Theorem subhalfnqq 7481

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 7477). (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 7477	. . . . . 6 |- ( A e. Q. -> E. y e. Q. ( y +Q y ) = A )
2		df-rex 2481	. . . . . . 7 |- ( E. y e. Q. ( y +Q y ) = A <-> E. y ( y e. Q. /\ ( y +Q y ) = A ) )
3		halfnqq 7477	. . . . . . . . . 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 1614	. . . . . . 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 2481	. . . . . . 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 1619	. . . . 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 1924	. . . 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 7474	. . . . . . . . 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 4037	. . . . . . . . 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 4036	. . . . . . . 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 1614	. . . 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 1612	. . 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 2481	. 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 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   Q.cnq 7347   +Q cplq 7349   <Q cltq 7352

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420

This theorem is referenced by:  prarloc  7570  cauappcvgprlemloc  7719  caucvgprlemloc  7742  caucvgprprlemml  7761  caucvgprprlemloc  7770