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Theorem subhalfnqq 6569
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 6565). (Contributed by Jim Kingdon, 25-Nov-2019.)
Assertion
Ref Expression
subhalfnqq (𝐴Q → ∃𝑥Q (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 6565 . . . . . 6 (𝐴Q → ∃𝑦Q (𝑦 +Q 𝑦) = 𝐴)
2 df-rex 2329 . . . . . . 7 (∃𝑦Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3 halfnqq 6565 . . . . . . . . . 10 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦)
43adantr 265 . . . . . . . . 9 ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦)
54ancli 310 . . . . . . . 8 ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
65eximi 1507 . . . . . . 7 (∃𝑦(𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
72, 6sylbi 118 . . . . . 6 (∃𝑦Q (𝑦 +Q 𝑦) = 𝐴 → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
81, 7syl 14 . . . . 5 (𝐴Q → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
9 df-rex 2329 . . . . . . 7 (∃𝑥Q (𝑥 +Q 𝑥) = 𝑦 ↔ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦))
109anbi2i 438 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦) ↔ ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
1110exbii 1512 . . . . 5 (∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦) ↔ ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
128, 11sylib 131 . . . 4 (𝐴Q → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
13 exdistr 1803 . . . 4 (∃𝑦𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) ↔ ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
1412, 13sylibr 141 . . 3 (𝐴Q → ∃𝑦𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
15 simprl 491 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑥Q)
16 simpll 489 . . . . . . . . 9 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦Q)
17 ltaddnq 6562 . . . . . . . . 9 ((𝑦Q𝑦Q) → 𝑦 <Q (𝑦 +Q 𝑦))
1816, 16, 17syl2anc 397 . . . . . . . 8 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q (𝑦 +Q 𝑦))
19 breq2 3795 . . . . . . . . 9 ((𝑦 +Q 𝑦) = 𝐴 → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
2019ad2antlr 466 . . . . . . . 8 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
2118, 20mpbid 139 . . . . . . 7 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q 𝐴)
22 breq1 3794 . . . . . . . 8 ((𝑥 +Q 𝑥) = 𝑦 → ((𝑥 +Q 𝑥) <Q 𝐴𝑦 <Q 𝐴))
2322ad2antll 468 . . . . . . 7 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ((𝑥 +Q 𝑥) <Q 𝐴𝑦 <Q 𝐴))
2421, 23mpbird 160 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥 +Q 𝑥) <Q 𝐴)
2515, 24jca 294 . . . . 5 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
2625eximi 1507 . . . 4 (∃𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
2726exlimiv 1505 . . 3 (∃𝑦𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
2814, 27syl 14 . 2 (𝐴Q → ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
29 df-rex 2329 . 2 (∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴 ↔ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
3028, 29sylibr 141 1 (𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  wrex 2324   class class class wbr 3791  (class class class)co 5539  Qcnq 6435   +Q cplq 6437   <Q cltq 6440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508
This theorem is referenced by:  prarloc  6658  cauappcvgprlemloc  6807  caucvgprlemloc  6830  caucvgprprlemml  6849  caucvgprprlemloc  6858
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