Theorem subhalfnqq 7547

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 7543). (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 7543	. . . . . 6 ⊢ (𝐴 ∈ Q → ∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴)
2		df-rex 2491	. . . . . . 7 ⊢ (∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3		halfnqq 7543	. . . . . . . . . 10 ⊢ (𝑦 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦)
4	3	adantr 276	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦)
5	4	ancli 323	. . . . . . . 8 ⊢ ((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) → ((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦))
6	5	eximi 1624	. . . . . . 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 121	. . . . . 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 2491	. . . . . . 7 ⊢ (∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝑦 ↔ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦))
10	9	anbi2i 457	. . . . . 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 1629	. . . . 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 122	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑦((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
13		exdistr 1934	. . . 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 134	. . 3 ⊢ (𝐴 ∈ Q → ∃𝑦∃𝑥((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)))
15		simprl 529	. . . . . 6 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑥 ∈ Q)
16		simpll 527	. . . . . . . . 9 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 ∈ Q)
17		ltaddnq 7540	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑦 ∈ Q) → 𝑦 <Q (𝑦 +Q 𝑦))
18	16, 16, 17	syl2anc 411	. . . . . . . 8 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q (𝑦 +Q 𝑦))
19		breq2 4055	. . . . . . . . 9 ⊢ ((𝑦 +Q 𝑦) = 𝐴 → (𝑦 <Q (𝑦 +Q 𝑦) ↔ 𝑦 <Q 𝐴))
20	19	ad2antlr 489	. . . . . . . 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 147	. . . . . . 7 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → 𝑦 <Q 𝐴)
22		breq1 4054	. . . . . . . 8 ⊢ ((𝑥 +Q 𝑥) = 𝑦 → ((𝑥 +Q 𝑥) <Q 𝐴 ↔ 𝑦 <Q 𝐴))
23	22	ad2antll 491	. . . . . . 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 167	. . . . . 6 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥 +Q 𝑥) <Q 𝐴)
25	15, 24	jca 306	. . . . 5 ⊢ (((𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴) ∧ (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) = 𝑦)) → (𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
26	25	eximi 1624	. . . 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 1622	. . 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 2491	. 2 ⊢ (∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴 ↔ ∃𝑥(𝑥 ∈ Q ∧ (𝑥 +Q 𝑥) <Q 𝐴))
30	28, 29	sylibr 134	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) <Q 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486   class class class wbr 4051  (class class class)co 5957  Qcnq 7413   +_Q cplq 7415   <_Q cltq 7418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-eprel 4344  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-lti 7440  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-rq 7485  df-ltnqqs 7486

This theorem is referenced by:  prarloc  7636  cauappcvgprlemloc  7785  caucvgprlemloc  7808  caucvgprprlemml  7827  caucvgprprlemloc  7836