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Theorem halfnq 8568
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 8553 . . . 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 8553 . . . . . . . 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 8520 . . . . . . . . . . 11  |-  1Q  e.  Q.
4 addclnq 8537 . . . . . . . . . . 11  |-  ( ( 1Q  e.  Q.  /\  1Q  e.  Q. )  -> 
( 1Q  +Q  1Q )  e.  Q. )
53, 3, 4mp2an 656 . . . . . . . . . 10  |-  ( 1Q 
+Q  1Q )  e. 
Q.
6 recidnq 8557 . . . . . . . . . 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 5804 . . . . . . . 8  |-  ( ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( ( 1Q  +Q  1Q )  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
92, 8eqtri 2278 . . . . . . 7  |-  ( ( 1Q  +Q  1Q )  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( 1Q  +Q  1Q )
109oveq1i 5802 . . . . . 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 5803 . . . . . . 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 8551 . . . . . . . 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 8545 . . . . . . . . 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 5802 . . . . . . . 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 2280 . . . . . . 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 8558 . . . . . . . . 9  |-  ( ( 1Q  +Q  1Q )  e.  Q.  ->  ( *Q `  ( 1Q  +Q  1Q ) )  e.  Q. )
17 addclnq 8537 . . . . . . . . 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 8555 . . . . . . . 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 2286 . . . . . 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 2286 . . . . 5  |-  ( ( *Q `  ( 1Q 
+Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  =  1Q
2322oveq2i 5803 . . . 4  |-  ( A  .Q  ( ( *Q
`  ( 1Q  +Q  1Q ) )  +Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
241, 23eqtr3i 2280 . . 3  |-  ( ( A  .Q  ( *Q
`  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  ( A  .Q  1Q )
25 mulidnq 8555 . . 3  |-  ( A  e.  Q.  ->  ( A  .Q  1Q )  =  A )
2624, 25syl5eq 2302 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  +Q  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) ) )  =  A )
27 ovex 5817 . . 3  |-  ( A  .Q  ( *Q `  ( 1Q  +Q  1Q ) ) )  e. 
_V
28 oveq12 5801 . . . . 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 2266 . . 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 2850 . 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 4673  (class class class)co 5792   Q.cnq 8442   1Qc1q 8443    +Q cplq 8445    .Q cmq 8446   *Qcrq 8447
This theorem is referenced by:  nsmallnq  8569
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-er 6628  df-ni 8464  df-pli 8465  df-mi 8466  df-lti 8467  df-plpq 8500  df-mpq 8501  df-enq 8503  df-nq 8504  df-erq 8505  df-plq 8506  df-mq 8507  df-1nq 8508  df-rq 8509
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