Theorem pitonnlem2 7655

Description: Lemma for pitonn 7656. 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 7628	. . . 4 |- 1 = <. 1R , 0R >.
2	1	oveq2i 5785	. . 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 7361	. . . . . . . 8 |- ( K e. N. -> <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. e. P. )
4		1pr 7362	. . . . . . . 8 |- 1P e. P.
5		addclpr 7345	. . . . . . . 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 409	. . . . . . 7 |- ( K e. N. -> ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
7		opelxpi 4571	. . . . . . 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 409	. . . . . 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 7545	. . . . . . 7 |- ~R e. _V
10	9	ecelqsi 6483	. . . . . 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 7535	. . . . 5 |- R. = ( ( P. X. P. ) /. ~R )
13	11, 12	eleqtrrdi 2233	. . . 4 |- ( K e. N. -> [ <. ( <. { l | l <Q [ <. K , 1o >. ] ~Q } , { u | [ <. K , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
14		1sr 7559	. . . 4 |- 1R e. R.
15		addresr 7645	. . . 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 409	. . 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	syl5eq 2184	. 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 7654	. . . . 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 7540	. . . . . 6 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
21	20	oveq2i 5785	. . . . 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 7345	. . . . . . . 8 |- ( ( 1P e. P. /\ 1P e. P. ) -> ( 1P +P. 1P ) e. P. )
23	4, 4, 22	mp2an 422	. . . . . . 7 |- ( 1P +P. 1P ) e. P.
24		addsrpr 7553	. . . . . . . 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 431	. . . . . . 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 435	. . . . . 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	syl5eq 2184	. . . 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 7272	. . . . . . . . . . 11 |- ( K e. N. -> [ <. ( K +N 1o ) , 1o >. ] ~Q = ( [ <. K , 1o >. ] ~Q +Q 1Q ) )
30	29	breq2d 3941	. . . . . . . . . 10 |- ( K e. N. -> ( l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q <-> l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) ) )
31	30	abbidv 2257	. . . . . . . . 9 |- ( K e. N. -> { l | l <Q [ <. ( K +N 1o ) , 1o >. ] ~Q } = { l | l <Q ( [ <. K , 1o >. ] ~Q +Q 1Q ) } )
32	29	breq1d 3939	. . . . . . . . . 10 |- ( K e. N. -> ( [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u <-> ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u ) )
33	32	abbidv 2257	. . . . . . . . 9 |- ( K e. N. -> { u | [ <. ( K +N 1o ) , 1o >. ] ~Q <Q u } = { u | ( [ <. K , 1o >. ] ~Q +Q 1Q ) <Q u } )
34	31, 33	opeq12d 3713	. . . . . . . 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 7230	. . . . . . . . 9 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
36		addnqpr1 7370	. . . . . . . . 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 2172	. . . . . . 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 5789	. . . . . 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 3711	. . . . 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 6465	. . . 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 2182	. . 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 3711	. 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 2172	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 103   = wceq 1331   e. wcel 1480   {cab 2125   <.cop 3530   class class class wbr 3929   X. cxp 4537  (class class class)co 5774   1oc1o 6306   [cec 6427   /.cqs 6428   N.cnpi 7080   +N cpli 7081   ~Q ceq 7087   Q.cnq 7088   1Qc1q 7089   +Q cplq 7090   <Q cltq 7093   P.cnp 7099   1Pc1p 7100   +P. cpp 7101   ~R cer 7104   R.cnr 7105   0Rc0r 7106   1Rc1r 7107   +R cplr 7109   1c1 7621   + caddc 7623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-enr 7534  df-nr 7535  df-plr 7536  df-0r 7539  df-1r 7540  df-c 7626  df-1 7628  df-add 7631

This theorem is referenced by:  pitonn  7656  nntopi  7702