Theorem pitonn 8215

Description: Mapping from N. to NN. (Contributed by Jim Kingdon, 22-Apr-2020.)

Assertion

Ref	Expression
pitonn	|- ( N e. N. -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )

Distinct variable groups:   N, l, u   y, l, u   x, y

Allowed substitution hints:   N( x, y)

Proof of Theorem pitonn

Dummy variables w z k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3904	. . . . . . . . . . . . . . 15 |- ( w = 1o -> <. w , 1o >. = <. 1o , 1o >. )
2	1	eceq1d 6843	. . . . . . . . . . . . . 14 |- ( w = 1o -> [ <. w , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
3	2	breq2d 4142	. . . . . . . . . . . . 13 |- ( w = 1o -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. 1o , 1o >. ] ~Q ) )
4	3	abbidv 2358	. . . . . . . . . . . 12 |- ( w = 1o -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q } )
5	2	breq1d 4140	. . . . . . . . . . . . 13 |- ( w = 1o -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u ) )
6	5	abbidv 2358	. . . . . . . . . . . 12 |- ( w = 1o -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u } )
7	4, 6	opeq12d 3912	. . . . . . . . . . 11 |- ( w = 1o -> <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. = <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. )
8	7	oveq1d 6100	. . . . . . . . . 10 |- ( w = 1o -> ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) )
9	8	opeq1d 3910	. . . . . . . . 9 |- ( w = 1o -> <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. )
10	9	eceq1d 6843	. . . . . . . 8 |- ( w = 1o -> [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
11	10	opeq1d 3910	. . . . . . 7 |- ( w = 1o -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
12	11	eleq1d 2307	. . . . . 6 |- ( w = 1o -> ( <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z <-> <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
13	12	imbi2d 230	. . . . 5 |- ( w = 1o -> ( ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) <-> ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) ) )
14		opeq1 3904	. . . . . . . . . . . . . . 15 |- ( w = k -> <. w , 1o >. = <. k , 1o >. )
15	14	eceq1d 6843	. . . . . . . . . . . . . 14 |- ( w = k -> [ <. w , 1o >. ] ~Q = [ <. k , 1o >. ] ~Q )
16	15	breq2d 4142	. . . . . . . . . . . . 13 |- ( w = k -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. k , 1o >. ] ~Q ) )
17	16	abbidv 2358	. . . . . . . . . . . 12 |- ( w = k -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. k , 1o >. ] ~Q } )
18	15	breq1d 4140	. . . . . . . . . . . . 13 |- ( w = k -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. k , 1o >. ] ~Q <Q u ) )
19	18	abbidv 2358	. . . . . . . . . . . 12 |- ( w = k -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. k , 1o >. ] ~Q <Q u } )
20	17, 19	opeq12d 3912	. . . . . . . . . . 11 |- ( w = k -> <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. = <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. )
21	20	oveq1d 6100	. . . . . . . . . 10 |- ( w = k -> ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) )
22	21	opeq1d 3910	. . . . . . . . 9 |- ( w = k -> <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. )
23	22	eceq1d 6843	. . . . . . . 8 |- ( w = k -> [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
24	23	opeq1d 3910	. . . . . . 7 |- ( w = k -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
25	24	eleq1d 2307	. . . . . 6 |- ( w = k -> ( <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z <-> <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
26	25	imbi2d 230	. . . . 5 |- ( w = k -> ( ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) <-> ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) ) )
27		opeq1 3904	. . . . . . . . . . . . . . 15 |- ( w = ( k +N 1o ) -> <. w , 1o >. = <. ( k +N 1o ) , 1o >. )
28	27	eceq1d 6843	. . . . . . . . . . . . . 14 |- ( w = ( k +N 1o ) -> [ <. w , 1o >. ] ~Q = [ <. ( k +N 1o ) , 1o >. ] ~Q )
29	28	breq2d 4142	. . . . . . . . . . . . 13 |- ( w = ( k +N 1o ) -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q ) )
30	29	abbidv 2358	. . . . . . . . . . . 12 |- ( w = ( k +N 1o ) -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } )
31	28	breq1d 4140	. . . . . . . . . . . . 13 |- ( w = ( k +N 1o ) -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u ) )
32	31	abbidv 2358	. . . . . . . . . . . 12 |- ( w = ( k +N 1o ) -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } )
33	30, 32	opeq12d 3912	. . . . . . . . . . 11 |- ( w = ( k +N 1o ) -> <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. = <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. )
34	33	oveq1d 6100	. . . . . . . . . 10 |- ( w = ( k +N 1o ) -> ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) )
35	34	opeq1d 3910	. . . . . . . . 9 |- ( w = ( k +N 1o ) -> <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. )
36	35	eceq1d 6843	. . . . . . . 8 |- ( w = ( k +N 1o ) -> [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
37	36	opeq1d 3910	. . . . . . 7 |- ( w = ( k +N 1o ) -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
38	37	eleq1d 2307	. . . . . 6 |- ( w = ( k +N 1o ) -> ( <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z <-> <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
39	38	imbi2d 230	. . . . 5 |- ( w = ( k +N 1o ) -> ( ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) <-> ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) ) )
40		opeq1 3904	. . . . . . . . . . . . . . 15 |- ( w = N -> <. w , 1o >. = <. N , 1o >. )
41	40	eceq1d 6843	. . . . . . . . . . . . . 14 |- ( w = N -> [ <. w , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
42	41	breq2d 4142	. . . . . . . . . . . . 13 |- ( w = N -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. N , 1o >. ] ~Q ) )
43	42	abbidv 2358	. . . . . . . . . . . 12 |- ( w = N -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. N , 1o >. ] ~Q } )
44	41	breq1d 4140	. . . . . . . . . . . . 13 |- ( w = N -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. N , 1o >. ] ~Q <Q u ) )
45	44	abbidv 2358	. . . . . . . . . . . 12 |- ( w = N -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. N , 1o >. ] ~Q <Q u } )
46	43, 45	opeq12d 3912	. . . . . . . . . . 11 |- ( w = N -> <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. = <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. )
47	46	oveq1d 6100	. . . . . . . . . 10 |- ( w = N -> ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) )
48	47	opeq1d 3910	. . . . . . . . 9 |- ( w = N -> <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. )
49	48	eceq1d 6843	. . . . . . . 8 |- ( w = N -> [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
50	49	opeq1d 3910	. . . . . . 7 |- ( w = N -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
51	50	eleq1d 2307	. . . . . 6 |- ( w = N -> ( <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z <-> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
52	51	imbi2d 230	. . . . 5 |- ( w = N -> ( ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) <-> ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) ) )
53		pitonnlem1 8212	. . . . . . . 8 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1
54	53	eleq1i 2304	. . . . . . 7 |- ( <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z <-> 1 e. z )
55	54	biimpri 133	. . . . . 6 |- ( 1 e. z -> <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z )
56	55	adantr 276	. . . . 5 |- ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z )
57		oveq1 6092	. . . . . . . . . . 11 |- ( y = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( y + 1 ) = ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) )
58	57	eleq1d 2307	. . . . . . . . . 10 |- ( y = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( y + 1 ) e. z <-> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) e. z ) )
59	58	rspccv 2926	. . . . . . . . 9 |- ( A. y e. z ( y + 1 ) e. z -> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z -> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) e. z ) )
60	59	ad2antll 495	. . . . . . . 8 |- ( ( k e. N. /\ ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) ) -> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z -> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) e. z ) )
61		pitonnlem2 8214	. . . . . . . . . 10 |- ( k e. N. -> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
62	61	eleq1d 2307	. . . . . . . . 9 |- ( k e. N. -> ( ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) e. z <-> <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
63	62	adantr 276	. . . . . . . 8 |- ( ( k e. N. /\ ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) ) -> ( ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) e. z <-> <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
64	60, 63	sylibd 149	. . . . . . 7 |- ( ( k e. N. /\ ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) ) -> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z -> <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
65	64	ex 115	. . . . . 6 |- ( k e. N. -> ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z -> <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) ) )
66	65	a2d 26	. . . . 5 |- ( k e. N. -> ( ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) -> ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) ) )
67	13, 26, 39, 52, 56, 66	indpi 7709	. . . 4 |- ( N e. N. -> ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
68	67	alrimiv 1927	. . 3 |- ( N e. N. -> A. z ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
69		eleq2 2302	. . . . 5 |- ( x = z -> ( 1 e. x <-> 1 e. z ) )
70		eleq2 2302	. . . . . 6 |- ( x = z -> ( ( y + 1 ) e. x <-> ( y + 1 ) e. z ) )
71	70	raleqbi1dv 2761	. . . . 5 |- ( x = z -> ( A. y e. x ( y + 1 ) e. x <-> A. y e. z ( y + 1 ) e. z ) )
72	69, 71	anbi12d 477	. . . 4 |- ( x = z -> ( ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) <-> ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) ) )
73	72	ralab 2986	. . 3 |- ( A. z e. { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z <-> A. z ( ( 1 e. z /\ A. y e. z ( y + 1 ) e. z ) -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
74	68, 73	sylibr 134	. 2 |- ( N e. N. -> A. z e. { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z )
75		nnprlu 7920	. . . . . . 7 |- ( N e. N. -> <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. e. P. )
76		1pr 7921	. . . . . . 7 |- 1P e. P.
77		addclpr 7904	. . . . . . 7 |- ( ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. e. P. /\ 1P e. P. ) -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
78	75, 76, 77	sylancl 417	. . . . . 6 |- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
79		opelxpi 4806	. . . . . 6 |- ( ( ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. /\ 1P e. P. ) -> <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. e. ( P. X. P. ) )
80	78, 76, 79	sylancl 417	. . . . 5 |- ( N e. N. -> <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. e. ( P. X. P. ) )
81		enrex 8104	. . . . . 6 |- ~R e. _V
82	81	ecelqsi 6863	. . . . 5 |- ( <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. e. ( P. X. P. ) -> [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
83	80, 82	syl 14	. . . 4 |- ( N e. N. -> [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
84		0r 8117	. . . 4 |- 0R e. R.
85		opelxpi 4806	. . . 4 |- ( ( [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R e. ( ( P. X. P. ) /. ~R ) /\ 0R e. R. ) -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. ( ( ( P. X. P. ) /. ~R ) X. R. ) )
86	83, 84, 85	sylancl 417	. . 3 |- ( N e. N. -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. ( ( ( P. X. P. ) /. ~R ) X. R. ) )
87		elintg 3978	. . 3 |- ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. ( ( ( P. X. P. ) /. ~R ) X. R. ) -> ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } <-> A. z e. { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
88	86, 87	syl 14	. 2 |- ( N e. N. -> ( <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } <-> A. z e. { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. z ) )
89	74, 88	mpbird 167	1 |- ( N e. N. -> <. [ <. ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   <.cop 3712   |^|cint 3970   class class class wbr 4130   X. cxp 4772  (class class class)co 6085   1oc1o 6680   [cec 6805   /.cqs 6806   N.cnpi 7639   +N cpli 7640   ~Q ceq 7646   <Q cltq 7652   P.cnp 7658   1Pc1p 7659   +P. cpp 7660   ~R cer 7663   R.cnr 7664   0Rc0r 7665   1c1 8180   + caddc 8182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  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-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-i1p 7834  df-iplp 7835  df-enr 8093  df-nr 8094  df-plr 8095  df-0r 8098  df-1r 8099  df-c 8185  df-1 8187  df-add 8190

This theorem is used by:  axarch  8258  axcaucvglemcl  8262  axcaucvglemval  8264  axcaucvglemcau  8265  axcaucvglemres  8266