Theorem subhalfnqq 7412

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 7408). (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 7408	. . . . . 6 ⊢ (𝐴 ∈ Q → ∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴)
2		df-rex 2461	. . . . . . 7 ⊢ (∃𝑦 ∈ Q (𝑦 +Q 𝑦) = 𝐴 ↔ ∃𝑦(𝑦 ∈ Q ∧ (𝑦 +Q 𝑦) = 𝐴))
3		halfnqq 7408	. . . . . . . . . 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 1600	. . . . . . 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 2461	. . . . . . 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 1605	. . . . 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 1909	. . . 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 7405	. . . . . . . . 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 4007	. . . . . . . . 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 4006	. . . . . . . 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 1600	. . . 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 1598	. . 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 2461	. 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 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456   class class class wbr 4003  (class class class)co 5874  Qcnq 7278   +_Q cplq 7280   <_Q cltq 7283

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351

This theorem is referenced by:  prarloc  7501  cauappcvgprlemloc  7650  caucvgprlemloc  7673  caucvgprprlemml  7692  caucvgprprlemloc  7701