Theorem nqprloc 7486

Description: A cut produced from a rational is located. Lemma for nqprlu 7488. (Contributed by Jim Kingdon, 8-Dec-2019.)

Assertion

Ref	Expression
nqprloc	|- ( A e. Q. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )

Distinct variable group:   x, A, r, q

Proof of Theorem nqprloc

Step	Hyp	Ref	Expression
1		nqtri3or 7337	. . . . . . 7 |- ( ( q e. Q. /\ A e. Q. ) -> ( q <Q A \/ q = A \/ A <Q q ) )
2	1	ancoms 266	. . . . . 6 |- ( ( A e. Q. /\ q e. Q. ) -> ( q <Q A \/ q = A \/ A <Q q ) )
3	2	ad2antrr 480	. . . . 5 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( q <Q A \/ q = A \/ A <Q q ) )
4		vex 2729	. . . . . . . . . 10 |- q e. _V
5		breq1 3985	. . . . . . . . . 10 |- ( x = q -> ( x <Q A <-> q <Q A ) )
6	4, 5	elab 2870	. . . . . . . . 9 |- ( q e. { x | x <Q A } <-> q <Q A )
7	6	biimpri 132	. . . . . . . 8 |- ( q <Q A -> q e. { x | x <Q A } )
8	7	orcd 723	. . . . . . 7 |- ( q <Q A -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) )
9	8	a1i 9	. . . . . 6 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( q <Q A -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
10		simpr 109	. . . . . . . 8 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> q <Q r )
11		breq1 3985	. . . . . . . 8 |- ( q = A -> ( q <Q r <-> A <Q r ) )
12	10, 11	syl5ibcom 154	. . . . . . 7 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( q = A -> A <Q r ) )
13		vex 2729	. . . . . . . . 9 |- r e. _V
14		breq2 3986	. . . . . . . . 9 |- ( x = r -> ( A <Q x <-> A <Q r ) )
15	13, 14	elab 2870	. . . . . . . 8 |- ( r e. { x | A <Q x } <-> A <Q r )
16		olc 701	. . . . . . . 8 |- ( r e. { x | A <Q x } -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) )
17	15, 16	sylbir 134	. . . . . . 7 |- ( A <Q r -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) )
18	12, 17	syl6 33	. . . . . 6 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( q = A -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
19		ltsonq 7339	. . . . . . . . . 10 |- <Q Or Q.
20		ltrelnq 7306	. . . . . . . . . 10 |- <Q C_ ( Q. X. Q. )
21	19, 20	sotri 4999	. . . . . . . . 9 |- ( ( A <Q q /\ q <Q r ) -> A <Q r )
22	21, 17	syl 14	. . . . . . . 8 |- ( ( A <Q q /\ q <Q r ) -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) )
23	22	expcom 115	. . . . . . 7 |- ( q <Q r -> ( A <Q q -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
24	23	adantl 275	. . . . . 6 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( A <Q q -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
25	9, 18, 24	3jaod 1294	. . . . 5 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( ( q <Q A \/ q = A \/ A <Q q ) -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
26	3, 25	mpd 13	. . . 4 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) )
27	26	ex 114	. . 3 |- ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) -> ( q <Q r -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
28	27	ralrimiva 2539	. 2 |- ( ( A e. Q. /\ q e. Q. ) -> A. r e. Q. ( q <Q r -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
29	28	ralrimiva 2539	1 |- ( A e. Q. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   \/ w3o 967   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   class class class wbr 3982   Q.cnq 7221   <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-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-mi 7247  df-lti 7248  df-enq 7288  df-nqqs 7289  df-ltnqqs 7294

This theorem is referenced by:  nqprxx  7487