Theorem recexprlemm 6814

Description: B is inhabited. Lemma for recexpr 6828. (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
recexprlemm	|- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

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

Proof of Theorem recexprlemm

Step	Hyp	Ref	Expression
1		prop 6665	. . 3 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		prmu 6668	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
3		recclnq 6582	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) e. Q. )
4		nsmallnqq 6602	. . . . . . 7 |- ( ( *Q ` x ) e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
5	3, 4	syl 14	. . . . . 6 |- ( x e. Q. -> E. q e. Q. q <Q ( *Q ` x ) )
6	5	adantr 270	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q <Q ( *Q ` x ) )
7		recrecnq 6584	. . . . . . . . . . . 12 |- ( x e. Q. -> ( *Q ` ( *Q ` x ) ) = x )
8	7	eleq1d 2147	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) <-> x e. ( 2nd ` A ) ) )
9	8	anbi2d 451	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) ) )
10		breq2 3789	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( q <Q y <-> q <Q ( *Q ` x ) ) )
11		fveq2 5198	. . . . . . . . . . . . . 14 |- ( y = ( *Q ` x ) -> ( *Q ` y ) = ( *Q ` ( *Q ` x ) ) )
12	11	eleq1d 2147	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 2nd ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) )
13	10, 12	anbi12d 456	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) <-> ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) ) )
14	13	spcegv 2686	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
15	3, 14	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ ( *Q ` ( *Q ` x ) ) e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
16	9, 15	sylbird 168	. . . . . . . . 9 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) ) )
17		recexpr.1	. . . . . . . . . 10 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
18	17	recexprlemell 6812	. . . . . . . . 9 |- ( q e. ( 1st ` B ) <-> E. y ( q <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) )
19	16, 18	syl6ibr 160	. . . . . . . 8 |- ( x e. Q. -> ( ( q <Q ( *Q ` x ) /\ x e. ( 2nd ` A ) ) -> q e. ( 1st ` B ) ) )
20	19	expcomd 1370	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 2nd ` A ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) ) )
21	20	imp 122	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( q <Q ( *Q ` x ) -> q e. ( 1st ` B ) ) )
22	21	reximdv 2462	. . . . 5 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> ( E. q e. Q. q <Q ( *Q ` x ) -> E. q e. Q. q e. ( 1st ` B ) ) )
23	6, 22	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 2nd ` A ) ) -> E. q e. Q. q e. ( 1st ` B ) )
24	23	rexlimiva 2472	. . 3 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. q e. Q. q e. ( 1st ` B ) )
25	1, 2, 24	3syl 17	. 2 |- ( A e. P. -> E. q e. Q. q e. ( 1st ` B ) )
26		prml 6667	. . 3 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 1st ` A ) )
27		1nq 6556	. . . . . . . 8 |- 1Q e. Q.
28		addclnq 6565	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
29	3, 27, 28	sylancl 404	. . . . . . 7 |- ( x e. Q. -> ( ( *Q ` x ) +Q 1Q ) e. Q. )
30		ltaddnq 6597	. . . . . . . 8 |- ( ( ( *Q ` x ) e. Q. /\ 1Q e. Q. ) -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
31	3, 27, 30	sylancl 404	. . . . . . 7 |- ( x e. Q. -> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) )
32		breq2 3789	. . . . . . . 8 |- ( r = ( ( *Q ` x ) +Q 1Q ) -> ( ( *Q ` x ) <Q r <-> ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) )
33	32	rspcev 2701	. . . . . . 7 |- ( ( ( ( *Q ` x ) +Q 1Q ) e. Q. /\ ( *Q ` x ) <Q ( ( *Q ` x ) +Q 1Q ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
34	29, 31, 33	syl2anc 403	. . . . . 6 |- ( x e. Q. -> E. r e. Q. ( *Q ` x ) <Q r )
35	34	adantr 270	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. ( *Q ` x ) <Q r )
36	7	eleq1d 2147	. . . . . . . . . . 11 |- ( x e. Q. -> ( ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) <-> x e. ( 1st ` A ) ) )
37	36	anbi2d 451	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) ) )
38		breq1 3788	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( y <Q r <-> ( *Q ` x ) <Q r ) )
39	11	eleq1d 2147	. . . . . . . . . . . . 13 |- ( y = ( *Q ` x ) -> ( ( *Q ` y ) e. ( 1st ` A ) <-> ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) )
40	38, 39	anbi12d 456	. . . . . . . . . . . 12 |- ( y = ( *Q ` x ) -> ( ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) <-> ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) ) )
41	40	spcegv 2686	. . . . . . . . . . 11 |- ( ( *Q ` x ) e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
42	3, 41	syl 14	. . . . . . . . . 10 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ ( *Q ` ( *Q ` x ) ) e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
43	37, 42	sylbird 168	. . . . . . . . 9 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) ) )
44	17	recexprlemelu 6813	. . . . . . . . 9 |- ( r e. ( 2nd ` B ) <-> E. y ( y <Q r /\ ( *Q ` y ) e. ( 1st ` A ) ) )
45	43, 44	syl6ibr 160	. . . . . . . 8 |- ( x e. Q. -> ( ( ( *Q ` x ) <Q r /\ x e. ( 1st ` A ) ) -> r e. ( 2nd ` B ) ) )
46	45	expcomd 1370	. . . . . . 7 |- ( x e. Q. -> ( x e. ( 1st ` A ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) ) )
47	46	imp 122	. . . . . 6 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( ( *Q ` x ) <Q r -> r e. ( 2nd ` B ) ) )
48	47	reximdv 2462	. . . . 5 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> ( E. r e. Q. ( *Q ` x ) <Q r -> E. r e. Q. r e. ( 2nd ` B ) ) )
49	35, 48	mpd 13	. . . 4 |- ( ( x e. Q. /\ x e. ( 1st ` A ) ) -> E. r e. Q. r e. ( 2nd ` B ) )
50	49	rexlimiva 2472	. . 3 |- ( E. x e. Q. x e. ( 1st ` A ) -> E. r e. Q. r e. ( 2nd ` B ) )
51	1, 26, 50	3syl 17	. 2 |- ( A e. P. -> E. r e. Q. r e. ( 2nd ` B ) )
52	25, 51	jca 300	1 |- ( A e. P. -> ( E. q e. Q. q e. ( 1st ` B ) /\ E. r e. Q. r e. ( 2nd ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   E.wrex 2349   <.cop 3401   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   Q.cnq 6470   1Qc1q 6471   +Q cplq 6472   *Qcrq 6474   <Q cltq 6475   P.cnp 6481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-inp 6656

This theorem is referenced by:  recexprlempr  6822