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Theorem subhalfnqq 6271
Description: There is a number which is less than half of any positive fraction. The case where A is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 6267). (Contributed by Jim Kingdon, 25-Nov-2019.)
Assertion
Ref Expression
subhalfnqq (A Qx Q (x +Q x) <Q A)
Distinct variable group:   x,A

Proof of Theorem subhalfnqq
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 halfnqq 6267 . . . . . 6 (A Qy Q (y +Q y) = A)
2 df-rex 2290 . . . . . . 7 (y Q (y +Q y) = Ay(y Q (y +Q y) = A))
3 halfnqq 6267 . . . . . . . . . 10 (y Qx Q (x +Q x) = y)
43adantr 261 . . . . . . . . 9 ((y Q (y +Q y) = A) → x Q (x +Q x) = y)
54ancli 306 . . . . . . . 8 ((y Q (y +Q y) = A) → ((y Q (y +Q y) = A) x Q (x +Q x) = y))
65eximi 1473 . . . . . . 7 (y(y Q (y +Q y) = A) → y((y Q (y +Q y) = A) x Q (x +Q x) = y))
72, 6sylbi 114 . . . . . 6 (y Q (y +Q y) = Ay((y Q (y +Q y) = A) x Q (x +Q x) = y))
81, 7syl 14 . . . . 5 (A Qy((y Q (y +Q y) = A) x Q (x +Q x) = y))
9 df-rex 2290 . . . . . . 7 (x Q (x +Q x) = yx(x Q (x +Q x) = y))
109anbi2i 433 . . . . . 6 (((y Q (y +Q y) = A) x Q (x +Q x) = y) ↔ ((y Q (y +Q y) = A) x(x Q (x +Q x) = y)))
1110exbii 1478 . . . . 5 (y((y Q (y +Q y) = A) x Q (x +Q x) = y) ↔ y((y Q (y +Q y) = A) x(x Q (x +Q x) = y)))
128, 11sylib 127 . . . 4 (A Qy((y Q (y +Q y) = A) x(x Q (x +Q x) = y)))
13 exdistr 1769 . . . 4 (yx((y Q (y +Q y) = A) (x Q (x +Q x) = y)) ↔ y((y Q (y +Q y) = A) x(x Q (x +Q x) = y)))
1412, 13sylibr 137 . . 3 (A Qyx((y Q (y +Q y) = A) (x Q (x +Q x) = y)))
15 simprl 471 . . . . . 6 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → x Q)
16 simpll 469 . . . . . . . . 9 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → y Q)
17 ltaddnq 6265 . . . . . . . . 9 ((y Q y Q) → y <Q (y +Q y))
1816, 16, 17syl2anc 393 . . . . . . . 8 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → y <Q (y +Q y))
19 breq2 3742 . . . . . . . . 9 ((y +Q y) = A → (y <Q (y +Q y) ↔ y <Q A))
2019ad2antlr 462 . . . . . . . 8 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → (y <Q (y +Q y) ↔ y <Q A))
2118, 20mpbid 135 . . . . . . 7 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → y <Q A)
22 breq1 3741 . . . . . . . 8 ((x +Q x) = y → ((x +Q x) <Q Ay <Q A))
2322ad2antll 464 . . . . . . 7 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → ((x +Q x) <Q Ay <Q A))
2421, 23mpbird 156 . . . . . 6 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → (x +Q x) <Q A)
2515, 24jca 290 . . . . 5 (((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → (x Q (x +Q x) <Q A))
2625eximi 1473 . . . 4 (x((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → x(x Q (x +Q x) <Q A))
2726exlimiv 1471 . . 3 (yx((y Q (y +Q y) = A) (x Q (x +Q x) = y)) → x(x Q (x +Q x) <Q A))
2814, 27syl 14 . 2 (A Qx(x Q (x +Q x) <Q A))
29 df-rex 2290 . 2 (x Q (x +Q x) <Q Ax(x Q (x +Q x) <Q A))
3028, 29sylibr 137 1 (A Qx Q (x +Q x) <Q A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228  wex 1362   wcel 1374  wrex 2285   class class class wbr 3738  (class class class)co 5436  Qcnq 6138   +Q cplq 6140   <Q cltq 6143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-eprel 4000  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-1o 5916  df-oadd 5920  df-omul 5921  df-er 6017  df-ec 6019  df-qs 6023  df-ni 6164  df-pli 6165  df-mi 6166  df-lti 6167  df-plpq 6203  df-mpq 6204  df-enq 6206  df-nqqs 6207  df-plqqs 6208  df-mqqs 6209  df-1nqqs 6210  df-rq 6211  df-ltnqqs 6212
This theorem is referenced by:  prarloc  6357
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