Theorem subhalfnqq 7601

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 7597). (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 7597	. . . . . 6 |- ( A e. Q. -> E. y e. Q. ( y +Q y ) = A )
2		df-rex 2514	. . . . . . 7 |- ( E. y e. Q. ( y +Q y ) = A <-> E. y ( y e. Q. /\ ( y +Q y ) = A ) )
3		halfnqq 7597	. . . . . . . . . 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 1646	. . . . . . 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 2514	. . . . . . 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 1651	. . . . 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 1956	. . . 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 529	. . . . . 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 7594	. . . . . . . . 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 4087	. . . . . . . . 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 4086	. . . . . . . 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 1646	. . . 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 1644	. . 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 2514	. 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 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   class class class wbr 4083  (class class class)co 6001   Q.cnq 7467   +Q cplq 7469   <Q cltq 7472

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540

This theorem is referenced by:  prarloc  7690  cauappcvgprlemloc  7839  caucvgprlemloc  7862  caucvgprprlemml  7881  caucvgprprlemloc  7890