Theorem subhalfnqq 7334

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 7330). (Contributed by Jim Kingdon, 25-Nov-2019.)

Assertion

Ref	Expression
subhalfnqq	⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem subhalfnqq

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		halfnqq 7330	. . . . . 6 ⊢ (𝐴 ∈ Q → ∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴)
2		df-rex 2441	. . . . . . 7 ⊢ (∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3		halfnqq 7330	. . . . . . . . . 10 ⊢ (𝑦 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦)
4	3	adantr 274	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦)
5	4	ancli 321	. . . . . . . 8 ⊢ ((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦))
6	5	eximi 1580	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑦((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦))
7	2, 6	sylbi 120	. . . . . 6 ⊢ (∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴 → ∃𝑦((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦))
8	1, 7	syl 14	. . . . 5 ⊢ (𝐴 ∈ Q → ∃𝑦((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦))
9		df-rex 2441	. . . . . . 7 ⊢ (∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦 ↔ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦))
10	9	anbi2i 453	. . . . . 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)))
11	10	exbii 1585	. . . . 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)))
12	8, 11	sylib 121	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑦((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
13		exdistr 1889	. . . 4 ⊢ (∃𝑦∃𝑥((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)))
14	12, 13	sylibr 133	. . 3 ⊢ (𝐴 ∈ Q → ∃𝑦∃𝑥((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
15		simprl 521	. . . . . 6 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑥 ∈ Q)
16		simpll 519	. . . . . . . . 9 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 ∈ Q)
17		ltaddnq 7327	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑦 ∈ Q) → 𝑦 <Q (𝑦 +Q 𝑦))
18	16, 16, 17	syl2anc 409	. . . . . . . 8 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q (𝑦 +Q 𝑦))
19		breq2 3969	. . . . . . . . 9 ⊢ ((𝑦 +Q 𝑦) = 𝐴 → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
20	19	ad2antlr 481	. . . . . . . 8 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
21	18, 20	mpbid 146	. . . . . . 7 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q 𝐴)
22		breq1 3968	. . . . . . . 8 ⊢ ((𝑥 +Q 𝑥) = 𝑦 → ((𝑥 +Q 𝑥) <Q 𝐴 ↔ 𝑦 <Q 𝐴))
23	22	ad2antll 483	. . . . . . 7 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ((𝑥 +Q 𝑥) <Q 𝐴 ↔ 𝑦 <Q 𝐴))
24	21, 23	mpbird 166	. . . . . 6 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥 +Q 𝑥) <Q 𝐴)
25	15, 24	jca 304	. . . . 5 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
26	25	eximi 1580	. . . 4 ⊢ (∃𝑥((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
27	26	exlimiv 1578	. . 3 ⊢ (∃𝑦∃𝑥((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
28	14, 27	syl 14	. 2 ⊢ (𝐴 ∈ Q → ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
29		df-rex 2441	. 2 ⊢ (∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴 ↔ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
30	28, 29	sylibr 133	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∃wrex 2436   class class class wbr 3965  (class class class)co 5824  Qcnq 7200   +_Q cplq 7202   <_Q cltq 7205

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273

This theorem is referenced by:  prarloc  7423  cauappcvgprlemloc  7572  caucvgprlemloc  7595  caucvgprprlemml  7614  caucvgprprlemloc  7623