Theorem subhalfnqq 7164

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 7160). (Contributed by Jim Kingdon, 25-Nov-2019.)

Assertion

Ref	Expression
subhalfnqq	|- ( A e. Q. -> E. x e. 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 7160	. . . . . 6 |- ( A e. Q. -> E. y e. Q. ( y +Q y ) = A )
2		df-rex 2394	. . . . . . 7 |- ( E. y e. Q. ( y +Q y ) = A <-> E. y ( y e. Q. /\ ( y +Q y ) = A ) )
3		halfnqq 7160	. . . . . . . . . 10 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) = y )
4	3	adantr 272	. . . . . . . . 9 |- ( ( y e. Q. /\ ( y +Q y ) = A ) -> E. x e. Q. ( x +Q x ) = y )
5	4	ancli 319	. . . . . . . 8 |- ( ( y e. Q. /\ ( y +Q y ) = A ) -> ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x e. Q. ( x +Q x ) = y ) )
6	5	eximi 1560	. . . . . . 7 |- ( E. y ( y e. Q. /\ ( y +Q y ) = A ) -> E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x e. Q. ( x +Q x ) = y ) )
7	2, 6	sylbi 120	. . . . . 6 |- ( E. y e. Q. ( y +Q y ) = A -> E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x e. Q. ( x +Q x ) = y ) )
8	1, 7	syl 14	. . . . 5 |- ( A e. Q. -> E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x e. Q. ( x +Q x ) = y ) )
9		df-rex 2394	. . . . . . 7 |- ( E. x e. Q. ( x +Q x ) = y <-> E. x ( x e. Q. /\ ( x +Q x ) = y ) )
10	9	anbi2i 450	. . . . . 6 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x e. Q. ( x +Q x ) = y ) <-> ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x ( x e. Q. /\ ( x +Q x ) = y ) ) )
11	10	exbii 1565	. . . . 5 |- ( E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x e. Q. ( x +Q x ) = y ) <-> E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x ( x e. Q. /\ ( x +Q x ) = y ) ) )
12	8, 11	sylib 121	. . . 4 |- ( A e. Q. -> E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x ( x e. Q. /\ ( x +Q x ) = y ) ) )
13		exdistr 1859	. . . 4 |- ( E. y E. x ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) <-> E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x ( x e. Q. /\ ( x +Q x ) = y ) ) )
14	12, 13	sylibr 133	. . 3 |- ( A e. Q. -> E. y E. x ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) )
15		simprl 503	. . . . . 6 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> x e. Q. )
16		simpll 501	. . . . . . . . 9 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y e. Q. )
17		ltaddnq 7157	. . . . . . . . 9 |- ( ( y e. Q. /\ y e. Q. ) -> y <Q ( y +Q y ) )
18	16, 16, 17	syl2anc 406	. . . . . . . 8 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y <Q ( y +Q y ) )
19		breq2 3897	. . . . . . . . 9 |- ( ( y +Q y ) = A -> ( y <Q ( y +Q y ) <-> y <Q A ) )
20	19	ad2antlr 478	. . . . . . . 8 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> ( y <Q ( y +Q y ) <-> y <Q A ) )
21	18, 20	mpbid 146	. . . . . . 7 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y <Q A )
22		breq1 3896	. . . . . . . 8 |- ( ( x +Q x ) = y -> ( ( x +Q x ) <Q A <-> y <Q A ) )
23	22	ad2antll 480	. . . . . . 7 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> ( ( x +Q x ) <Q A <-> y <Q A ) )
24	21, 23	mpbird 166	. . . . . 6 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> ( x +Q x ) <Q A )
25	15, 24	jca 302	. . . . 5 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> ( x e. Q. /\ ( x +Q x ) <Q A ) )
26	25	eximi 1560	. . . 4 |- ( E. x ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> E. x ( x e. Q. /\ ( x +Q x ) <Q A ) )
27	26	exlimiv 1558	. . 3 |- ( E. y E. x ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> E. x ( x e. Q. /\ ( x +Q x ) <Q A ) )
28	14, 27	syl 14	. 2 |- ( A e. Q. -> E. x ( x e. Q. /\ ( x +Q x ) <Q A ) )
29		df-rex 2394	. 2 |- ( E. x e. Q. ( x +Q x ) <Q A <-> E. x ( x e. Q. /\ ( x +Q x ) <Q A ) )
30	28, 29	sylibr 133	1 |- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1312   E.wex 1449   e. wcel 1461   E.wrex 2389   class class class wbr 3893  (class class class)co 5726   Q.cnq 7030   +Q cplq 7032   <Q cltq 7035

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103

This theorem is referenced by:  prarloc  7253  cauappcvgprlemloc  7402  caucvgprlemloc  7425  caucvgprprlemml  7444  caucvgprprlemloc  7453