Theorem ltbtwnnqq 7416

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 7366	. . . . 5 |- <Q C_ ( Q. X. Q. )
2	1	brel 4680	. . . 4 |- ( A <Q B -> ( A e. Q. /\ B e. Q. ) )
3	2	simpld 112	. . 3 |- ( A <Q B -> A e. Q. )
4		ltexnqi 7410	. . 3 |- ( A <Q B -> E. y e. Q. ( A +Q y ) = B )
5		nsmallnq 7414	. . . . . 6 |- ( y e. Q. -> E. z z <Q y )
6	1	brel 4680	. . . . . . . . . . . . . . 15 |- ( z <Q y -> ( z e. Q. /\ y e. Q. ) )
7	6	simpld 112	. . . . . . . . . . . . . 14 |- ( z <Q y -> z e. Q. )
8		ltaddnq 7408	. . . . . . . . . . . . . 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 7403	. . . . . . . . . . . . 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 4009	. . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 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 4009	. . . . . . . . . . . . . 14 |- ( x = ( A +Q z ) -> ( A <Q x <-> A <Q ( A +Q z ) ) )
22		breq1 4008	. . . . . . . . . . . . . 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 2847	. . . . . . . . . . 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 1194	. . . . . . . . 9 |- ( ( z <Q y /\ A e. Q. /\ ( A +Q y ) = B ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
28	27	3coml 1210	. . . . . . . 8 |- ( ( A e. Q. /\ ( A +Q y ) = B /\ z <Q y ) -> E. x e. Q. ( A <Q x /\ x <Q B ) )
29	28	3expia 1205	. . . . . . 7 |- ( ( A e. Q. /\ ( A +Q y ) = B ) -> ( z <Q y -> E. x e. Q. ( A <Q x /\ x <Q B ) ) )
30	29	exlimdv 1819	. . . . . 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 2594	. . 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 7399	. . . 4 |- <Q Or Q.
36	35, 1	sotri 5026	. . 3 |- ( ( A <Q x /\ x <Q B ) -> A <Q B )
37	36	rexlimivw 2590	. 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 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   class class class wbr 4005  (class class class)co 5877   Q.cnq 7281   +Q cplq 7283   <Q cltq 7286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354

This theorem is referenced by:  ltbtwnnq  7417  nqprrnd  7544  appdivnq  7564  ltnqpr  7594  ltnqpri  7595  recexprlemopl  7626  recexprlemopu  7628  cauappcvgprlemopl  7647  cauappcvgprlemopu  7649  cauappcvgprlem2  7661  caucvgprlemopl  7670  caucvgprlemopu  7672  caucvgprlem2  7681  suplocexprlemru  7720  suplocexprlemloc  7722