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Theorem halfnq 8616
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 8601 . . . 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 8601 . . . . . . . 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 8568 . . . . . . . . . . 11  |-  1Q  e.  Q.
4 addclnq 8585 . . . . . . . . . . 11  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  e.  Q. )
53, 3, 4mp2an 653 . . . . . . . . . 10  |-  ( 1Q 
+Q  1Q )  e. 
Q.
6 recidnq 8605 . . . . . . . . . 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 5886 . . . . . . . 8  |-  ( ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
92, 8eqtri 2316 . . . . . . 7  |-  ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
109oveq1i 5884 . . . . . 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 5885 . . . . . . 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 8599 . . . . . . . 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 8593 . . . . . . . . 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 5884 . . . . . . . 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 2318 . . . . . . 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 8606 . . . . . . . . 9  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  ( *Q `  ( 1Q  +Q  1Q ) )  e.  Q. )
17 addclnq 8585 . . . . . . . . 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 642 . . . . . . . 8  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  (
( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
Q. )
19 mulidnq 8603 . . . . . . . 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 9 . . . . . . 7  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  1Q )  =  ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )
2111, 15, 203eqtr3i 2324 . . . . . 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 2324 . . . . 5  |-  ( ( *Q `  ( 1Q 
+Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q
2322oveq2i 5885 . . . 4  |-  ( A  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
241, 23eqtr3i 2318 . . 3  |-  ( ( A  .Q  ( *Q
`  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
25 mulidnq 8603 . . 3  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
2624, 25syl5eq 2340 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A )
27 ovex 5899 . . 3  |-  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
_V
28 oveq12 5883 . . . . 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 626 . . . 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 2304 . . 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 2888 . 2  |-  ( ( ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A  ->  E. x
( x  +Q  x
)  =  A )
3226, 31syl 15 1  |-  ( A  e.  Q.  ->  E. x
( x  +Q  x
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Q.cnq 8490   1Qc1q 8491    +Q cplq 8493    .Q cmq 8494   *Qcrq 8495
This theorem is referenced by:  nsmallnq  8617
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-ni 8512  df-pli 8513  df-mi 8514  df-lti 8515  df-plpq 8548  df-mpq 8549  df-enq 8551  df-nq 8552  df-erq 8553  df-plq 8554  df-mq 8555  df-1nq 8556  df-rq 8557
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