Theorem subhalfnqq 7500

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 7496). (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 7496	. . . . . 6 ⊢ (𝐴 ∈ Q → ∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴)
2		df-rex 2481	. . . . . . 7 ⊢ (∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3		halfnqq 7496	. . . . . . . . . 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 1614	. . . . . . 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 2481	. . . . . . 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 1619	. . . . 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 1924	. . . 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 7493	. . . . . . . . 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 4038	. . . . . . . . 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 4037	. . . . . . . 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 1614	. . . 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 1612	. . 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 2481	. 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 1364  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476   class class class wbr 4034  (class class class)co 5925  Qcnq 7366   +_Q cplq 7368   <_Q cltq 7371

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7390  df-pli 7391  df-mi 7392  df-lti 7393  df-plpq 7430  df-mpq 7431  df-enq 7433  df-nqqs 7434  df-plqqs 7435  df-mqqs 7436  df-1nqqs 7437  df-rq 7438  df-ltnqqs 7439

This theorem is referenced by:  prarloc  7589  cauappcvgprlemloc  7738  caucvgprlemloc  7761  caucvgprprlemml  7780  caucvgprprlemloc  7789