Theorem pitonnlem1 8064

Description: Lemma for pitonn 8067. 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 8039	. 2 |- 1 = <. 1R , 0R >.
2		df-1r 7951	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
3		df-i1p 7686	. . . . . . . 8 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
4		df-1nqqs 7570	. . . . . . . . . . 11 |- 1Q = [ <. 1o , 1o >. ] ~Q
5	4	breq2i 4096	. . . . . . . . . 10 |- ( l <Q 1Q <-> l <Q [ <. 1o , 1o >. ] ~Q )
6	5	abbii 2347	. . . . . . . . 9 |- { l | l <Q 1Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q }
7	4	breq1i 4095	. . . . . . . . . 10 |- ( 1Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u )
8	7	abbii 2347	. . . . . . . . 9 |- { u | 1Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u }
9	6, 8	opeq12i 3867	. . . . . . . 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 2252	. . . . . . 7 |- 1P = <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >.
11	10	oveq1i 6027	. . . . . 6 |- ( 1P +P. 1P ) = ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P )
12	11	opeq1i 3865	. . . . 5 |- <. ( 1P +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >.
13		eceq1 6736	. . . . 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 2252	. . 3 |- 1R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R
16	15	opeq1i 3865	. 2 |- <. 1R , 0R >. = <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
17	1, 16	eqtr2i 2253	1 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1

Syntax hints:   = wceq 1397   {cab 2217   <.cop 3672   class class class wbr 4088  (class class class)co 6017   1oc1o 6574   [cec 6699   ~Q ceq 7498   1Qc1q 7500   <Q cltq 7504   1Pc1p 7511   +P. cpp 7512   ~R cer 7515   0Rc0r 7517   1Rc1r 7518   1c1 8032

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fv 5334  df-ov 6020  df-ec 6703  df-1nqqs 7570  df-i1p 7686  df-1r 7951  df-1 8039

This theorem is referenced by:  pitonn  8067