Theorem pitonn 7680

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 3713	. . . . . . . . . . . . . . 15 |- ( w = 1o -> <. w , 1o >. = <. 1o , 1o >. )
2	1	eceq1d 6473	. . . . . . . . . . . . . 14 |- ( w = 1o -> [ <. w , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
3	2	breq2d 3949	. . . . . . . . . . . . 13 |- ( w = 1o -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. 1o , 1o >. ] ~Q ) )
4	3	abbidv 2258	. . . . . . . . . . . 12 |- ( w = 1o -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. 1o , 1o >. ] ~Q } )
5	2	breq1d 3947	. . . . . . . . . . . . 13 |- ( w = 1o -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. 1o , 1o >. ] ~Q <Q u ) )
6	5	abbidv 2258	. . . . . . . . . . . 12 |- ( w = 1o -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. 1o , 1o >. ] ~Q <Q u } )
7	4, 6	opeq12d 3721	. . . . . . . . . . 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 5797	. . . . . . . . . 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 3719	. . . . . . . . 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 6473	. . . . . . . 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 3719	. . . . . . 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 2209	. . . . . 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 229	. . . . 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 3713	. . . . . . . . . . . . . . 15 |- ( w = k -> <. w , 1o >. = <. k , 1o >. )
15	14	eceq1d 6473	. . . . . . . . . . . . . 14 |- ( w = k -> [ <. w , 1o >. ] ~Q = [ <. k , 1o >. ] ~Q )
16	15	breq2d 3949	. . . . . . . . . . . . 13 |- ( w = k -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. k , 1o >. ] ~Q ) )
17	16	abbidv 2258	. . . . . . . . . . . 12 |- ( w = k -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. k , 1o >. ] ~Q } )
18	15	breq1d 3947	. . . . . . . . . . . . 13 |- ( w = k -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. k , 1o >. ] ~Q <Q u ) )
19	18	abbidv 2258	. . . . . . . . . . . 12 |- ( w = k -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. k , 1o >. ] ~Q <Q u } )
20	17, 19	opeq12d 3721	. . . . . . . . . . 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 5797	. . . . . . . . . 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 3719	. . . . . . . . 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 6473	. . . . . . . 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 3719	. . . . . . 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 2209	. . . . . 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 229	. . . . 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 3713	. . . . . . . . . . . . . . 15 |- ( w = ( k +N 1o ) -> <. w , 1o >. = <. ( k +N 1o ) , 1o >. )
28	27	eceq1d 6473	. . . . . . . . . . . . . 14 |- ( w = ( k +N 1o ) -> [ <. w , 1o >. ] ~Q = [ <. ( k +N 1o ) , 1o >. ] ~Q )
29	28	breq2d 3949	. . . . . . . . . . . . 13 |- ( w = ( k +N 1o ) -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q ) )
30	29	abbidv 2258	. . . . . . . . . . . 12 |- ( w = ( k +N 1o ) -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. ( k +N 1o ) , 1o >. ] ~Q } )
31	28	breq1d 3947	. . . . . . . . . . . . 13 |- ( w = ( k +N 1o ) -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u ) )
32	31	abbidv 2258	. . . . . . . . . . . 12 |- ( w = ( k +N 1o ) -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. ( k +N 1o ) , 1o >. ] ~Q <Q u } )
33	30, 32	opeq12d 3721	. . . . . . . . . . 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 5797	. . . . . . . . . 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 3719	. . . . . . . . 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 6473	. . . . . . . 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 3719	. . . . . . 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 2209	. . . . . 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 229	. . . . 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 3713	. . . . . . . . . . . . . . 15 |- ( w = N -> <. w , 1o >. = <. N , 1o >. )
41	40	eceq1d 6473	. . . . . . . . . . . . . 14 |- ( w = N -> [ <. w , 1o >. ] ~Q = [ <. N , 1o >. ] ~Q )
42	41	breq2d 3949	. . . . . . . . . . . . 13 |- ( w = N -> ( l <Q [ <. w , 1o >. ] ~Q <-> l <Q [ <. N , 1o >. ] ~Q ) )
43	42	abbidv 2258	. . . . . . . . . . . 12 |- ( w = N -> { l | l <Q [ <. w , 1o >. ] ~Q } = { l | l <Q [ <. N , 1o >. ] ~Q } )
44	41	breq1d 3947	. . . . . . . . . . . . 13 |- ( w = N -> ( [ <. w , 1o >. ] ~Q <Q u <-> [ <. N , 1o >. ] ~Q <Q u ) )
45	44	abbidv 2258	. . . . . . . . . . . 12 |- ( w = N -> { u | [ <. w , 1o >. ] ~Q <Q u } = { u | [ <. N , 1o >. ] ~Q <Q u } )
46	43, 45	opeq12d 3721	. . . . . . . . . . 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 5797	. . . . . . . . . 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 3719	. . . . . . . . 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 6473	. . . . . . . 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 3719	. . . . . . 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 2209	. . . . . 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 229	. . . . 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 7677	. . . . . . . 8 |- <. [ <. ( <. { l | l <Q [ <. 1o , 1o >. ] ~Q } , { u | [ <. 1o , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1
54	53	eleq1i 2206	. . . . . . 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 132	. . . . . 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 274	. . . . 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 5789	. . . . . . . . . . 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 2209	. . . . . . . . . 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 2790	. . . . . . . . 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 483	. . . . . . . 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 7679	. . . . . . . . . 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 2209	. . . . . . . . 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 274	. . . . . . . 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 148	. . . . . . 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 114	. . . . . 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 7174	. . . 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 1847	. . 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 2204	. . . . 5 |- ( x = z -> ( 1 e. x <-> 1 e. z ) )
70		eleq2 2204	. . . . . 6 |- ( x = z -> ( ( y + 1 ) e. x <-> ( y + 1 ) e. z ) )
71	70	raleqbi1dv 2637	. . . . 5 |- ( x = z -> ( A. y e. x ( y + 1 ) e. x <-> A. y e. z ( y + 1 ) e. z ) )
72	69, 71	anbi12d 465	. . . 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 2848	. . 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 133	. 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 7385	. . . . . . 7 |- ( N e. N. -> <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. e. P. )
76		1pr 7386	. . . . . . 7 |- 1P e. P.
77		addclpr 7369	. . . . . . 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 410	. . . . . 6 |- ( N e. N. -> ( <. { l | l <Q [ <. N , 1o >. ] ~Q } , { u | [ <. N , 1o >. ] ~Q <Q u } >. +P. 1P ) e. P. )
79		opelxpi 4579	. . . . . 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 410	. . . . 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 7569	. . . . . 6 |- ~R e. _V
82	81	ecelqsi 6491	. . . . 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 7582	. . . 4 |- 0R e. R.
85		opelxpi 4579	. . . 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 410	. . 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 3787	. . 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 166	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 103   <-> wb 104   A.wal 1330   = wceq 1332   e. wcel 1481   {cab 2126   A.wral 2417   <.cop 3535   |^|cint 3779   class class class wbr 3937   X. cxp 4545  (class class class)co 5782   1oc1o 6314   [cec 6435   /.cqs 6436   N.cnpi 7104   +N cpli 7105   ~Q ceq 7111   <Q cltq 7117   P.cnp 7123   1Pc1p 7124   +P. cpp 7125   ~R cer 7128   R.cnr 7129   0Rc0r 7130   1c1 7645   + caddc 7647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-i1p 7299  df-iplp 7300  df-enr 7558  df-nr 7559  df-plr 7560  df-0r 7563  df-1r 7564  df-c 7650  df-1 7652  df-add 7655

This theorem is referenced by:  axarch  7723  axcaucvglemcl  7727  axcaucvglemval  7729  axcaucvglemcau  7730  axcaucvglemres  7731