Theorem pitonn 7910

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 3805	. . . . . . . . . . . . . . 15 |- ( w = 1o -> <. w , 1o >. = <. 1o , 1o >. )
2	1	eceq1d 6625	. . . . . . . . . . . . . 14 |- ( w = 1o -> [ <. w , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
3	2	breq2d 4042	. . . . . . . . . . . . 13 |- ( w = 1o -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. 1o , 1o >. ] ~Q ) )
4	3	abbidv 2311	. . . . . . . . . . . 12 |- ( w = 1o -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q } )
5	2	breq1d 4040	. . . . . . . . . . . . 13 |- ( w = 1o -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u ) )
6	5	abbidv 2311	. . . . . . . . . . . 12 |- ( w = 1o -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u } )
7	4, 6	opeq12d 3813	. . . . . . . . . . 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 5934	. . . . . . . . . 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 3811	. . . . . . . . 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 6625	. . . . . . . 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 3811	. . . . . . 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 2262	. . . . . 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 3805	. . . . . . . . . . . . . . 15 |- ( w = k -> <. w , 1o >. = <. k , 1o >. )
15	14	eceq1d 6625	. . . . . . . . . . . . . 14 |- ( w = k -> [ <. w , 1o >. ] ~Q = [ <. k , 1o >. ] ~Q )
16	15	breq2d 4042	. . . . . . . . . . . . 13 |- ( w = k -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. k , 1o >. ] ~Q ) )
17	16	abbidv 2311	. . . . . . . . . . . 12 |- ( w = k -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. k , 1o >. ] ~Q } )
18	15	breq1d 4040	. . . . . . . . . . . . 13 |- ( w = k -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. k , 1o >. ] ~Q <Q u ) )
19	18	abbidv 2311	. . . . . . . . . . . 12 |- ( w = k -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. k , 1o >. ] ~Q <Q u } )
20	17, 19	opeq12d 3813	. . . . . . . . . . 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 5934	. . . . . . . . . 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 3811	. . . . . . . . 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 6625	. . . . . . . 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 3811	. . . . . . 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 2262	. . . . . 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 3805	. . . . . . . . . . . . . . 15 |- ( w = ( k +N 1o ) -> <. w , 1o >. = <. ( k +N 1o ) , 1o >. )
28	27	eceq1d 6625	. . . . . . . . . . . . . 14 |- ( w = ( k +N 1o ) -> [ <. w , 1o >. ] ~Q = [ <. ( k +N 1o ) , 1o >. ] ~Q )
29	28	breq2d 4042	. . . . . . . . . . . . 13 |- ( w = ( k +N 1o ) -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q ) )
30	29	abbidv 2311	. . . . . . . . . . . 12 |- ( w = ( k +N 1o ) -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } )
31	28	breq1d 4040	. . . . . . . . . . . . 13 |- ( w = ( k +N 1o ) -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u ) )
32	31	abbidv 2311	. . . . . . . . . . . 12 |- ( w = ( k +N 1o ) -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } )
33	30, 32	opeq12d 3813	. . . . . . . . . . 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 5934	. . . . . . . . . 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 3811	. . . . . . . . 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 6625	. . . . . . . 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 3811	. . . . . . 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 2262	. . . . . 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 3805	. . . . . . . . . . . . . . 15 |- ( w = N -> <. w , 1o >. = <. N , 1o >. )
41	40	eceq1d 6625	. . . . . . . . . . . . . 14 |- ( w = N -> [ <. w , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
42	41	breq2d 4042	. . . . . . . . . . . . 13 |- ( w = N -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. N , 1o >. ] ~Q ) )
43	42	abbidv 2311	. . . . . . . . . . . 12 |- ( w = N -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. N , 1o >. ] ~Q } )
44	41	breq1d 4040	. . . . . . . . . . . . 13 |- ( w = N -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. N , 1o >. ] ~Q <Q u ) )
45	44	abbidv 2311	. . . . . . . . . . . 12 |- ( w = N -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. N , 1o >. ] ~Q <Q u } )
46	43, 45	opeq12d 3813	. . . . . . . . . . 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 5934	. . . . . . . . . 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 3811	. . . . . . . . 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 6625	. . . . . . . 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 3811	. . . . . . 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 2262	. . . . . 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 7907	. . . . . . . 8 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1
54	53	eleq1i 2259	. . . . . . 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 5926	. . . . . . . . . . 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 2262	. . . . . . . . . 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 2862	. . . . . . . . 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 491	. . . . . . . 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 7909	. . . . . . . . . 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 2262	. . . . . . . . 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 7404	. . . 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 1885	. . 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 2257	. . . . 5 |- ( x = z -> ( 1 e. x <-> 1 e. z ) )
70		eleq2 2257	. . . . . 6 |- ( x = z -> ( ( y + 1 ) e. x <-> ( y + 1 ) e. z ) )
71	70	raleqbi1dv 2702	. . . . 5 |- ( x = z -> ( A. y e. x ( y + 1 ) e. x <-> A. y e. z ( y + 1 ) e. z ) )
72	69, 71	anbi12d 473	. . . 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 2921	. . 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 7615	. . . . . . 7 |- ( N e. N. -> <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. e. P. )
76		1pr 7616	. . . . . . 7 |- 1P e. P.
77		addclpr 7599	. . . . . . 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 413	. . . . . 6 |- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
79		opelxpi 4692	. . . . . 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 413	. . . . 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 7799	. . . . . 6 |- ~R e. _V
82	81	ecelqsi 6645	. . . . 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 7812	. . . 4 |- 0R e. R.
85		opelxpi 4692	. . . 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 413	. . 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 3879	. . 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 ) } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   <.cop 3622   |^|cint 3871   class class class wbr 4030   X. cxp 4658  (class class class)co 5919   1oc1o 6464   [cec 6587   /.cqs 6588   N.cnpi 7334   +N cpli 7335   ~Q ceq 7341   <Q cltq 7347   P.cnp 7353   1Pc1p 7354   +P. cpp 7355   ~R cer 7358   R.cnr 7359   0Rc0r 7360   1c1 7875   + caddc 7877

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-iplp 7530  df-enr 7788  df-nr 7789  df-plr 7790  df-0r 7793  df-1r 7794  df-c 7880  df-1 7882  df-add 7885

This theorem is referenced by:  axarch  7953  axcaucvglemcl  7957  axcaucvglemval  7959  axcaucvglemcau  7960  axcaucvglemres  7961