Theorem ltbtwnnqq 7788

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 7738	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4829	. . . 4 |- ( A <Q B -> ( A e. Q. /\ B e. Q. ) )
3	2	simpld 112	. . 3 |- ( A <Q B -> A e. Q. )
4		ltexnqi 7782	. . 3 |- ( A <Q B -> E. y e. Q. ( A +Q y ) = B )
5		nsmallnq 7786	. . . . . 6 |- ( y e. Q. -> E. z z <Q y )
6	1	brel 4829	. . . . . . . . . . . . . . 15 |- ( z <Q y -> ( z e. Q. /\ y e. Q. ) )
7	6	simpld 112	. . . . . . . . . . . . . 14 |- ( z <Q y -> z e. Q. )
8		ltaddnq 7780	. . . . . . . . . . . . . 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 7775	. . . . . . . . . . . . 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 4135	. . . . . . . . . . . . 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 7748	. . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . 14 |- ( x = ( A +Q z ) -> ( A <Q x <-> A <Q ( A +Q z ) ) )
22		breq1 4134	. . . . . . . . . . . . . 14 |- ( x = ( A +Q z ) -> ( x <Q B <-> ( A +Q z ) <Q B ) )
23	21, 22	anbi12d 477	. . . . . . . . . . . . 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 2933	. . . . . . . . . . 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 437	. . . . . . . . . 10 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
27	26	3impa 1225	. . . . . . . . 9 |- ( ( z <Q y /\ A e. Q. /\ ( A +Q y ) = B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
28	27	3coml 1241	. . . . . . . 8 |- ( ( A e. Q. /\ ( A +Q y ) = B /\ z <Q y ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
29	28	3expia 1236	. . . . . . 7 |- ( ( A e. Q. /\ ( A +Q y ) = B ) -> ( z <Q y -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
30	29	exlimdv 1872	. . . . . 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 2668	. . 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 7771	. . . 4 |- <Q Or Q.
36	35, 1	sotri 5186	. . 3 |- ( ( A <Q x /\ x <Q B ) -> A <Q B )
37	36	rexlimivw 2664	. 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 ) )

This proof depends on syntax axioms:   /\ 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-po 4443  df-iso 4444  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:  ltbtwnnq  7789  nqprrnd  7916  appdivnq  7936  ltnqpr  7966  ltnqpri  7967  recexprlemopl  7998  recexprlemopu  8000  cauappcvgprlemopl  8019  cauappcvgprlemopu  8021  cauappcvgprlem2  8033  caucvgprlemopl  8042  caucvgprlemopu  8044  caucvgprlem2  8053  suplocexprlemru  8092  suplocexprlemloc  8094