Theorem subhalfnqq 7745

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 7741). (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 7741	. . . . . 6 |- ( A e. Q. -> E. y e. Q. ( y +Q y ) = A )
2		df-rex 2528	. . . . . . 7 |- ( E. y e. Q. ( y +Q y ) = A <-> E. y ( y e. Q. /\ ( y +Q y ) = A ) )
3		halfnqq 7741	. . . . . . . . . 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 1649	. . . . . . 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 2528	. . . . . . 7 |- ( E. x e. Q. ( x +Q x ) = y <-> E. x ( x e. Q. /\ ( x +Q x ) = y ) )
10	9	anbi2i 457	. . . . . 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 1654	. . . . 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 1961	. . . 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 531	. . . . . 6 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> x e. Q. )
16		simpll 527	. . . . . . . . 9 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y e. Q. )
17		ltaddnq 7738	. . . . . . . . 9 |- ( ( y e. Q. /\ y e. Q. ) -> y <Q ( y +Q y ) )
18	16, 16, 17	syl2anc 411	. . . . . . . 8 |- ( ( ( y e. Q. /\ ( y +Q y ) = A ) /\ ( x e. Q. /\ ( x +Q x ) = y ) ) -> y <Q ( y +Q y ) )
19		breq2 4118	. . . . . . . . 9 |- ( ( y +Q y ) = A -> ( y <Q ( y +Q y ) <-> y <Q A ) )
20	19	ad2antlr 489	. . . . . . . 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 4117	. . . . . . . 8 |- ( ( x +Q x ) = y -> ( ( x +Q x ) <Q A <-> y <Q A ) )
23	22	ad2antll 491	. . . . . . 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 1649	. . . 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 1647	. . 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 2528	. 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 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   Q.cnq 7611   +Q cplq 7613   <Q cltq 7616

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684

This theorem is referenced by:  prarloc  7834  cauappcvgprlemloc  7983  caucvgprlemloc  8006  caucvgprprlemml  8025  caucvgprprlemloc  8034