Theorem nntopi 7892

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 2187	. . 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 2478	. 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 2187	. . 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 2478	. 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 2187	. . 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 2478	. 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 2187	. . 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 2478	. 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 7313	. . 3 |- 1o e. N.
11		eqid 2177	. . 3 |- 1 = 1
12		opeq1 3778	. . . . . . . . . . . . . . . . 17 |- ( z = 1o -> <. z , 1o >. = <. 1o , 1o >. )
13	12	eceq1d 6570	. . . . . . . . . . . . . . . 16 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
14		df-1nqqs 7349	. . . . . . . . . . . . . . . 16 |- 1Q = [ <. 1o , 1o >. ] ~Q
15	13, 14	eqtr4di 2228	. . . . . . . . . . . . . . 15 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = 1Q )
16	15	breq2d 4015	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( l <Q [ <. z , 1o >. ] ~Q <-> l <Q 1Q ) )
17	16	abbidv 2295	. . . . . . . . . . . . 13 |- ( z = 1o -> { l | l <Q [ <. z , 1o >. ] ~Q } = { l | l <Q 1Q } )
18	15	breq1d 4013	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( [ <. z , 1o >. ] ~Q <Q u <-> 1Q <Q u ) )
19	18	abbidv 2295	. . . . . . . . . . . . 13 |- ( z = 1o -> { u | [ <. z , 1o >. ] ~Q <Q u } = { u | 1Q <Q u } )
20	17, 19	opeq12d 3786	. . . . . . . . . . . 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 7465	. . . . . . . . . . . 12 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
22	20, 21	eqtr4di 2228	. . . . . . . . . . 11 |- ( z = 1o -> <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. = 1P )
23	22	oveq1d 5889	. . . . . . . . . 10 |- ( z = 1o -> ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( 1P +P. 1P ) )
24	23	opeq1d 3784	. . . . . . . . 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 6570	. . . . . . . 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 7730	. . . . . . . 8 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
27	25, 26	eqtr4di 2228	. . . . . . 7 |- ( z = 1o -> [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R )
28	27	opeq1d 3784	. . . . . 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 7818	. . . . . 6 |- 1 = <. 1R , 0R >.
30	28, 29	eqtr4di 2228	. . . . 5 |- ( z = 1o -> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 )
31	30	eqeq1d 2186	. . . 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 2841	. . 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 528	. . . . . . 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 7325	. . . . . . 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 7845	. . . . . . . 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 5889	. . . . . . 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 2212	. . . . . 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 3778	. . . . . . . . . . . . . . . 16 |- ( v = ( z +N 1o ) -> <. v , 1o >. = <. ( z +N 1o ) , 1o >. )
43	42	eceq1d 6570	. . . . . . . . . . . . . . 15 |- ( v = ( z +N 1o ) -> [ <. v , 1o >. ] ~Q = [ <. ( z +N 1o ) , 1o >. ] ~Q )
44	43	breq2d 4015	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q ) )
45	44	abbidv 2295	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } )
46	43	breq1d 4013	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u ) )
47	46	abbidv 2295	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } )
48	45, 47	opeq12d 3786	. . . . . . . . . . . 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 5889	. . . . . . . . . . 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 3784	. . . . . . . . . 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 6570	. . . . . . . . 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 3784	. . . . . . . 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 2186	. . . . . . 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 2841	. . . . . 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 2594	. . 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 3778	. . . . . . . . . . . . 13 |- ( v = z -> <. v , 1o >. = <. z , 1o >. )
59	58	eceq1d 6570	. . . . . . . . . . . 12 |- ( v = z -> [ <. v , 1o >. ] ~Q = [ <. z , 1o >. ] ~Q )
60	59	breq2d 4015	. . . . . . . . . . 11 |- ( v = z -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. z , 1o >. ] ~Q ) )
61	60	abbidv 2295	. . . . . . . . . 10 |- ( v = z -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. z , 1o >. ] ~Q } )
62	59	breq1d 4013	. . . . . . . . . . 11 |- ( v = z -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. z , 1o >. ] ~Q <Q u ) )
63	62	abbidv 2295	. . . . . . . . . 10 |- ( v = z -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. z , 1o >. ] ~Q <Q u } )
64	61, 63	opeq12d 3786	. . . . . . . . 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 5889	. . . . . . . 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 3784	. . . . . . 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 6570	. . . . . 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 3784	. . . . 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 2186	. . . 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 2704	. . 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 7891	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 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   <.cop 3595   |^|cint 3844   class class class wbr 4003  (class class class)co 5874   1oc1o 6409   [cec 6532   N.cnpi 7270   +N cpli 7271   ~Q ceq 7277   1Qc1q 7279   <Q cltq 7283   1Pc1p 7290   +P. cpp 7291   ~R cer 7294   0Rc0r 7296   1Rc1r 7297   1c1 7811   + caddc 7813

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-iplp 7466  df-enr 7724  df-nr 7725  df-plr 7726  df-0r 7729  df-1r 7730  df-c 7816  df-1 7818  df-r 7820  df-add 7821

This theorem is referenced by:  axcaucvglemres  7897