Theorem subhalfnqq 7376

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 7372). (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 7372	. . . . . 6 ⊢ (𝐴 ∈ Q → ∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴)
2		df-rex 2454	. . . . . . 7 ⊢ (∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3		halfnqq 7372	. . . . . . . . . 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 1593	. . . . . . 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 2454	. . . . . . 7 ⊢ (∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦 ↔ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦))
10	9	anbi2i 454	. . . . . 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 1598	. . . . 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 1902	. . . 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 526	. . . . . 6 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑥 ∈ Q)
16		simpll 524	. . . . . . . . 9 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 ∈ Q)
17		ltaddnq 7369	. . . . . . . . 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 3993	. . . . . . . . 9 ⊢ ((𝑦 +Q 𝑦) = 𝐴 → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
20	19	ad2antlr 486	. . . . . . . 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 3992	. . . . . . . 8 ⊢ ((𝑥 +Q 𝑥) = 𝑦 → ((𝑥 +Q 𝑥) <Q 𝐴 ↔ 𝑦 <Q 𝐴))
23	22	ad2antll 488	. . . . . . 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 1593	. . . 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 1591	. . 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 2454	. 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 1348  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449   class class class wbr 3989  (class class class)co 5853  Qcnq 7242   +_Q cplq 7244   <_Q cltq 7247

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315

This theorem is referenced by:  prarloc  7465  cauappcvgprlemloc  7614  caucvgprlemloc  7637  caucvgprprlemml  7656  caucvgprprlemloc  7665