Theorem pitonnlem1 7653

Description: Lemma for pitonn 7656. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)

Assertion

Ref	Expression
pitonnlem1	|- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1

Distinct variable group:   u, l

Proof of Theorem pitonnlem1

Step	Hyp	Ref	Expression
1		df-1 7628	. 2 |- 1 = <. 1R , 0R >.
2		df-1r 7540	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
3		df-i1p 7275	. . . . . . . 8 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
4		df-1nqqs 7159	. . . . . . . . . . 11 |- 1Q = [ <. 1o , 1o >. ] ~Q
5	4	breq2i 3937	. . . . . . . . . 10 |- ( l <Q 1Q <-> l <Q [ <. 1o , 1o >. ] ~Q )
6	5	abbii 2255	. . . . . . . . 9 |- { l | l <Q 1Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q }
7	4	breq1i 3936	. . . . . . . . . 10 |- ( 1Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u )
8	7	abbii 2255	. . . . . . . . 9 |- { u | 1Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u }
9	6, 8	opeq12i 3710	. . . . . . . 8 |- <. { l | l <Q 1Q } , { u | 1Q <Q u } >. = <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >.
10	3, 9	eqtri 2160	. . . . . . 7 |- 1P = <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >.
11	10	oveq1i 5784	. . . . . 6 |- ( 1P +P. 1P ) = ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P )
12	11	opeq1i 3708	. . . . 5 |- <. ( 1P +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >.
13		eceq1 6464	. . . . 5 |- ( <. ( 1P +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. -> [ <. ( 1P +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
14	12, 13	ax-mp 5	. . . 4 |- [ <. ( 1P +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R
15	2, 14	eqtri 2160	. . 3 |- 1R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R
16	15	opeq1i 3708	. 2 |- <. 1R , 0R >. = <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
17	1, 16	eqtr2i 2161	1 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1

Syntax hints:   = wceq 1331   {cab 2125   <.cop 3530   class class class wbr 3929  (class class class)co 5774   1oc1o 6306   [cec 6427   ~Q ceq 7087   1Qc1q 7089   <Q cltq 7093   1Pc1p 7100   +P. cpp 7101   ~R cer 7104   0Rc0r 7106   1Rc1r 7107   1c1 7621

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-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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fv 5131  df-ov 5777  df-ec 6431  df-1nqqs 7159  df-i1p 7275  df-1r 7540  df-1 7628

This theorem is referenced by:  pitonn  7656