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Theorem halfnq 8533
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 8518 . . . 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 8518 . . . . . . . 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 8485 . . . . . . . . . . 11  |-  1Q  e.  Q.
4 addclnq 8502 . . . . . . . . . . 11  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  e.  Q. )
53, 3, 4mp2an 656 . . . . . . . . . 10  |-  ( 1Q 
+Q  1Q )  e. 
Q.
6 recidnq 8522 . . . . . . . . . 10  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  (
( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q )
75, 6ax-mp 10 . . . . . . . . 9  |-  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q
87, 7oveq12i 5769 . . . . . . . 8  |-  ( ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
92, 8eqtri 2276 . . . . . . 7  |-  ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
109oveq1i 5767 . . . . . 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 5768 . . . . . . 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 8516 . . . . . . . 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 8510 . . . . . . . . 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 5767 . . . . . . . 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 2278 . . . . . . 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 8523 . . . . . . . . 9  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  ( *Q `  ( 1Q  +Q  1Q ) )  e.  Q. )
17 addclnq 8502 . . . . . . . . 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 645 . . . . . . . 8  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  (
( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
Q. )
19 mulidnq 8520 . . . . . . . 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 11 . . . . . . 7  |-  ( ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  .Q  1Q )  =  ( ( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )
2111, 15, 203eqtr3i 2284 . . . . . 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 2284 . . . . 5  |-  ( ( *Q `  ( 1Q 
+Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q
2322oveq2i 5768 . . . 4  |-  ( A  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
241, 23eqtr3i 2278 . . 3  |-  ( ( A  .Q  ( *Q
`  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
25 mulidnq 8520 . . 3  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
2624, 25syl5eq 2300 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A )
27 ovex 5782 . . 3  |-  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
_V
28 oveq12 5766 . . . . 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 629 . . . 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 2264 . . 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, 30cla4ev 2826 . 2  |-  ( ( ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A  ->  E. x
( x  +Q  x
)  =  A )
3226, 31syl 17 1  |-  ( A  e.  Q.  ->  E. x
( x  +Q  x
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6   E.wex 1537    = wceq 1619    e. wcel 1621   ` cfv 4638  (class class class)co 5757   Q.cnq 8407   1Qc1q 8408    +Q cplq 8410    .Q cmq 8411   *Qcrq 8412
This theorem is referenced by:  nsmallnq  8534
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-omul 6417  df-er 6593  df-ni 8429  df-pli 8430  df-mi 8431  df-lti 8432  df-plpq 8465  df-mpq 8466  df-enq 8468  df-nq 8469  df-erq 8470  df-plq 8471  df-mq 8472  df-1nq 8473  df-rq 8474
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