Theorem ltbtwnnqq 7732

Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by Jim Kingdon, 24-Sep-2019.)

Assertion

Ref	Expression
ltbtwnnqq	|- ( A <Q B <-> E. x e. Q. ( A <Q x /\ x <Q B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem ltbtwnnqq

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 7682	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4804	. . . 4 |- ( A <Q B -> ( A e. Q. /\ B e. Q. ) )
3	2	simpld 112	. . 3 |- ( A <Q B -> A e. Q. )
4		ltexnqi 7726	. . 3 |- ( A <Q B -> E. y e. Q. ( A +Q y ) = B )
5		nsmallnq 7730	. . . . . 6 |- ( y e. Q. -> E. z z <Q y )
6	1	brel 4804	. . . . . . . . . . . . . . 15 |- ( z <Q y -> ( z e. Q. /\ y e. Q. ) )
7	6	simpld 112	. . . . . . . . . . . . . 14 |- ( z <Q y -> z e. Q. )
8		ltaddnq 7724	. . . . . . . . . . . . . 14 |- ( ( A e. Q. /\ z e. Q. ) -> A <Q ( A +Q z ) )
9	7, 8	sylan2 286	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ z <Q y ) -> A <Q ( A +Q z ) )
10	9	ancoms 268	. . . . . . . . . . . 12 |- ( ( z <Q y /\ A e. Q. ) -> A <Q ( A +Q z ) )
11	10	adantr 276	. . . . . . . . . . 11 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> A <Q ( A +Q z ) )
12		ltanqi 7719	. . . . . . . . . . . . 13 |- ( ( z <Q y /\ A e. Q. ) -> ( A +Q z ) <Q ( A +Q y ) )
13	12	adantr 276	. . . . . . . . . . . 12 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( A +Q z ) <Q ( A +Q y ) )
14		breq2 4115	. . . . . . . . . . . . 13 |- ( ( A +Q y ) = B -> ( ( A +Q z ) <Q ( A +Q y ) <-> ( A +Q z ) <Q B ) )
15	14	adantl 277	. . . . . . . . . . . 12 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( ( A +Q z ) <Q ( A +Q y ) <-> ( A +Q z ) <Q B ) )
16	13, 15	mpbid 147	. . . . . . . . . . 11 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( A +Q z ) <Q B )
17		addclnq 7692	. . . . . . . . . . . . . . 15 |- ( ( A e. Q. /\ z e. Q. ) -> ( A +Q z ) e. Q. )
18	7, 17	sylan2 286	. . . . . . . . . . . . . 14 |- ( ( A e. Q. /\ z <Q y ) -> ( A +Q z ) e. Q. )
19	18	ancoms 268	. . . . . . . . . . . . 13 |- ( ( z <Q y /\ A e. Q. ) -> ( A +Q z ) e. Q. )
20	19	adantr 276	. . . . . . . . . . . 12 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( A +Q z ) e. Q. )
21		breq2 4115	. . . . . . . . . . . . . 14 |- ( x = ( A +Q z ) -> ( A <Q x <-> A <Q ( A +Q z ) ) )
22		breq1 4114	. . . . . . . . . . . . . 14 |- ( x = ( A +Q z ) -> ( x <Q B <-> ( A +Q z ) <Q B ) )
23	21, 22	anbi12d 473	. . . . . . . . . . . . 13 |- ( x = ( A +Q z ) -> ( ( A <Q x /\ x <Q B ) <-> ( A <Q ( A +Q z ) /\ ( A +Q z ) <Q B ) ) )
24	23	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) /\ x = ( A +Q z ) ) -> ( ( A <Q x /\ x <Q B ) <-> ( A <Q ( A +Q z ) /\ ( A +Q z ) <Q B ) ) )
25	20, 24	rspcedv 2927	. . . . . . . . . . 11 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( ( A <Q ( A +Q z ) /\ ( A +Q z ) <Q B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
26	11, 16, 25	mp2and 433	. . . . . . . . . 10 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
27	26	3impa 1221	. . . . . . . . 9 |- ( ( z <Q y /\ A e. Q. /\ ( A +Q y ) = B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
28	27	3coml 1237	. . . . . . . 8 |- ( ( A e. Q. /\ ( A +Q y ) = B /\ z <Q y ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
29	28	3expia 1232	. . . . . . 7 |- ( ( A e. Q. /\ ( A +Q y ) = B ) -> ( z <Q y -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
30	29	exlimdv 1868	. . . . . 6 |- ( ( A e. Q. /\ ( A +Q y ) = B ) -> ( E. z z <Q y -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
31	5, 30	syl5 32	. . . . 5 |- ( ( A e. Q. /\ ( A +Q y ) = B ) -> ( y e. Q. -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
32	31	impancom 260	. . . 4 |- ( ( A e. Q. /\ y e. Q. ) -> ( ( A +Q y ) = B -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
33	32	rexlimdva 2662	. . 3 |- ( A e. Q. -> ( E. y e. Q. ( A +Q y ) = B -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
34	3, 4, 33	sylc 62	. 2 |- ( A <Q B -> E. x e. Q. ( A <Q x /\ x <Q B ) )
35		ltsonq 7715	. . . 4 |- <Q Or Q.
36	35, 1	sotri 5160	. . 3 |- ( ( A <Q x /\ x <Q B ) -> A <Q B )
37	36	rexlimivw 2658	. 2 |- ( E. x e. Q. ( A <Q x /\ x <Q B ) -> A <Q B )
38	34, 37	impbii 126	1 |- ( A <Q B <-> E. x e. Q. ( A <Q x /\ x <Q B ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   class class class wbr 4111  (class class class)co 6052   Q.cnq 7597   +Q cplq 7599   <Q cltq 7602

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 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

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 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-mpq 7662  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-mqqs 7667  df-1nqqs 7668  df-rq 7669  df-ltnqqs 7670

This theorem is referenced by:  ltbtwnnq  7733  nqprrnd  7860  appdivnq  7880  ltnqpr  7910  ltnqpri  7911  recexprlemopl  7942  recexprlemopu  7944  cauappcvgprlemopl  7963  cauappcvgprlemopu  7965  cauappcvgprlem2  7977  caucvgprlemopl  7986  caucvgprlemopu  7988  caucvgprlem2  7997  suplocexprlemru  8036  suplocexprlemloc  8038