Theorem recexprlemloc 7463

Description: B is located. Lemma for recexpr 7470. (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 7307	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prnmaxl 7320	. . . . . . . . 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 281	. . . . . . . 8 |- ( ( A e. P. /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
4	3	adantlr 469	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
5		simprr 522	. . . . . . . . . 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 7313	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
7	1, 6	sylan 281	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ u e. ( 1st ` A ) ) -> u e. Q. )
8	7	ad2ant2r 501	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> u e. Q. )
9	8	adantlr 469	. . . . . . . . . . 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 7226	. . . . . . . . . . 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 3964	. . . . . . . . 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 7224	. . . . . . . . . . 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 7197	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
16	15	brel 4599	. . . . . . . . . . . . 13 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
17	16	adantl 275	. . . . . . . . . . . 12 |- ( ( A e. P. /\ q <Q r ) -> ( q e. Q. /\ r e. Q. ) )
18	17	ad2antrr 480	. . . . . . . . . . 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 113	. . . . . . . . . 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 7252	. . . . . . . . . 10 |- ( ( ( *Q ` u ) e. Q. /\ r e. Q. ) -> ( ( *Q ` u ) <Q r <-> ( *Q ` r ) <Q ( *Q ` ( *Q ` u ) ) ) )
21	14, 19, 20	syl2anc 409	. . . . . . . . 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 166	. . . . . . . 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 521	. . . . . . . . 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 2217	. . . . . . . 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 3940	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( y <Q r <-> ( *Q ` u ) <Q r ) )
26		fveq2 5429	. . . . . . . . . . . . 13 |- ( y = ( *Q ` u ) -> ( *Q ` y ) = ( *Q ` ( *Q ` u ) ) )
27	26	eleq1d 2209	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) )
28	25, 27	anbi12d 465	. . . . . . . . . . 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 2777	. . . . . . . . . 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 7455	. . . . . . . . . 10 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
32	29, 31	syl6ibr 161	. . . . . . . . 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 430	. . . . . . 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 2557	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) )
36	35	olcd 724	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
37		prnminu 7321	. . . . . . . . 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 281	. . . . . . . 8 |- ( ( A e. P. /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
39	38	adantlr 469	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
40		elprnqu 7314	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ v e. ( 2nd ` A ) ) -> v e. Q. )
41	1, 40	sylan 281	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ v e. ( 2nd ` A ) ) -> v e. Q. )
42	41	adantlr 469	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ v e. ( 2nd ` A ) ) -> v e. Q. )
43	42	ad2ant2r 501	. . . . . . . . . . 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 7226	. . . . . . . . . . 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 522	. . . . . . . . . 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 3958	. . . . . . . . 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 480	. . . . . . . . . . 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 111	. . . . . . . . . 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 7224	. . . . . . . . . . 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 7252	. . . . . . . . . 10 |- ( ( q e. Q. /\ ( *Q ` v ) e. Q. ) -> ( q <Q ( *Q ` v ) <-> ( *Q ` ( *Q ` v ) ) <Q ( *Q ` q ) ) )
53	49, 51, 52	syl2anc 409	. . . . . . . . 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 166	. . . . . . . 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 521	. . . . . . . . 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 2217	. . . . . . . 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 3941	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( q <Q y <-> q <Q ( *Q ` v ) ) )
58		fveq2 5429	. . . . . . . . . . . . 13 |- ( y = ( *Q ` v ) -> ( *Q ` y ) = ( *Q ` ( *Q ` v ) ) )
59	58	eleq1d 2209	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) )
60	57, 59	anbi12d 465	. . . . . . . . . . 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 2777	. . . . . . . . . 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 7454	. . . . . . . . . 10 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
63	61, 62	syl6ibr 161	. . . . . . . . 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 430	. . . . . . 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 2557	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) )
67	66	orcd 723	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
68		ltrnqi 7253	. . . . . 6 |- ( q <Q r -> ( *Q ` r ) <Q ( *Q ` q ) )
69		prloc 7323	. . . . . 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 287	. . . . 5 |- ( ( A e. P. /\ q <Q r ) -> ( ( *Q ` r ) e. ( 1st ` A ) \/ ( *Q ` q ) e. ( 2nd ` A ) ) )
71	36, 67, 70	mpjaodan 788	. . . 4 |- ( ( A e. P. /\ q <Q r ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
72	71	ex 114	. . 3 |- ( A e. P. -> ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
73	72	ralrimivw 2509	. 2 |- ( A e. P. -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
74	73	ralrimivw 2509	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 103   <-> wb 104   \/ wo 698   = wceq 1332   E.wex 1469   e. wcel 1481   {cab 2126   A.wral 2417   E.wrex 2418   <.cop 3535   class class class wbr 3937   ` cfv 5131   1stc1st 6044   2ndc2nd 6045   Q.cnq 7112   *Qcrq 7116   <Q cltq 7117   P.cnp 7123

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-mi 7138  df-lti 7139  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-inp 7298

This theorem is referenced by:  recexprlempr  7464