Theorem recexprlemex 7438

Description: B is the reciprocal of A. Lemma for recexpr 7439. (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
recexprlemex	|- ( A e. P. -> ( A .P. B ) = 1P )

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

Proof of Theorem recexprlemex

Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 |- B = <. { x | E. y ( x <Q y /\ ( *Q ` y ) e. ( 2nd ` A ) ) } , { x | E. y ( y <Q x /\ ( *Q ` y ) e. ( 1st ` A ) ) } >.
2	1	recexprlemss1l 7436	. . 3 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )
3	1	recexprlem1ssl 7434	. . 3 |- ( A e. P. -> ( 1st ` 1P ) C_ ( 1st ` ( A .P. B ) ) )
4	2, 3	eqssd 3109	. 2 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) = ( 1st ` 1P ) )
5	1	recexprlemss1u 7437	. . 3 |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) )
6	1	recexprlem1ssu 7435	. . 3 |- ( A e. P. -> ( 2nd ` 1P ) C_ ( 2nd ` ( A .P. B ) ) )
7	5, 6	eqssd 3109	. 2 |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) = ( 2nd ` 1P ) )
8	1	recexprlempr 7433	. . . 4 |- ( A e. P. -> B e. P. )
9		mulclpr 7373	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
10	8, 9	mpdan 417	. . 3 |- ( A e. P. -> ( A .P. B ) e. P. )
11		1pr 7355	. . 3 |- 1P e. P.
12		preqlu 7273	. . 3 |- ( ( ( A .P. B ) e. P. /\ 1P e. P. ) -> ( ( A .P. B ) = 1P <-> ( ( 1st ` ( A .P. B ) ) = ( 1st ` 1P ) /\ ( 2nd ` ( A .P. B ) ) = ( 2nd ` 1P ) ) ) )
13	10, 11, 12	sylancl 409	. 2 |- ( A e. P. -> ( ( A .P. B ) = 1P <-> ( ( 1st ` ( A .P. B ) ) = ( 1st ` 1P ) /\ ( 2nd ` ( A .P. B ) ) = ( 2nd ` 1P ) ) ) )
14	4, 7, 13	mpbir2and 928	1 |- ( A e. P. -> ( A .P. B ) = 1P )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123   <.cop 3525   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   *Qcrq 7085   <Q cltq 7086   P.cnp 7092   1Pc1p 7093   .P. cmp 7095

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-imp 7270

This theorem is referenced by:  recexpr  7439