Theorem pitonnlem2 8110

Description: Lemma for pitonn 8111. 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 8083	. . . 4 |- 1 = <. 1R , 0R >.
2	1	oveq2i 6039	. . 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 7816	. . . . . . . 8 |- ( K e. N. -> <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. e. P. )
4		1pr 7817	. . . . . . . 8 |- 1P e. P.
5		addclpr 7800	. . . . . . . 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 4763	. . . . . . 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 8000	. . . . . . 7 |- ~R e. _V
10	9	ecelqsi 6801	. . . . . 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 7990	. . . . 5 |- R. = ( ( P. X. P. ) /. ~R )
13	11, 12	eleqtrrdi 2325	. . . 4 |- ( K e. N. -> [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
14		1sr 8014	. . . 4 |- 1R e. R.
15		addresr 8100	. . . 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 2276	. 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 8109	. . . . 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 7995	. . . . . 6 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
21	20	oveq2i 6039	. . . . 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 7800	. . . . . . . 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 8008	. . . . . . . 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 2276	. . . 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 7727	. . . . . . . . . . 11 |- ( K e. N. -> [ <. ( K +N 1o ) , 1o >. ] ~Q = ( [ <. K , 1o >. ] ~Q +Q 1Q ) )
30	29	breq2d 4105	. . . . . . . . . 10 |- ( K e. N. -> ( l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q <-> l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) ) )
31	30	abbidv 2350	. . . . . . . . 9 |- ( K e. N. -> { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } = { l | l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) } )
32	29	breq1d 4103	. . . . . . . . . 10 |- ( K e. N. -> ( [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u <-> ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u ) )
33	32	abbidv 2350	. . . . . . . . 9 |- ( K e. N. -> { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } = { u | ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u } )
34	31, 33	opeq12d 3875	. . . . . . . 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 7685	. . . . . . . . 9 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
36		addnqpr1 7825	. . . . . . . . 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 2264	. . . . . . 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 6043	. . . . . 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 3873	. . . . 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 6781	. . . 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 2274	. . 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 3873	. 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 2264	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 1398   e. wcel 2202   {cab 2217   <.cop 3676   class class class wbr 4093   X. cxp 4729  (class class class)co 6028   1oc1o 6618   [cec 6743   /.cqs 6744   N.cnpi 7535   +N cpli 7536   ~Q ceq 7542   Q.cnq 7543   1Qc1q 7544   +Q cplq 7545   <Q cltq 7548   P.cnp 7554   1Pc1p 7555   +P. cpp 7556   ~R cer 7559   R.cnr 7560   0Rc0r 7561   1Rc1r 7562   +R cplr 7564   1c1 8076   + caddc 8078

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-i1p 7730  df-iplp 7731  df-enr 7989  df-nr 7990  df-plr 7991  df-0r 7994  df-1r 7995  df-c 8081  df-1 8083  df-add 8086

This theorem is referenced by:  pitonn  8111  nntopi  8157