Theorem recidpipr 8075

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 7641	. . 3 |- ( N e. N. -> [ <. N , 1o >. ] ~Q e. Q. )
2		recclnq 7611	. . . 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 7796	. . 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 7612	. . . . . . 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 4100	. . . . 5 |- ( N e. N. -> ( l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <-> l <Q 1Q ) )
9	8	abbidv 2349	. . . 4 |- ( N e. N. -> { l | l <Q ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) } = { l | l <Q 1Q } )
10	7	breq1d 4098	. . . . 5 |- ( N e. N. -> ( ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u <-> 1Q <Q u ) )
11	10	abbidv 2349	. . . 4 |- ( N e. N. -> { u | ( [ <. N , 1o >. ] ~Q .Q ( *Q ` [ <. N , 1o >. ] ~Q ) ) <Q u } = { u | 1Q <Q u } )
12	9, 11	opeq12d 3870	. . 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 7686	. . 3 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
14	12, 13	eqtr4di 2282	. 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 2266	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 1397   e. wcel 2202   {cab 2217   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   1oc1o 6574   [cec 6699   N.cnpi 7491   ~Q ceq 7498   Q.cnq 7499   1Qc1q 7500   .Q cmq 7502   *Qcrq 7503   <Q cltq 7504   1Pc1p 7511   .P. cmp 7513

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-i1p 7686  df-imp 7688

This theorem is referenced by:  recidpirq  8077