Theorem pitonnlem2 8214

Description: Lemma for pitonn 8215. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)

Assertion

Ref	Expression
pitonnlem2	|- ( K e. N. -> ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = <. [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )

Distinct variable group:   K, l, u

Proof of Theorem pitonnlem2

Step	Hyp	Ref	Expression
1		df-1 8187	. . . 4 |- 1 = <. 1R , 0R >.
2	1	oveq2i 6096	. . 3 |- ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + <. 1R , 0R >. )
3		nnprlu 7920	. . . . . . . 8 |- ( K e. N. -> <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. e. P. )
4		1pr 7921	. . . . . . . 8 |- 1P e. P.
5		addclpr 7904	. . . . . . . 8 |- ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. e. P. /\ 1P e. P. ) -> ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
6	3, 4, 5	sylancl 417	. . . . . . 7 |- ( K e. N. -> ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
7		opelxpi 4806	. . . . . . 7 |- ( ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. /\ 1P e. P. ) -> <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. e. ( P. X. P. ) )
8	6, 4, 7	sylancl 417	. . . . . 6 |- ( K e. N. -> <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. e. ( P. X. P. ) )
9		enrex 8104	. . . . . . 7 |- ~R e. _V
10	9	ecelqsi 6863	. . . . . 6 |- ( <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. e. ( P. X. P. ) -> [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
11	8, 10	syl 14	. . . . 5 |- ( K e. N. -> [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
12		df-nr 8094	. . . . 5 |- R. = ( ( P. X. P. ) /. ~R )
13	11, 12	eleqtrrdi 2332	. . . 4 |- ( K e. N. -> [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
14		1sr 8118	. . . 4 |- 1R e. R.
15		addresr 8204	. . . 4 |- ( ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. /\ 1R e. R. ) -> ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + <. 1R , 0R >. ) = <. ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R 1R ) , 0R >. )
16	13, 14, 15	sylancl 417	. . 3 |- ( K e. N. -> ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + <. 1R , 0R >. ) = <. ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R 1R ) , 0R >. )
17	2, 16	eqtrid 2283	. 2 |- ( K e. N. -> ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = <. ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R 1R ) , 0R >. )
18		pitonnlem1p1 8213	. . . . 5 |- ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. -> [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. 1P ) , 1P >. ] ~R )
19	6, 18	syl 14	. . . 4 |- ( K e. N. -> [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. 1P ) , 1P >. ] ~R )
20		df-1r 8099	. . . . . 6 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
21	20	oveq2i 6096	. . . . 5 |- ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R 1R ) = ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
22		addclpr 7904	. . . . . . . 8 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
23	4, 4, 22	mp2an 430	. . . . . . 7 |- ( 1P +P. 1P ) e. P.
24		addsrpr 8112	. . . . . . . 8 |- ( ( ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. /\ 1P e. P. ) /\ ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) ) -> ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R )
25	4, 24	mpanl2 439	. . . . . . 7 |- ( ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. /\ ( ( 1P +P. 1P ) e. P. /\ 1P e. P. ) ) -> ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R )
26	23, 4, 25	mpanr12 443	. . . . . 6 |- ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. -> ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R )
27	6, 26	syl 14	. . . . 5 |- ( K e. N. -> ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R [ <. ( 1P +P. 1P ) , 1P >. ] ~R ) = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R )
28	21, 27	eqtrid 2283	. . . 4 |- ( K e. N. -> ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R 1R ) = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. ( 1P +P. 1P ) ) , ( 1P +P. 1P ) >. ] ~R )
29		addpinq1 7831	. . . . . . . . . . 11 |- ( K e. N. -> [ <. ( K +N 1o ) , 1o >. ] ~Q = ( [ <. K , 1o >. ] ~Q +Q 1Q ) )
30	29	breq2d 4142	. . . . . . . . . 10 |- ( K e. N. -> ( l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q <-> l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) ) )
31	30	abbidv 2358	. . . . . . . . 9 |- ( K e. N. -> { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } = { l | l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) } )
32	29	breq1d 4140	. . . . . . . . . 10 |- ( K e. N. -> ( [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u <-> ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u ) )
33	32	abbidv 2358	. . . . . . . . 9 |- ( K e. N. -> { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } = { u | ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u } )
34	31, 33	opeq12d 3912	. . . . . . . 8 |- ( K e. N. -> <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. = <. { l | l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) } , { u | ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u } >. )
35		nnnq 7789	. . . . . . . . 9 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
36		addnqpr1 7929	. . . . . . . . 9 |- ( [ <. K , 1o >. ] ~Q e. Q. -> <. { l | l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) } , { u | ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u } >. = ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) )
37	35, 36	syl 14	. . . . . . . 8 |- ( K e. N. -> <. { l | l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) } , { u | ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u } >. = ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) )
38	34, 37	eqtrd 2271	. . . . . . 7 |- ( K e. N. -> <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. = ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) )
39	38	oveq1d 6100	. . . . . 6 |- ( K e. N. -> ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. 1P ) )
40	39	opeq1d 3910	. . . . 5 |- ( K e. N. -> <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. 1P ) , 1P >. )
41	40	eceq1d 6843	. . . 4 |- ( K e. N. -> [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) +P. 1P ) , 1P >. ] ~R )
42	19, 28, 41	3eqtr4d 2281	. . 3 |- ( K e. N. -> ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R 1R ) = [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
43	42	opeq1d 3910	. 2 |- ( K e. N. -> <. ( [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R +R 1R ) , 0R >. = <. [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
44	17, 43	eqtrd 2271	1 |- ( K e. N. -> ( <. [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = <. [ <. ( <. { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   <.cop 3712   class class class wbr 4130   X. cxp 4772  (class class class)co 6085   1oc1o 6680   [cec 6805   /.cqs 6806   N.cnpi 7639   +N cpli 7640   ~Q ceq 7646   Q.cnq 7647   1Qc1q 7648   +Q cplq 7649   <Q cltq 7652   P.cnp 7658   1Pc1p 7659   +P. cpp 7660   ~R cer 7663   R.cnr 7664   0Rc0r 7665   1Rc1r 7666   +R cplr 7668   1c1 8180   + caddc 8182

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-i1p 7834  df-iplp 7835  df-enr 8093  df-nr 8094  df-plr 8095  df-0r 8098  df-1r 8099  df-c 8185  df-1 8187  df-add 8190

This theorem is used by:  pitonn  8215  nntopi  8261