Theorem pitonnlem1 7929

Description: Lemma for pitonn 7932. 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 7904	. 2 |- 1 = <. 1R , 0R >.
2		df-1r 7816	. . . 4 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
3		df-i1p 7551	. . . . . . . 8 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
4		df-1nqqs 7435	. . . . . . . . . . 11 |- 1Q = [ <. 1o , 1o >. ] ~Q
5	4	breq2i 4042	. . . . . . . . . 10 |- ( l <Q 1Q <-> l <Q [ <. 1o , 1o >. ] ~Q )
6	5	abbii 2312	. . . . . . . . 9 |- { l | l <Q 1Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q }
7	4	breq1i 4041	. . . . . . . . . 10 |- ( 1Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u )
8	7	abbii 2312	. . . . . . . . 9 |- { u | 1Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u }
9	6, 8	opeq12i 3814	. . . . . . . 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 2217	. . . . . . 7 |- 1P = <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >.
11	10	oveq1i 5935	. . . . . 6 |- ( 1P +P. 1P ) = ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P )
12	11	opeq1i 3812	. . . . 5 |- <. ( 1P +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >.
13		eceq1 6636	. . . . 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 2217	. . 3 |- 1R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R
16	15	opeq1i 3812	. 2 |- <. 1R , 0R >. = <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >.
17	1, 16	eqtr2i 2218	1 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1

Syntax hints:   = wceq 1364   {cab 2182   <.cop 3626   class class class wbr 4034  (class class class)co 5925   1oc1o 6476   [cec 6599   ~Q ceq 7363   1Qc1q 7365   <Q cltq 7369   1Pc1p 7376   +P. cpp 7377   ~R cer 7380   0Rc0r 7382   1Rc1r 7383   1c1 7897

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fv 5267  df-ov 5928  df-ec 6603  df-1nqqs 7435  df-i1p 7551  df-1r 7816  df-1 7904

This theorem is referenced by:  pitonn  7932