Theorem ltbtwnnqq 7356

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 7306	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4656	. . . 4 |- ( A <Q B -> ( A e. Q. /\ B e. Q. ) )
3	2	simpld 111	. . 3 |- ( A <Q B -> A e. Q. )
4		ltexnqi 7350	. . 3 |- ( A <Q B -> E. y e. Q. ( A +Q y ) = B )
5		nsmallnq 7354	. . . . . 6 |- ( y e. Q. -> E. z z <Q y )
6	1	brel 4656	. . . . . . . . . . . . . . 15 |- ( z <Q y -> ( z e. Q. /\ y e. Q. ) )
7	6	simpld 111	. . . . . . . . . . . . . 14 |- ( z <Q y -> z e. Q. )
8		ltaddnq 7348	. . . . . . . . . . . . . 14 |- ( ( A e. Q. /\ z e. Q. ) -> A <Q ( A +Q z ) )
9	7, 8	sylan2 284	. . . . . . . . . . . . 13 |- ( ( A e. Q. /\ z <Q y ) -> A <Q ( A +Q z ) )
10	9	ancoms 266	. . . . . . . . . . . 12 |- ( ( z <Q y /\ A e. Q. ) -> A <Q ( A +Q z ) )
11	10	adantr 274	. . . . . . . . . . 11 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> A <Q ( A +Q z ) )
12		ltanqi 7343	. . . . . . . . . . . . 13 |- ( ( z <Q y /\ A e. Q. ) -> ( A +Q z ) <Q ( A +Q y ) )
13	12	adantr 274	. . . . . . . . . . . 12 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( A +Q z ) <Q ( A +Q y ) )
14		breq2 3986	. . . . . . . . . . . . 13 |- ( ( A +Q y ) = B -> ( ( A +Q z ) <Q ( A +Q y ) <-> ( A +Q z ) <Q B ) )
15	14	adantl 275	. . . . . . . . . . . 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 146	. . . . . . . . . . 11 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( A +Q z ) <Q B )
17		addclnq 7316	. . . . . . . . . . . . . . 15 |- ( ( A e. Q. /\ z e. Q. ) -> ( A +Q z ) e. Q. )
18	7, 17	sylan2 284	. . . . . . . . . . . . . 14 |- ( ( A e. Q. /\ z <Q y ) -> ( A +Q z ) e. Q. )
19	18	ancoms 266	. . . . . . . . . . . . 13 |- ( ( z <Q y /\ A e. Q. ) -> ( A +Q z ) e. Q. )
20	19	adantr 274	. . . . . . . . . . . 12 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> ( A +Q z ) e. Q. )
21		breq2 3986	. . . . . . . . . . . . . 14 |- ( x = ( A +Q z ) -> ( A <Q x <-> A <Q ( A +Q z ) ) )
22		breq1 3985	. . . . . . . . . . . . . 14 |- ( x = ( A +Q z ) -> ( x <Q B <-> ( A +Q z ) <Q B ) )
23	21, 22	anbi12d 465	. . . . . . . . . . . . 13 |- ( x = ( A +Q z ) -> ( ( A <Q x /\ x <Q B ) <-> ( A <Q ( A +Q z ) /\ ( A +Q z ) <Q B ) ) )
24	23	adantl 275	. . . . . . . . . . . 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 2834	. . . . . . . . . . 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 430	. . . . . . . . . 10 |- ( ( ( z <Q y /\ A e. Q. ) /\ ( A +Q y ) = B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
27	26	3impa 1184	. . . . . . . . 9 |- ( ( z <Q y /\ A e. Q. /\ ( A +Q y ) = B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
28	27	3coml 1200	. . . . . . . 8 |- ( ( A e. Q. /\ ( A +Q y ) = B /\ z <Q y ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
29	28	3expia 1195	. . . . . . 7 |- ( ( A e. Q. /\ ( A +Q y ) = B ) -> ( z <Q y -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
30	29	exlimdv 1807	. . . . . 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 258	. . . 4 |- ( ( A e. Q. /\ y e. Q. ) -> ( ( A +Q y ) = B -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
33	32	rexlimdva 2583	. . 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 7339	. . . 4 |- <Q Or Q.
36	35, 1	sotri 4999	. . 3 |- ( ( A <Q x /\ x <Q B ) -> A <Q B )
37	36	rexlimivw 2579	. 2 |- ( E. x e. Q. ( A <Q x /\ x <Q B ) -> A <Q B )
38	34, 37	impbii 125	1 |- ( A <Q B <-> E. x e. Q. ( A <Q x /\ x <Q B ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294

This theorem is referenced by:  ltbtwnnq  7357  nqprrnd  7484  appdivnq  7504  ltnqpr  7534  ltnqpri  7535  recexprlemopl  7566  recexprlemopu  7568  cauappcvgprlemopl  7587  cauappcvgprlemopu  7589  cauappcvgprlem2  7601  caucvgprlemopl  7610  caucvgprlemopu  7612  caucvgprlem2  7621  suplocexprlemru  7660  suplocexprlemloc  7662