Theorem pitonnlem2 8030

Description: Lemma for pitonn 8031. 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 8003	. . . 4 |- 1 = <. 1R , 0R >.
2	1	oveq2i 6011	. . 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 7736	. . . . . . . 8 |- ( K e. N. -> <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. e. P. )
4		1pr 7737	. . . . . . . 8 |- 1P e. P.
5		addclpr 7720	. . . . . . . 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 413	. . . . . . 7 |- ( K e. N. -> ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
7		opelxpi 4750	. . . . . . 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 413	. . . . . 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 7920	. . . . . . 7 |- ~R e. _V
10	9	ecelqsi 6734	. . . . . 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 7910	. . . . 5 |- R. = ( ( P. X. P. ) /. ~R )
13	11, 12	eleqtrrdi 2323	. . . 4 |- ( K e. N. -> [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
14		1sr 7934	. . . 4 |- 1R e. R.
15		addresr 8020	. . . 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 413	. . 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 2274	. 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 8029	. . . . 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 7915	. . . . . 6 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
21	20	oveq2i 6011	. . . . 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 7720	. . . . . . . 8 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
23	4, 4, 22	mp2an 426	. . . . . . 7 |- ( 1P +P. 1P ) e. P.
24		addsrpr 7928	. . . . . . . 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 435	. . . . . . 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 439	. . . . . 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 2274	. . . 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 7647	. . . . . . . . . . 11 |- ( K e. N. -> [ <. ( K +N 1o ) , 1o >. ] ~Q = ( [ <. K , 1o >. ] ~Q +Q 1Q ) )
30	29	breq2d 4094	. . . . . . . . . 10 |- ( K e. N. -> ( l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q <-> l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) ) )
31	30	abbidv 2347	. . . . . . . . 9 |- ( K e. N. -> { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } = { l | l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) } )
32	29	breq1d 4092	. . . . . . . . . 10 |- ( K e. N. -> ( [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u <-> ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u ) )
33	32	abbidv 2347	. . . . . . . . 9 |- ( K e. N. -> { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } = { u | ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u } )
34	31, 33	opeq12d 3864	. . . . . . . 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 7605	. . . . . . . . 9 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
36		addnqpr1 7745	. . . . . . . . 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 2262	. . . . . . 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 6015	. . . . . 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 3862	. . . . 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 6714	. . . 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 2272	. . 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 3862	. 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 2262	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 >. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   <.cop 3669   class class class wbr 4082   X. cxp 4716  (class class class)co 6000   1oc1o 6553   [cec 6676   /.cqs 6677   N.cnpi 7455   +N cpli 7456   ~Q ceq 7462   Q.cnq 7463   1Qc1q 7464   +Q cplq 7465   <Q cltq 7468   P.cnp 7474   1Pc1p 7475   +P. cpp 7476   ~R cer 7479   R.cnr 7480   0Rc0r 7481   1Rc1r 7482   +R cplr 7484   1c1 7996   + caddc 7998

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-i1p 7650  df-iplp 7651  df-enr 7909  df-nr 7910  df-plr 7911  df-0r 7914  df-1r 7915  df-c 8001  df-1 8003  df-add 8006

This theorem is referenced by:  pitonn  8031  nntopi  8077