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Theorem halfnq 8809
Description: One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
halfnq  |-  ( A  e.  Q.  ->  E. x
( x  +Q  x
)  =  A )
Distinct variable group:    x, A

Proof of Theorem halfnq
StepHypRef Expression
1 distrnq 8794 . . . 4  |-  ( A  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )
2 distrnq 8794 . . . . . . . 8  |-  ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )
3 1nq 8761 . . . . . . . . . . 11  |-  1Q  e.  Q.
4 addclnq 8778 . . . . . . . . . . 11  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  e.  Q. )
53, 3, 4mp2an 654 . . . . . . . . . 10  |-  ( 1Q 
+Q  1Q )  e. 
Q.
6 recidnq 8798 . . . . . . . . . 10  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  (
( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q )
75, 6ax-mp 8 . . . . . . . . 9  |-  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q
87, 7oveq12i 6052 . . . . . . . 8  |-  ( ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
92, 8eqtri 2424 . . . . . . 7  |-  ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
109oveq1i 6050 . . . . . 6  |-  ( ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )
117oveq2i 6051 . . . . . . 7  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( ( ( *Q `  ( 1Q 
+Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  1Q )
12 mulassnq 8792 . . . . . . . 8  |-  ( ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  ( 1Q  +Q  1Q ) )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  ( ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )
13 mulcomnq 8786 . . . . . . . . 9  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  ( 1Q  +Q  1Q ) )  =  ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )
1413oveq1i 6050 . . . . . . . 8  |-  ( ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  ( 1Q  +Q  1Q ) )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  ( ( ( 1Q 
+Q  1Q )  .Q  ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )
1512, 14eqtr3i 2426 . . . . . . 7  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )
16 recclnq 8799 . . . . . . . . 9  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  ( *Q `  ( 1Q  +Q  1Q ) )  e.  Q. )
17 addclnq 8778 . . . . . . . . 9  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  e.  Q.  /\  ( *Q `  ( 1Q  +Q  1Q ) )  e.  Q. )  ->  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
Q. )
1816, 16, 17syl2anc 643 . . . . . . . 8  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  (
( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
Q. )
19 mulidnq 8796 . . . . . . . 8  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
Q.  ->  ( ( ( *Q `  ( 1Q 
+Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  1Q )  =  ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )
205, 18, 19mp2b 10 . . . . . . 7  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  1Q )  =  ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )
2111, 15, 203eqtr3i 2432 . . . . . 6  |-  ( ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )
2210, 21, 73eqtr3i 2432 . . . . 5  |-  ( ( *Q `  ( 1Q 
+Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q
2322oveq2i 6051 . . . 4  |-  ( A  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
241, 23eqtr3i 2426 . . 3  |-  ( ( A  .Q  ( *Q
`  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
25 mulidnq 8796 . . 3  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
2624, 25syl5eq 2448 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A )
27 ovex 6065 . . 3  |-  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
_V
28 oveq12 6049 . . . . 5  |-  ( ( x  =  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  /\  x  =  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  ->  (
x  +Q  x )  =  ( ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) ) )
2928anidms 627 . . . 4  |-  ( x  =  ( A  .Q  ( *Q `  ( 1Q 
+Q  1Q ) ) )  ->  ( x  +Q  x )  =  ( ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) ) )
3029eqeq1d 2412 . . 3  |-  ( x  =  ( A  .Q  ( *Q `  ( 1Q 
+Q  1Q ) ) )  ->  ( (
x  +Q  x )  =  A  <->  ( ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A ) )
3127, 30spcev 3003 . 2  |-  ( ( ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A  ->  E. x
( x  +Q  x
)  =  A )
3226, 31syl 16 1  |-  ( A  e.  Q.  ->  E. x
( x  +Q  x
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040   Q.cnq 8683   1Qc1q 8684    +Q cplq 8686    .Q cmq 8687   *Qcrq 8688
This theorem is referenced by:  nsmallnq  8810
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-mpq 8742  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-mq 8748  df-1nq 8749  df-rq 8750
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