Theorem pitonnlem1 8032

Description: Lemma for pitonn 8035. 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 8007	. 2 |- 1 = <. 1R , 0R >.
2		df-1r 7919	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
3		df-i1p 7654	. . . . . . . 8 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
4		df-1nqqs 7538	. . . . . . . . . . 11 |- 1Q = [ <. 1o , 1o >. ] ~Q
5	4	breq2i 4091	. . . . . . . . . 10 |- ( l <Q 1Q <-> l <Q [ <. 1o , 1o >. ] ~Q )
6	5	abbii 2345	. . . . . . . . 9 |- { l | l <Q 1Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q }
7	4	breq1i 4090	. . . . . . . . . 10 |- ( 1Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u )
8	7	abbii 2345	. . . . . . . . 9 |- { u | 1Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u }
9	6, 8	opeq12i 3862	. . . . . . . 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 2250	. . . . . . 7 |- 1P = <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >.
11	10	oveq1i 6011	. . . . . 6 |- ( 1P +P. 1P ) = ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P )
12	11	opeq1i 3860	. . . . 5 |- <. ( 1P +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >.
13		eceq1 6715	. . . . 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 2250	. . 3 |- 1R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R
16	15	opeq1i 3860	. 2 |- <. 1R , 0R >. = <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
17	1, 16	eqtr2i 2251	1 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1

Syntax hints:   = wceq 1395   {cab 2215   <.cop 3669   class class class wbr 4083  (class class class)co 6001   1oc1o 6555   [cec 6678   ~Q ceq 7466   1Qc1q 7468   <Q cltq 7472   1Pc1p 7479   +P. cpp 7480   ~R cer 7483   0Rc0r 7485   1Rc1r 7486   1c1 8000

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-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-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fv 5326  df-ov 6004  df-ec 6682  df-1nqqs 7538  df-i1p 7654  df-1r 7919  df-1 8007

This theorem is referenced by:  pitonn  8035