Theorem pitonnlem1 8212

Description: Lemma for pitonn 8215. 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 8187	. 2 |- 1 = <. 1R , 0R >.
2		df-1r 8099	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
3		df-i1p 7834	. . . . . . . 8 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
4		df-1nqqs 7718	. . . . . . . . . . 11 |- 1Q = [ <. 1o , 1o >. ] ~Q
5	4	breq2i 4138	. . . . . . . . . 10 |- ( l <Q 1Q <-> l <Q [ <. 1o , 1o >. ] ~Q )
6	5	abbii 2354	. . . . . . . . 9 |- { l | l <Q 1Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q }
7	4	breq1i 4137	. . . . . . . . . 10 |- ( 1Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u )
8	7	abbii 2354	. . . . . . . . 9 |- { u | 1Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u }
9	6, 8	opeq12i 3909	. . . . . . . 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 2259	. . . . . . 7 |- 1P = <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >.
11	10	oveq1i 6095	. . . . . 6 |- ( 1P +P. 1P ) = ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P )
12	11	opeq1i 3907	. . . . 5 |- <. ( 1P +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >.
13		eceq1 6842	. . . . 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 2259	. . 3 |- 1R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R
16	15	opeq1i 3907	. 2 |- <. 1R , 0R >. = <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
17	1, 16	eqtr2i 2260	1 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1

This proof depends on syntax axioms:   = wceq 1402   {cab 2224   <.cop 3712   class class class wbr 4130  (class class class)co 6085   1oc1o 6680   [cec 6805   ~Q ceq 7646   1Qc1q 7648   <Q cltq 7652   1Pc1p 7659   +P. cpp 7660   ~R cer 7663   0Rc0r 7665   1Rc1r 7666   1c1 8180

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fv 5385  df-ov 6088  df-ec 6809  df-1nqqs 7718  df-i1p 7834  df-1r 8099  df-1 8187

This theorem is used by:  pitonn  8215