Theorem recexprlemloc 7988

Description: B is located. Lemma for recexpr 7995. (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 7832	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prnmaxl 7845	. . . . . . . . 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 481	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> E. u e. ( 1st ` A ) ( *Q ` r ) <Q u )
5		simprr 537	. . . . . . . . . 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 7838	. . . . . . . . . . . . . 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 513	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ ( u e. ( 1st ` A ) /\ ( *Q ` r ) <Q u ) ) -> u e. Q. )
9	8	adantlr 481	. . . . . . . . . . 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 7751	. . . . . . . . . . 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 4153	. . . . . . . . 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 7749	. . . . . . . . . . 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 7722	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
16	15	brel 4822	. . . . . . . . . . . . 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 492	. . . . . . . . . . 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 7777	. . . . . . . . . 10 |- ( ( ( *Q ` u ) e. Q. /\ r e. Q. ) -> ( ( *Q ` u ) <Q r <-> ( *Q ` r ) <Q ( *Q ` ( *Q ` u ) ) ) )
21	14, 19, 20	syl2anc 415	. . . . . . . . 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 535	. . . . . . . . 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 2315	. . . . . . . 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 4128	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( y <Q r <-> ( *Q ` u ) <Q r ) )
26		fveq2 5690	. . . . . . . . . . . . 13 |- ( y = ( *Q ` u ) -> ( *Q ` y ) = ( *Q ` ( *Q ` u ) ) )
27	26	eleq1d 2307	. . . . . . . . . . . 12 |- ( y = ( *Q ` u ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` u ) ) e. ( 1st ` A ) ) )
28	25, 27	anbi12d 477	. . . . . . . . . . 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 2913	. . . . . . . . . 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 7980	. . . . . . . . . 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 437	. . . . . . 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 2673	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) )
36	35	olcd 746	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` r ) e. ( 1st ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
37		prnminu 7846	. . . . . . . . 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 481	. . . . . . 7 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> E. v e. ( 2nd ` A ) v <Q ( *Q ` q ) )
40		elprnqu 7839	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ q <Q r ) /\ v e. ( 2nd ` A ) ) -> v e. Q. )
43	42	ad2ant2r 513	. . . . . . . . . . 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 7751	. . . . . . . . . . 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 537	. . . . . . . . . 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 4147	. . . . . . . . 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 492	. . . . . . . . . . 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 7749	. . . . . . . . . . 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 7777	. . . . . . . . . 10 |- ( ( q e. Q. /\ ( *Q ` v ) e. Q. ) -> ( q <Q ( *Q ` v ) <-> ( *Q ` ( *Q ` v ) ) <Q ( *Q ` q ) ) )
53	49, 51, 52	syl2anc 415	. . . . . . . . 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 535	. . . . . . . . 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 2315	. . . . . . . 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 4129	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( q <Q y <-> q <Q ( *Q ` v ) ) )
58		fveq2 5690	. . . . . . . . . . . . 13 |- ( y = ( *Q ` v ) -> ( *Q ` y ) = ( *Q ` ( *Q ` v ) ) )
59	58	eleq1d 2307	. . . . . . . . . . . 12 |- ( y = ( *Q ` v ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` v ) ) e. ( 2nd ` A ) ) )
60	57, 59	anbi12d 477	. . . . . . . . . . 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 2913	. . . . . . . . . 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 7979	. . . . . . . . . 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 437	. . . . . . 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 2673	. . . . . 6 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) )
67	66	orcd 745	. . . . 5 |- ( ( ( A e. P. /\ q <Q r ) /\ ( *Q ` q ) e. ( 2nd ` A ) ) -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) )
68		ltrnqi 7778	. . . . . 6 |- ( q <Q r -> ( *Q ` r ) <Q ( *Q ` q ) )
69		prloc 7848	. . . . . 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 810	. . . 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 2624	. 2 |- ( A e. P. -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` B ) \/ r e. ( 2nd ` B ) ) ) )
74	73	ralrimivw 2624	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 720   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   <.cop 3708   class class class wbr 4125   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   *Qcrq 7641   <Q cltq 7642   P.cnp 7648

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-mi 7663  df-lti 7664  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-inp 7823

This theorem is referenced by:  recexprlempr  7989