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Theorem subhalfnqq 7215
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 7211). (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 7211 . . . . . 6 (𝐴Q → ∃𝑦Q (𝑦 +Q 𝑦) = 𝐴)
2 df-rex 2420 . . . . . . 7 (∃𝑦Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3 halfnqq 7211 . . . . . . . . . 10 (𝑦Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦)
43adantr 274 . . . . . . . . 9 ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦)
54ancli 321 . . . . . . . 8 ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
65eximi 1579 . . . . . . 7 (∃𝑦(𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦))
72, 6sylbi 120 . . . . . 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 2420 . . . . . . 7 (∃𝑥Q (𝑥 +Q 𝑥) = 𝑦 ↔ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦))
109anbi2i 452 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥Q (𝑥 +Q 𝑥) = 𝑦) ↔ ((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
1110exbii 1584 . . . . 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 121 . . . 4 (𝐴Q → ∃𝑦((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
13 exdistr 1881 . . . 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 133 . . 3 (𝐴Q → ∃𝑦𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
15 simprl 520 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑥Q)
16 simpll 518 . . . . . . . . 9 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦Q)
17 ltaddnq 7208 . . . . . . . . 9 ((𝑦Q𝑦Q) → 𝑦 <Q (𝑦 +Q 𝑦))
1816, 16, 17syl2anc 408 . . . . . . . 8 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q (𝑦 +Q 𝑦))
19 breq2 3928 . . . . . . . . 9 ((𝑦 +Q 𝑦) = 𝐴 → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
2019ad2antlr 480 . . . . . . . 8 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
2118, 20mpbid 146 . . . . . . 7 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q 𝐴)
22 breq1 3927 . . . . . . . 8 ((𝑥 +Q 𝑥) = 𝑦 → ((𝑥 +Q 𝑥) <Q 𝐴𝑦 <Q 𝐴))
2322ad2antll 482 . . . . . . 7 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ((𝑥 +Q 𝑥) <Q 𝐴𝑦 <Q 𝐴))
2421, 23mpbird 166 . . . . . 6 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥 +Q 𝑥) <Q 𝐴)
2515, 24jca 304 . . . . 5 (((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
2625eximi 1579 . . . 4 (∃𝑥((𝑦Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
2726exlimiv 1577 . . 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 2420 . 2 (∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴 ↔ ∃𝑥(𝑥Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
3028, 29sylibr 133 1 (𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) <Q 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wrex 2415   class class class wbr 3924  (class class class)co 5767  Qcnq 7081   +Q cplq 7083   <Q cltq 7086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154
This theorem is referenced by:  prarloc  7304  cauappcvgprlemloc  7453  caucvgprlemloc  7476  caucvgprprlemml  7495  caucvgprprlemloc  7504
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