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Theorem subhalfnqq 6963
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 6959). (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 6959 . . . . . 6 (𝐴Q → ∃𝑦Q (𝑦 +Q 𝑦) = 𝐴)
2 df-rex 2365 . . . . . . 7 (∃𝑦Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3 halfnqq 6959 . . . . . . . . . 10 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦)
43adantr 270 . . . . . . . . 9 ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦)
54ancli 316 . . . . . . . 8 ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
65eximi 1536 . . . . . . 7 (∃𝑦(𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
72, 6sylbi 119 . . . . . 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 2365 . . . . . . 7 (∃𝑥Q (𝑥 +Q 𝑥) = 𝑦 ↔ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦))
109anbi2i 445 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦) ↔ ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
1110exbii 1541 . . . . 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 120 . . . 4 (𝐴Q → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
13 exdistr 1835 . . . 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 132 . . 3 (𝐴Q → ∃𝑦𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
15 simprl 498 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑥Q)
16 simpll 496 . . . . . . . . 9 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦Q)
17 ltaddnq 6956 . . . . . . . . 9 ((𝑦Q𝑦Q) → 𝑦 <Q (𝑦 +Q 𝑦))
1816, 16, 17syl2anc 403 . . . . . . . 8 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q (𝑦 +Q 𝑦))
19 breq2 3847 . . . . . . . . 9 ((𝑦 +Q 𝑦) = 𝐴 → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
2019ad2antlr 473 . . . . . . . 8 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
2118, 20mpbid 145 . . . . . . 7 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q 𝐴)
22 breq1 3846 . . . . . . . 8 ((𝑥 +Q 𝑥) = 𝑦 → ((𝑥 +Q 𝑥) <Q 𝐴𝑦 <Q 𝐴))
2322ad2antll 475 . . . . . . 7 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ((𝑥 +Q 𝑥) <Q 𝐴𝑦 <Q 𝐴))
2421, 23mpbird 165 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥 +Q 𝑥) <Q 𝐴)
2515, 24jca 300 . . . . 5 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
2625eximi 1536 . . . 4 (∃𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
2726exlimiv 1534 . . 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 2365 . 2 (∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴 ↔ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
3028, 29sylibr 132 1 (𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  wrex 2360   class class class wbr 3843  (class class class)co 5644  Qcnq 6829   +Q cplq 6831   <Q cltq 6834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-eprel 4114  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-irdg 6127  df-1o 6173  df-oadd 6177  df-omul 6178  df-er 6282  df-ec 6284  df-qs 6288  df-ni 6853  df-pli 6854  df-mi 6855  df-lti 6856  df-plpq 6893  df-mpq 6894  df-enq 6896  df-nqqs 6897  df-plqqs 6898  df-mqqs 6899  df-1nqqs 6900  df-rq 6901  df-ltnqqs 6902
This theorem is referenced by:  prarloc  7052  cauappcvgprlemloc  7201  caucvgprlemloc  7224  caucvgprprlemml  7243  caucvgprprlemloc  7252
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