MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  halfnq Unicode version

Theorem halfnq 8595
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 8580 . . . 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 8580 . . . . . . . 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 8547 . . . . . . . . . . 11  |-  1Q  e.  Q.
4 addclnq 8564 . . . . . . . . . . 11  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  e.  Q. )
53, 3, 4mp2an 655 . . . . . . . . . 10  |-  ( 1Q 
+Q  1Q )  e. 
Q.
6 recidnq 8584 . . . . . . . . . 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 5831 . . . . . . . 8  |-  ( ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
92, 8eqtri 2304 . . . . . . 7  |-  ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
109oveq1i 5829 . . . . . 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 5830 . . . . . . 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 8578 . . . . . . . 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 8572 . . . . . . . . 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 5829 . . . . . . . 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 2306 . . . . . . 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 8585 . . . . . . . . 9  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  ( *Q `  ( 1Q  +Q  1Q ) )  e.  Q. )
17 addclnq 8564 . . . . . . . . 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 644 . . . . . . . 8  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  (
( *Q `  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
Q. )
19 mulidnq 8582 . . . . . . . 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 2312 . . . . . 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 2312 . . . . 5  |-  ( ( *Q `  ( 1Q 
+Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q
2322oveq2i 5830 . . . 4  |-  ( A  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
241, 23eqtr3i 2306 . . 3  |-  ( ( A  .Q  ( *Q
`  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
25 mulidnq 8582 . . 3  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
2624, 25syl5eq 2328 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A )
27 ovex 5844 . . 3  |-  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
_V
28 oveq12 5828 . . . . 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 628 . . . 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 2292 . . 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 2876 . 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 1529    = wceq 1624    e. wcel 1685   ` cfv 5221  (class class class)co 5819   Q.cnq 8469   1Qc1q 8470    +Q cplq 8472    .Q cmq 8473   *Qcrq 8474
This theorem is referenced by:  nsmallnq  8596
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-pli 8492  df-mi 8493  df-lti 8494  df-plpq 8527  df-mpq 8528  df-enq 8530  df-nq 8531  df-erq 8532  df-plq 8533  df-mq 8534  df-1nq 8535  df-rq 8536
  Copyright terms: Public domain W3C validator