Theorem subhalfnqq 7787

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 7783). (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 7783	. . . . . 6 |- ( A e. Q. -> E. y e. Q. ( y +Q y ) = A )
2		df-rex 2534	. . . . . . 7 |- ( E. y e. Q. ( y +Q y ) = A <-> E. y ( y e. Q. /\ ( y +Q y ) = A ) )
3		halfnqq 7783	. . . . . . . . . 10 |- ( y e. Q. -> E. x e. Q. ( x +Q x ) = y )
4	3	adantr 276	. . . . . . . . 9 |- ( ( y e. Q. /\ ( y +Q y ) = A ) -> E. x e. Q. ( x +Q x ) = y )
5	4	ancli 323	. . . . . . . 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 1653	. . . . . . 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 121	. . . . . 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 2534	. . . . . . 7 |- ( E. x e. Q. ( x +Q x ) = y <-> E. x ( x e. Q. /\ ( x +Q x ) = y ) )
10	9	anbi2i 461	. . . . . 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 1658	. . . . 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 122	. . . 4 |- ( A e. Q. -> E. y ( ( y e. Q. /\ ( y +Q y ) = A ) /\ E. x ( x e. Q. /\ ( x +Q x ) = y ) ) )
13		exdistr 1965	. . . 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 134	. . 3 |- ( A e. Q. -> E. y E. x ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) )
15		simprl 535	. . . . . 6 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> x e. Q. )
16		simpll 531	. . . . . . . . 9 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y e. Q. )
17		ltaddnq 7780	. . . . . . . . 9 |- ( ( y e. Q. /\ y e. Q. ) -> y <Q ( y +Q y ) )
18	16, 16, 17	syl2anc 415	. . . . . . . 8 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y <Q ( y +Q y ) )
19		breq2 4135	. . . . . . . . 9 |- ( ( y +Q y ) = A -> ( y <Q ( y +Q y ) <-> y <Q A ) )
20	19	ad2antlr 493	. . . . . . . 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 147	. . . . . . 7 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y <Q A )
22		breq1 4134	. . . . . . . 8 |- ( ( x +Q x ) = y -> ( ( x +Q x ) <Q A <-> y <Q A ) )
23	22	ad2antll 495	. . . . . . 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 167	. . . . . 6 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> ( x +Q x ) <Q A )
25	15, 24	jca 306	. . . . 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 1653	. . . 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 1651	. . 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 2534	. 2 |- ( E. x e. Q. ( x +Q x ) <Q A <-> E. x ( x e. Q. /\ ( x +Q x ) <Q A ) )
30	28, 29	sylibr 134	1 |- ( A e. Q. -> E. x e. Q. ( x +Q x ) <Q A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   class class class wbr 4131  (class class class)co 6088   Q.cnq 7653   +Q cplq 7655   <Q cltq 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726

This theorem is used by:  prarloc  7876  cauappcvgprlemloc  8025  caucvgprlemloc  8048  caucvgprprlemml  8067  caucvgprprlemloc  8076