Theorem recexprlemloc 6729

Description: B is located. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	|- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.

Assertion

Ref	Expression
recexprlemloc	|- ( A e. P. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )

Distinct variable groups:   r, q, x, y, A   B, q, r, x, y

Proof of Theorem recexprlemloc

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 6573	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prnmaxl 6586	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
3	1, 2	sylan 267	. . . . . . . 8 |- ( ( A e. P. /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
4	3	adantlr 446	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
5		simprr 484	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( *Q ` r ) <Q u )
6		elprnql 6579	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
7	1, 6	sylan 267	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
8	7	ad2ant2r 478	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> u e. Q. )
9	8	adantlr 446	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> u e. Q. )
10		recrecnq 6492	. . . . . . . . . . 11 |- ( u e. Q. -> ( *Q ` ( *Q ` u ) ) = u )
11	9, 10	syl 14	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( *Q ` ( *Q ` u ) ) = u )
12	5, 11	breqtrrd 3790	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( *Q ` r ) <Q ( *Q ` ( *Q ` u ) ) )
13		recclnq 6490	. . . . . . . . . . 11 |- ( u e. Q. -> ( *Q ` u ) e. Q. )
14	9, 13	syl 14	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( *Q ` u ) e. Q. )
15		ltrelnq 6463	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
16	15	brel 4392	. . . . . . . . . . . . 13 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
17	16	adantl 262	. . . . . . . . . . . 12 |- ( ( A e. P. /\ q <Q r ) -> ( q e. Q. /\ r e. Q. ) )
18	17	ad2antrr 457	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( q e. Q. /\ r e. Q. ) )
19	18	simprd 107	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> r e. Q. )
20		ltrnqg 6518	. . . . . . . . . 10 |- ( ( ( *Q ` u ) e. Q. /\ r e. Q. ) -> ( ( *Q ` u ) <Q r <-> ( *Q ` r ) <Q ( *Q ` ( *Q ` u ) ) ) )
21	14, 19, 20	syl2anc 391	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( ( *Q ` u ) <Q r <-> ( *Q ` r ) <Q ( *Q ` ( *Q ` u ) ) ) )
22	12, 21	mpbird 156	. . . . . . . 8 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( *Q ` u ) <Q r )
23		simprl 483	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> u e. ( 1st ` A ) )
24	11, 23	eqeltrd 2114	. . . . . . . 8 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) )
25		breq1 3767	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( y <Q r <-> ( *Q ` u ) <Q r ) )
26		fveq2 5178	. . . . . . . . . . . . 13 |- ( y = ( *Q ` u ) -> ( *Q ` y ) = ( *Q ` ( *Q ` u ) ) )
27	26	eleq1d 2106	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) )
28	25, 27	anbi12d 442	. . . . . . . . . . 11 |- ( y = ( *Q ` u ) -> ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( ( *Q ` u ) <Q r /\ ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) ) )
29	28	spcegv 2641	. . . . . . . . . 10 |- ( ( *Q ` u ) e. Q. -> ( ( ( *Q ` u ) <Q r /\ ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
30		recexpr.1	. . . . . . . . . . 11 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
31	30	recexprlemelu 6721	. . . . . . . . . 10 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
32	29, 31	syl6ibr 151	. . . . . . . . 9 |- ( ( *Q ` u ) e. Q. -> ( ( ( *Q ` u ) <Q r /\ ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) ) )
33	14, 32	syl 14	. . . . . . . 8 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> ( ( ( *Q ` u ) <Q r /\ ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) ) )
34	22, 24, 33	mp2and 409	. . . . . . 7 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> r e. ( 2nd ` B ) )
35	4, 34	rexlimddv 2437	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) )
36	35	olcd 653	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
37		prnminu 6587	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
38	1, 37	sylan 267	. . . . . . . 8 |- ( ( A e. P. /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
39	38	adantlr 446	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
40		elprnqu 6580	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ v e. ( 2nd ` A ) ) -> v e. Q. )
41	1, 40	sylan 267	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ v e. ( 2nd ` A ) ) -> v e. Q. )
42	41	adantlr 446	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ v e. ( 2nd ` A ) ) -> v e. Q. )
43	42	ad2ant2r 478	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> v e. Q. )
44		recrecnq 6492	. . . . . . . . . . 11 |- ( v e. Q. -> ( *Q ` ( *Q ` v ) ) = v )
45	43, 44	syl 14	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> ( *Q ` ( *Q ` v ) ) = v )
46		simprr 484	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> v <Q ( *Q ` q ) )
47	45, 46	eqbrtrd 3784	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> ( *Q ` ( *Q ` v ) ) <Q ( *Q ` q ) )
48	17	ad2antrr 457	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> ( q e. Q. /\ r e. Q. ) )
49	48	simpld 105	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> q e. Q. )
50		recclnq 6490	. . . . . . . . . . 11 |- ( v e. Q. -> ( *Q ` v ) e. Q. )
51	43, 50	syl 14	. . . . . . . . . 10 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> ( *Q ` v ) e. Q. )
52		ltrnqg 6518	. . . . . . . . . 10 |- ( ( q e. Q. /\ ( *Q ` v ) e. Q. ) -> ( q <Q ( *Q ` v ) <-> ( *Q ` ( *Q ` v ) ) <Q ( *Q ` q ) ) )
53	49, 51, 52	syl2anc 391	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> ( q <Q ( *Q ` v ) <-> ( *Q ` ( *Q ` v ) ) <Q ( *Q ` q ) ) )
54	47, 53	mpbird 156	. . . . . . . 8 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> q <Q ( *Q ` v ) )
55		simprl 483	. . . . . . . . 9 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> v e. ( 2nd ` A ) )
56	45, 55	eqeltrd 2114	. . . . . . . 8 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) )
57		breq2 3768	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( q <Q y <-> q <Q ( *Q ` v ) ) )
58		fveq2 5178	. . . . . . . . . . . . 13 |- ( y = ( *Q ` v ) -> ( *Q ` y ) = ( *Q ` ( *Q ` v ) ) )
59	58	eleq1d 2106	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) )
60	57, 59	anbi12d 442	. . . . . . . . . . 11 |- ( y = ( *Q ` v ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` v ) /\ ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) ) )
61	60	spcegv 2641	. . . . . . . . . 10 |- ( ( *Q ` v ) e. Q. -> ( ( q <Q ( *Q ` v ) /\ ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
62	30	recexprlemell 6720	. . . . . . . . . 10 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
63	61, 62	syl6ibr 151	. . . . . . . . 9 |- ( ( *Q ` v ) e. Q. -> ( ( q <Q ( *Q ` v ) /\ ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) ) )
64	51, 63	syl 14	. . . . . . . 8 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> ( ( q <Q ( *Q ` v ) /\ ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) ) )
65	54, 56, 64	mp2and 409	. . . . . . 7 |- ( ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) /\ ( v e. ( 2nd ` A ) /\ v <Q ( *Q ` q ) ) ) -> q e. ( 1st ` B ) )
66	39, 65	rexlimddv 2437	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) )
67	66	orcd 652	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
68		ltrnqi 6519	. . . . . 6 |- ( q <Q r -> ( *Q ` r ) <Q ( *Q ` q ) )
69		prloc 6589	. . . . . 6 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ ( *Q ` r ) <Q ( *Q ` q ) ) -> ( ( *Q ` r ) e. ( 1st ` A ) \/ ( *Q ` q ) e. ( 2nd ` A ) ) )
70	1, 68, 69	syl2an 273	. . . . 5 |- ( ( A e. P. /\ q <Q r ) -> ( ( *Q ` r ) e. ( 1st ` A ) \/ ( *Q ` q ) e. ( 2nd ` A ) ) )
71	36, 67, 70	mpjaodan 711	. . . 4 |- ( ( A e. P. /\ q <Q r ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
72	71	ex 108	. . 3 |- ( A e. P. -> ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
73	72	ralrimivw 2393	. 2 |- ( A e. P. -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
74	73	ralrimivw 2393	1 |- ( A e. P. -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   <.cop 3378   class class class wbr 3764   ` cfv 4902   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   *Qcrq 6382   <Q cltq 6383   P.cnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-mi 6404  df-lti 6405  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  recexprlempr  6730