Theorem recexprlemex 7785

Description: B is the reciprocal of A. Lemma for recexpr 7786. (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 7783	. . 3 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) C_ ( 1st ` 1P ) )
3	1	recexprlem1ssl 7781	. . 3 |- ( A e. P. -> ( 1st ` 1P ) C_ ( 1st ` ( A .P. B ) ) )
4	2, 3	eqssd 3218	. 2 |- ( A e. P. -> ( 1st ` ( A .P. B ) ) = ( 1st ` 1P ) )
5	1	recexprlemss1u 7784	. . 3 |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) C_ ( 2nd ` 1P ) )
6	1	recexprlem1ssu 7782	. . 3 |- ( A e. P. -> ( 2nd ` 1P ) C_ ( 2nd ` ( A .P. B ) ) )
7	5, 6	eqssd 3218	. 2 |- ( A e. P. -> ( 2nd ` ( A .P. B ) ) = ( 2nd ` 1P ) )
8	1	recexprlempr 7780	. . . 4 |- ( A e. P. -> B e. P. )
9		mulclpr 7720	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )
10	8, 9	mpdan 421	. . 3 |- ( A e. P. -> ( A .P. B ) e. P. )
11		1pr 7702	. . 3 |- 1P e. P.
12		preqlu 7620	. . 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 413	. 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 947	1 |- ( A e. P. -> ( A .P. B ) = 1P )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   <.cop 3646   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   1stc1st 6247   2ndc2nd 6248   *Qcrq 7432   <Q cltq 7433   P.cnp 7439   1Pc1p 7440   .P. cmp 7442

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-i1p 7615  df-imp 7617

This theorem is referenced by:  recexpr  7786