Theorem nqprloc 7912

Description: A cut produced from a rational is located. Lemma for nqprlu 7914. (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 7763	. . . . . . 7 |- ( ( q e. Q. /\ A e. Q. ) -> ( q <Q A \/ q = A \/ A <Q q ) )
2	1	ancoms 268	. . . . . 6 |- ( ( A e. Q. /\ q e. Q. ) -> ( q <Q A \/ q = A \/ A <Q q ) )
3	2	ad2antrr 492	. . . . 5 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( q <Q A \/ q = A \/ A <Q q ) )
4		vex 2824	. . . . . . . . . 10 |- q e. _V
5		breq1 4133	. . . . . . . . . 10 |- ( x = q -> ( x <Q A <-> q <Q A ) )
6	4, 5	elab 2970	. . . . . . . . 9 |- ( q e. { x | x <Q A } <-> q <Q A )
7	6	biimpri 133	. . . . . . . 8 |- ( q <Q A -> q e. { x | x <Q A } )
8	7	orcd 745	. . . . . . 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 110	. . . . . . . 8 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> q <Q r )
11		breq1 4133	. . . . . . . 8 |- ( q = A -> ( q <Q r <-> A <Q r ) )
12	10, 11	syl5ibcom 155	. . . . . . 7 |- ( ( ( ( A e. Q. /\ q e. Q. ) /\ r e. Q. ) /\ q <Q r ) -> ( q = A -> A <Q r ) )
13		vex 2824	. . . . . . . . 9 |- r e. _V
14		breq2 4134	. . . . . . . . 9 |- ( x = r -> ( A <Q x <-> A <Q r ) )
15	13, 14	elab 2970	. . . . . . . 8 |- ( r e. { x | A <Q x } <-> A <Q r )
16		olc 723	. . . . . . . 8 |- ( r e. { x | A <Q x } -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) )
17	15, 16	sylbir 135	. . . . . . 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 7765	. . . . . . . . . 10 |- <Q Or Q.
20		ltrelnq 7732	. . . . . . . . . 10 |- <Q C_ ( Q. X. Q. )
21	19, 20	sotri 5183	. . . . . . . . 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 116	. . . . . . 7 |- ( q <Q r -> ( A <Q q -> ( q e. { x | x <Q A } \/ r e. { x | A <Q x } ) ) )
24	23	adantl 277	. . . . . 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 1345	. . . . 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 115	. . 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 2623	. 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 2623	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 } ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   \/ w3o 1008   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   class class class wbr 4130   Q.cnq 7647   <Q cltq 7652

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-mi 7673  df-lti 7674  df-enq 7714  df-nqqs 7715  df-ltnqqs 7720

This theorem is used by:  nqprxx  7913