Theorem recidpipr 8171

Description: Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)

Assertion

Ref	Expression
recidpipr	|- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. .P. <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) = 1P )

Distinct variable group:   N, l, u

Proof of Theorem recidpipr

Step	Hyp	Ref	Expression
1		nnnq 7737	. . 3 |- ( N e. N. -> [ <. N , 1o >. ] ~Q e. Q. )
2		recclnq 7707	. . . 4 |- ( [ <. N , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. N , 1o >. ] ~Q ) e. Q. )
3	1, 2	syl 14	. . 3 |- ( N e. N. -> ( *Q ` [ <. N , 1o >. ] ~Q ) e. Q. )
4		mulnqpr 7892	. . 3 |- ( ( [ <. N , 1o >. ] ~Q e. Q. /\ ( *Q ` [ <. N , 1o >. ] ~Q ) e. Q. ) -> <. { l | l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) } , { u | ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u } >. = ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. .P. <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) )
5	1, 3, 4	syl2anc 411	. 2 |- ( N e. N. -> <. { l | l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) } , { u | ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u } >. = ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. .P. <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) )
6		recidnq 7708	. . . . . . 7 |- ( [ <. N , 1o >. ] ~Q e. Q. -> ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) = 1Q )
7	1, 6	syl 14	. . . . . 6 |- ( N e. N. -> ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) = 1Q )
8	7	breq2d 4121	. . . . 5 |- ( N e. N. -> ( l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <-> l <Q 1Q ) )
9	8	abbidv 2352	. . . 4 |- ( N e. N. -> { l | l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) } = { l | l <Q 1Q } )
10	7	breq1d 4119	. . . . 5 |- ( N e. N. -> ( ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u <-> 1Q <Q u ) )
11	10	abbidv 2352	. . . 4 |- ( N e. N. -> { u | ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u } = { u | 1Q <Q u } )
12	9, 11	opeq12d 3891	. . 3 |- ( N e. N. -> <. { l | l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) } , { u | ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u } >. = <. { l | l <Q 1Q } , { u | 1Q <Q u } >. )
13		df-i1p 7782	. . 3 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
14	12, 13	eqtr4di 2283	. 2 |- ( N e. N. -> <. { l | l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) } , { u | ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u } >. = 1P )
15	5, 14	eqtr3d 2267	1 |- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. .P. <. { l | l <Q ( *Q ` [ <. N , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. N , 1o >. ] ~Q ) <Q u } >. ) = 1P )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   {cab 2218   <.cop 3692   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   1oc1o 6640   [cec 6765   N.cnpi 7587   ~Q ceq 7594   Q.cnq 7595   1Qc1q 7596   .Q cmq 7598   *Qcrq 7599   <Q cltq 7600   1Pc1p 7607   .P. cmp 7609

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-eprel 4410  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-1o 6647  df-2o 6648  df-oadd 6651  df-omul 6652  df-er 6767  df-ec 6769  df-qs 6773  df-ni 7619  df-pli 7620  df-mi 7621  df-lti 7622  df-plpq 7659  df-mpq 7660  df-enq 7662  df-nqqs 7663  df-plqqs 7664  df-mqqs 7665  df-1nqqs 7666  df-rq 7667  df-ltnqqs 7668  df-enq0 7739  df-nq0 7740  df-0nq0 7741  df-plq0 7742  df-mq0 7743  df-inp 7781  df-i1p 7782  df-imp 7784

This theorem is referenced by:  recidpirq  8173