Theorem nntopi 8113

Description: Mapping from NN to N.. (Contributed by Jim Kingdon, 13-Jul-2021.)

Hypothesis

Ref	Expression
nntopi.n	|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }

Assertion

Ref	Expression
nntopi	|- ( A e. N -> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = A )

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

Allowed substitution hints:   A( x, y, u, l)   N( u, l)

Proof of Theorem nntopi

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

Step	Hyp	Ref	Expression
1		nntopi.n	. 2 |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
2		eqeq2 2241	. . 3 |- ( w = 1 -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 ) )
3	2	rexbidv 2533	. 2 |- ( w = 1 -> ( E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 ) )
4		eqeq2 2241	. . 3 |- ( w = k -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) )
5	4	rexbidv 2533	. 2 |- ( w = k -> ( E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) )
6		eqeq2 2241	. . 3 |- ( w = ( k + 1 ) -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) )
7	6	rexbidv 2533	. 2 |- ( w = ( k + 1 ) -> ( E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) )
8		eqeq2 2241	. . 3 |- ( w = A -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = A ) )
9	8	rexbidv 2533	. 2 |- ( w = A -> ( E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = w <-> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = A ) )
10		1pi 7534	. . 3 |- 1o e. N.
11		eqid 2231	. . 3 |- 1 = 1
12		opeq1 3862	. . . . . . . . . . . . . . . . 17 |- ( z = 1o -> <. z , 1o >. = <. 1o , 1o >. )
13	12	eceq1d 6737	. . . . . . . . . . . . . . . 16 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
14		df-1nqqs 7570	. . . . . . . . . . . . . . . 16 |- 1Q = [ <. 1o , 1o >. ] ~Q
15	13, 14	eqtr4di 2282	. . . . . . . . . . . . . . 15 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = 1Q )
16	15	breq2d 4100	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( l <Q [ <. z , 1o >. ] ~Q <-> l <Q 1Q ) )
17	16	abbidv 2349	. . . . . . . . . . . . 13 |- ( z = 1o -> { l | l <Q [ <. z , 1o >. ] ~Q } = { l | l <Q 1Q } )
18	15	breq1d 4098	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( [ <. z , 1o >. ] ~Q <Q u <-> 1Q <Q u ) )
19	18	abbidv 2349	. . . . . . . . . . . . 13 |- ( z = 1o -> { u | [ <. z , 1o >. ] ~Q <Q u } = { u | 1Q <Q u } )
20	17, 19	opeq12d 3870	. . . . . . . . . . . 12 |- ( z = 1o -> <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. = <. { l | l <Q 1Q } , { u | 1Q <Q u } >. )
21		df-i1p 7686	. . . . . . . . . . . 12 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
22	20, 21	eqtr4di 2282	. . . . . . . . . . 11 |- ( z = 1o -> <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. = 1P )
23	22	oveq1d 6032	. . . . . . . . . 10 |- ( z = 1o -> ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( 1P +P. 1P ) )
24	23	opeq1d 3868	. . . . . . . . 9 |- ( z = 1o -> <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( 1P +P. 1P ) , 1P >. )
25	24	eceq1d 6737	. . . . . . . 8 |- ( z = 1o -> [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R )
26		df-1r 7951	. . . . . . . 8 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
27	25, 26	eqtr4di 2282	. . . . . . 7 |- ( z = 1o -> [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R )
28	27	opeq1d 3868	. . . . . 6 |- ( z = 1o -> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. 1R , 0R >. )
29		df-1 8039	. . . . . 6 |- 1 = <. 1R , 0R >.
30	28, 29	eqtr4di 2282	. . . . 5 |- ( z = 1o -> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 )
31	30	eqeq1d 2240	. . . 4 |- ( z = 1o -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 <-> 1 = 1 ) )
32	31	rspcev 2910	. . 3 |- ( ( 1o e. N. /\ 1 = 1 ) -> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 )
33	10, 11, 32	mp2an 426	. 2 |- E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1
34		simplr 529	. . . . . . 7 |- ( ( ( k e. N /\ z e. N. ) /\ <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) -> z e. N. )
35		addclpi 7546	. . . . . . 7 |- ( ( z e. N. /\ 1o e. N. ) -> ( z +N 1o ) e. N. )
36	34, 10, 35	sylancl 413	. . . . . 6 |- ( ( ( k e. N /\ z e. N. ) /\ <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) -> ( z +N 1o ) e. N. )
37		pitonnlem2 8066	. . . . . . . 8 |- ( z e. N. -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = <. [ <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
38	34, 37	syl 14	. . . . . . 7 |- ( ( ( k e. N /\ z e. N. ) /\ <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = <. [ <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
39		simpr 110	. . . . . . . 8 |- ( ( ( k e. N /\ z e. N. ) /\ <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) -> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k )
40	39	oveq1d 6032	. . . . . . 7 |- ( ( ( k e. N /\ z e. N. ) /\ <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. + 1 ) = ( k + 1 ) )
41	38, 40	eqtr3d 2266	. . . . . 6 |- ( ( ( k e. N /\ z e. N. ) /\ <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) -> <. [ <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) )
42		opeq1 3862	. . . . . . . . . . . . . . . 16 |- ( v = ( z +N 1o ) -> <. v , 1o >. = <. ( z +N 1o ) , 1o >. )
43	42	eceq1d 6737	. . . . . . . . . . . . . . 15 |- ( v = ( z +N 1o ) -> [ <. v , 1o >. ] ~Q = [ <. ( z +N 1o ) , 1o >. ] ~Q )
44	43	breq2d 4100	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q ) )
45	44	abbidv 2349	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } )
46	43	breq1d 4098	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u ) )
47	46	abbidv 2349	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } )
48	45, 47	opeq12d 3870	. . . . . . . . . . . 12 |- ( v = ( z +N 1o ) -> <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. = <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. )
49	48	oveq1d 6032	. . . . . . . . . . 11 |- ( v = ( z +N 1o ) -> ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) )
50	49	opeq1d 3868	. . . . . . . . . 10 |- ( v = ( z +N 1o ) -> <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. )
51	50	eceq1d 6737	. . . . . . . . 9 |- ( v = ( z +N 1o ) -> [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
52	51	opeq1d 3868	. . . . . . . 8 |- ( v = ( z +N 1o ) -> <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. [ <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
53	52	eqeq1d 2240	. . . . . . 7 |- ( v = ( z +N 1o ) -> ( <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) <-> <. [ <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) )
54	53	rspcev 2910	. . . . . 6 |- ( ( ( z +N 1o ) e. N. /\ <. [ <. ( <. { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } , { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) -> E. v e. N. <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) )
55	36, 41, 54	syl2anc 411	. . . . 5 |- ( ( ( k e. N /\ z e. N. ) /\ <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k ) -> E. v e. N. <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) )
56	55	ex 115	. . . 4 |- ( ( k e. N /\ z e. N. ) -> ( <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k -> E. v e. N. <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) )
57	56	rexlimdva 2650	. . 3 |- ( k e. N -> ( E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k -> E. v e. N. <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) )
58		opeq1 3862	. . . . . . . . . . . . 13 |- ( v = z -> <. v , 1o >. = <. z , 1o >. )
59	58	eceq1d 6737	. . . . . . . . . . . 12 |- ( v = z -> [ <. v , 1o >. ] ~Q = [ <. z , 1o >. ] ~Q )
60	59	breq2d 4100	. . . . . . . . . . 11 |- ( v = z -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. z , 1o >. ] ~Q ) )
61	60	abbidv 2349	. . . . . . . . . 10 |- ( v = z -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. z , 1o >. ] ~Q } )
62	59	breq1d 4098	. . . . . . . . . . 11 |- ( v = z -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. z , 1o >. ] ~Q <Q u ) )
63	62	abbidv 2349	. . . . . . . . . 10 |- ( v = z -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. z , 1o >. ] ~Q <Q u } )
64	61, 63	opeq12d 3870	. . . . . . . . 9 |- ( v = z -> <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. = <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. )
65	64	oveq1d 6032	. . . . . . . 8 |- ( v = z -> ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) )
66	65	opeq1d 3868	. . . . . . 7 |- ( v = z -> <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. )
67	66	eceq1d 6737	. . . . . 6 |- ( v = z -> [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
68	67	opeq1d 3868	. . . . 5 |- ( v = z -> <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
69	68	eqeq1d 2240	. . . 4 |- ( v = z -> ( <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) <-> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) )
70	69	cbvrexv 2768	. . 3 |- ( E. v e. N. <. [ <. ( <. { l | l <Q [ <. v , 1o >. ] ~Q } , { u | [ <. v , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) <-> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) )
71	57, 70	imbitrdi 161	. 2 |- ( k e. N -> ( E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = k -> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = ( k + 1 ) ) )
72	1, 3, 5, 7, 9, 33, 71	nnindnn 8112	1 |- ( A e. N -> E. z e. N. <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   <.cop 3672   |^|cint 3928   class class class wbr 4088  (class class class)co 6017   1oc1o 6574   [cec 6699   N.cnpi 7491   +N cpli 7492   ~Q ceq 7498   1Qc1q 7500   <Q cltq 7504   1Pc1p 7511   +P. cpp 7512   ~R cer 7515   0Rc0r 7517   1Rc1r 7518   1c1 8032   + caddc 8034

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 619  ax-in2 620  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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-i1p 7686  df-iplp 7687  df-enr 7945  df-nr 7946  df-plr 7947  df-0r 7950  df-1r 7951  df-c 8037  df-1 8039  df-r 8041  df-add 8042

This theorem is referenced by:  axcaucvglemres  8118