Theorem recexprlemloc 7698

Description: B is located. Lemma for recexpr 7705. (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 7542	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prnmaxl 7555	. . . . . . . . 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 283	. . . . . . . 8 |- ( ( A e. P. /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
4	3	adantlr 477	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
5		simprr 531	. . . . . . . . . 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 7548	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
7	1, 6	sylan 283	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
8	7	ad2ant2r 509	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> u e. Q. )
9	8	adantlr 477	. . . . . . . . . . 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 7461	. . . . . . . . . . 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 4061	. . . . . . . . 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 7459	. . . . . . . . . . 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 7432	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
16	15	brel 4715	. . . . . . . . . . . . 13 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
17	16	adantl 277	. . . . . . . . . . . 12 |- ( ( A e. P. /\ q <Q r ) -> ( q e. Q. /\ r e. Q. ) )
18	17	ad2antrr 488	. . . . . . . . . . 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 114	. . . . . . . . . 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 7487	. . . . . . . . . 10 |- ( ( ( *Q ` u ) e. Q. /\ r e. Q. ) -> ( ( *Q ` u ) <Q r <-> ( *Q ` r ) <Q ( *Q ` ( *Q ` u ) ) ) )
21	14, 19, 20	syl2anc 411	. . . . . . . . 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 167	. . . . . . . 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 529	. . . . . . . . 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 2273	. . . . . . . 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 4036	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( y <Q r <-> ( *Q ` u ) <Q r ) )
26		fveq2 5558	. . . . . . . . . . . . 13 |- ( y = ( *Q ` u ) -> ( *Q ` y ) = ( *Q ` ( *Q ` u ) ) )
27	26	eleq1d 2265	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) )
28	25, 27	anbi12d 473	. . . . . . . . . . 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 2852	. . . . . . . . . 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 7690	. . . . . . . . . 10 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
32	29, 31	imbitrrdi 162	. . . . . . . . 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 433	. . . . . . 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 2619	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) )
36	35	olcd 735	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
37		prnminu 7556	. . . . . . . . 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 283	. . . . . . . 8 |- ( ( A e. P. /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
39	38	adantlr 477	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
40		elprnqu 7549	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ v e. ( 2nd ` A ) ) -> v e. Q. )
41	1, 40	sylan 283	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ v e. ( 2nd ` A ) ) -> v e. Q. )
42	41	adantlr 477	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ v e. ( 2nd ` A ) ) -> v e. Q. )
43	42	ad2ant2r 509	. . . . . . . . . . 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 7461	. . . . . . . . . . 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 531	. . . . . . . . . 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 4055	. . . . . . . . 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 488	. . . . . . . . . . 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 112	. . . . . . . . . 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 7459	. . . . . . . . . . 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 7487	. . . . . . . . . 10 |- ( ( q e. Q. /\ ( *Q ` v ) e. Q. ) -> ( q <Q ( *Q ` v ) <-> ( *Q ` ( *Q ` v ) ) <Q ( *Q ` q ) ) )
53	49, 51, 52	syl2anc 411	. . . . . . . . 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 167	. . . . . . . 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 529	. . . . . . . . 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 2273	. . . . . . . 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 4037	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( q <Q y <-> q <Q ( *Q ` v ) ) )
58		fveq2 5558	. . . . . . . . . . . . 13 |- ( y = ( *Q ` v ) -> ( *Q ` y ) = ( *Q ` ( *Q ` v ) ) )
59	58	eleq1d 2265	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) )
60	57, 59	anbi12d 473	. . . . . . . . . . 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 2852	. . . . . . . . . 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 7689	. . . . . . . . . 10 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
63	61, 62	imbitrrdi 162	. . . . . . . . 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 433	. . . . . . 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 2619	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) )
67	66	orcd 734	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
68		ltrnqi 7488	. . . . . 6 |- ( q <Q r -> ( *Q ` r ) <Q ( *Q ` q ) )
69		prloc 7558	. . . . . 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 289	. . . . 5 |- ( ( A e. P. /\ q <Q r ) -> ( ( *Q ` r ) e. ( 1st ` A ) \/ ( *Q ` q ) e. ( 2nd ` A ) ) )
71	36, 67, 70	mpjaodan 799	. . . 4 |- ( ( A e. P. /\ q <Q r ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
72	71	ex 115	. . 3 |- ( A e. P. -> ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
73	72	ralrimivw 2571	. 2 |- ( A e. P. -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
74	73	ralrimivw 2571	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 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1506   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   <.cop 3625   class class class wbr 4033   ` cfv 5258   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   *Qcrq 7351   <Q cltq 7352   P.cnp 7358

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-mi 7373  df-lti 7374  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-inp 7533

This theorem is referenced by:  recexprlempr  7699