Theorem recidpipr 8213

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 7779	. . 3 |- ( N e. N. -> [ <. N , 1o >. ] ~Q e. Q. )
2		recclnq 7749	. . . 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 7934	. . 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 415	. 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 7750	. . . . . . 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 4137	. . . . 5 |- ( N e. N. -> ( l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <-> l <Q 1Q ) )
9	8	abbidv 2358	. . . 4 |- ( N e. N. -> { l | l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) } = { l | l <Q 1Q } )
10	7	breq1d 4135	. . . . 5 |- ( N e. N. -> ( ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u <-> 1Q <Q u ) )
11	10	abbidv 2358	. . . 4 |- ( N e. N. -> { u | ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u } = { u | 1Q <Q u } )
12	9, 11	opeq12d 3907	. . 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 7824	. . 3 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
14	12, 13	eqtr4di 2289	. 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 2273	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 1402   e. wcel 2209   {cab 2224   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1oc1o 6670   [cec 6795   N.cnpi 7629   ~Q ceq 7636   Q.cnq 7637   1Qc1q 7638   .Q cmq 7640   *Qcrq 7641   <Q cltq 7642   1Pc1p 7649   .P. cmp 7651

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-po 4436  df-iso 4437  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-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-i1p 7824  df-imp 7826

This theorem is referenced by:  recidpirq  8215