 0 [,] 1
) )
6 unitssre 11671 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  C_  RR
76sseli 3482 . . . . . . . . . 10  |-  ( u  e.  ( 0 [,] 1 )  ->  u  e.  RR )
87recnd 9620 . . . . . . . . 9  |-  ( u  e.  ( 0 [,] 1 )  ->  u  e.  CC )
98ad2antrr 725 . . . . . . . 8  |-  ( ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ )  ->  u  e.  CC )
106sseli 3482 . . . . . . . . . 10  |-  ( v  e.  ( 0 [,] 1 )  ->  v  e.  RR )
1110recnd 9620 . . . . . . . . 9  |-  ( v  e.  ( 0 [,] 1 )  ->  v  e.  CC )
1211ad2antlr 726 . . . . . . . 8  |-  ( ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ )  ->  v  e.  CC )
139, 12negsubdi2d 9947 . . . . . . 7  |-  ( ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ )  ->  -u (
u  -  v )  =  ( v  -  u ) )
14 qnegcl 11203 . . . . . . . 8  |-  ( ( u  -  v )  e.  QQ  ->  -u (
u  -  v )  e.  QQ )
1514adantl 466 . . . . . . 7  |-  ( ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ )  ->  -u (
u  -  v )  e.  QQ )
1613, 15eqeltrrd 2530 . . . . . 6  |-  ( ( ( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ )  ->  (
v  -  u )  e.  QQ )
174, 5, 16jca31 534 . . . . 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 6286 . . . . . . 7  |-  ( ( x  =  u  /\  y  =  v )  ->  ( x  -  y
)  =  ( u  -  v ) )
1918eleq1d 2510 . . . . . 6  |-  ( ( x  =  u  /\  y  =  v )  ->  ( ( x  -  y )  e.  QQ  <->  ( u  -  v )  e.  QQ ) )
2019, 1brab2ga 5061 . . . . 5  |-  ( u  .~  v  <->  ( (
u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) )  /\  ( u  -  v )  e.  QQ ) )
21 oveq12 6286 . . . . . . 7  |-  ( ( x  =  v  /\  y  =  u )  ->  ( x  -  y
)  =  ( v  -  u ) )
2221eleq1d 2510 . . . . . 6  |-  ( ( x  =  v  /\  y  =  u )  ->  ( ( x  -  y )  e.  QQ  <->  ( v  -  u )  e.  QQ ) )
2322, 1brab2ga 5061 . . . . 5  |-  ( v  .~  u  <->  ( (
v  e.  ( 0 [,] 1 )  /\  u  e.  ( 0 [,] 1 ) )  /\  ( v  -  u )  e.  QQ ) )
2417, 20, 233imtr4i 266 . . . 4  |-  ( u  .~  v  ->  v  .~  u )
2524adantl 466 . . 3  |-  ( ( T.  /\  u  .~  v )  ->  v  .~  u )
26 simpl 457 . . . . . . . . 9  |-  ( ( u  .~  v  /\  v  .~  w )  ->  u  .~  v )
2726, 20sylib 196 . . . . . . . 8  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( ( u  e.  ( 0 [,] 1
)  /\  v  e.  ( 0 [,] 1
) )  /\  (
u  -  v )  e.  QQ ) )
2827simpld 459 . . . . . . 7  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( u  e.  ( 0 [,] 1 )  /\  v  e.  ( 0 [,] 1 ) ) )
2928simpld 459 . . . . . 6  |-  ( ( u  .~  v  /\  v  .~  w )  ->  u  e.  ( 0 [,] 1 ) )
30 simpr 461 . . . . . . . . 9  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
v  .~  w )
31 oveq12 6286 . . . . . . . . . . 11  |-  ( ( x  =  v  /\  y  =  w )  ->  ( x  -  y
)  =  ( v  -  w ) )
3231eleq1d 2510 . . . . . . . . . 10  |-  ( ( x  =  v  /\  y  =  w )  ->  ( ( x  -  y )  e.  QQ  <->  ( v  -  w )  e.  QQ ) )
3332, 1brab2ga 5061 . . . . . . . . 9  |-  ( v  .~  w  <->  ( (
v  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  /\  ( v  -  w )  e.  QQ ) )
3430, 33sylib 196 . . . . . . . 8  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( ( v  e.  ( 0 [,] 1
)  /\  w  e.  ( 0 [,] 1
) )  /\  (
v  -  w )  e.  QQ ) )
3534simpld 459 . . . . . . 7  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( v  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) ) )
3635simprd 463 . . . . . 6  |-  ( ( u  .~  v  /\  v  .~  w )  ->  w  e.  ( 0 [,] 1 ) )
3729, 36jca 532 . . . . 5  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( u  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) ) )
3829, 8syl 16 . . . . . . 7  |-  ( ( u  .~  v  /\  v  .~  w )  ->  u  e.  CC )
3927, 12syl 16 . . . . . . 7  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
v  e.  CC )
406, 36sseldi 3484 . . . . . . . 8  |-  ( ( u  .~  v  /\  v  .~  w )  ->  w  e.  RR )
4140recnd 9620 . . . . . . 7  |-  ( ( u  .~  v  /\  v  .~  w )  ->  w  e.  CC )
4238, 39, 41npncand 9955 . . . . . 6  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( ( u  -  v )  +  ( v  -  w ) )  =  ( u  -  w ) )
4327simprd 463 . . . . . . 7  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( u  -  v
)  e.  QQ )
4434simprd 463 . . . . . . 7  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( v  -  w
)  e.  QQ )
45 qaddcl 11202 . . . . . . 7  |-  ( ( ( u  -  v
)  e.  QQ  /\  ( v  -  w
)  e.  QQ )  ->  ( ( u  -  v )  +  ( v  -  w
) )  e.  QQ )
4643, 44, 45syl2anc 661 . . . . . 6  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( ( u  -  v )  +  ( v  -  w ) )  e.  QQ )
4742, 46eqeltrrd 2530 . . . . 5  |-  ( ( u  .~  v  /\  v  .~  w )  -> 
( u  -  w
)  e.  QQ )
48 oveq12 6286 . . . . . . 7  |-  ( ( x  =  u  /\  y  =  w )  ->  ( x  -  y
)  =  ( u  -  w ) )
4948eleq1d 2510 . . . . . 6  |-  ( ( x  =  u  /\  y  =  w )  ->  ( ( x  -  y )  e.  QQ  <->  ( u  -  w )  e.  QQ ) )
5049, 1brab2ga 5061 . . . . 5  |-  ( u  .~  w  <->  ( (
u  e.  ( 0 [,] 1 )  /\  w  e.  ( 0 [,] 1 ) )  /\  ( u  -  w )  e.  QQ ) )
5137, 47, 50sylanbrc 664 . . . 4  |-  ( ( u  .~  v  /\  v  .~  w )  ->  u  .~  w )
5251adantl 466 . . 3  |-  ( ( T.  /\  ( u  .~  v  /\  v  .~  w ) )  ->  u  .~  w )
538subidd 9919 . . . . . . . 8  |-  ( u  e.  ( 0 [,] 1 )  ->  (
u  -  u )  =  0 )
54 0z 10876 . . . . . . . . 9  |-  0  e.  ZZ
55 zq 11192 . . . . . . . . 9  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
5654, 55ax-mp 5 . . . . . . . 8  |-  0  e.  QQ
5753, 56syl6eqel 2537 . . . . . . 7  |-  ( u  e.  ( 0 [,] 1 )  ->  (
u  -  u )  e.  QQ )
5857adantr 465 . . . . . 6  |-  ( ( u  e.  ( 0 [,] 1 )  /\  u  e.  ( 0 [,] 1 ) )  ->  ( u  -  u )  e.  QQ )
5958pm4.71i 632 . . . . 5  |-  ( ( u  e.  ( 0 [,] 1 )  /\  u  e.  ( 0 [,] 1 ) )  <-> 
( ( u  e.  ( 0 [,] 1
)  /\  u  e.  ( 0 [,] 1
) )  /\  (
u  -  u )  e.  QQ ) )
60 pm4.24 643 . . . . 5  |-  ( u  e.  ( 0 [,] 1 )  <->  ( u  e.  ( 0 [,] 1
)  /\  u  e.  ( 0 [,] 1
) ) )
61 oveq12 6286 . . . . . . 7  |-  ( ( x  =  u  /\  y  =  u )  ->  ( x  -  y
)  =  ( u  -  u ) )
6261eleq1d 2510 . . . . . 6  |-  ( ( x  =  u  /\  y  =  u )  ->  ( ( x  -  y )  e.  QQ  <->  ( u  -  u )  e.  QQ ) )
6362, 1brab2ga 5061 . . . . 5  |-  ( u  .~  u  <->  ( (
u  e.  ( 0 [,] 1 )  /\  u  e.  ( 0 [,] 1 ) )  /\  ( u  -  u )  e.  QQ ) )
6459, 60, 633bitr4i 277 . . . 4  |-  ( u  e.  ( 0 [,] 1 )  <->  u  .~  u )
6564a1i 11 . . 3  |-  ( T. 
->  ( u  e.  ( 0 [,] 1 )  <-> 
u  .~  u )
)
663, 25, 52, 65iserd 7335 . 2  |-  ( T. 
->  .~  Er  ( 0 [,] 1 ) )
6766trud 1390 1  |-  .~  Er  ( 0 [,] 1
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1381   T. wtru 1382    e. wcel 1802   class class class wbr 4433   {copab 4490   Rel wrel 4990  (class class class)co 6277    Er wer 7306   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493    - cmin 9805   -ucneg 9806   ZZcz 10865   QQcq 11186   [,]cicc 11536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-n0 10797  df-z 10866  df-q 11187  df-icc 11540
This theorem is referenced by:  vitalilem2  21884  vitalilem3  21885  vitalilem5  21887  vitali  21888
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