Theorem nntopi 8261

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 2248	. . 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 2551	. 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 2248	. . 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 2551	. 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 2248	. . 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 2551	. 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 2248	. . 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 2551	. 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 7682	. . 3 |- 1o e. N.
11		eqid 2238	. . 3 |- 1 = 1
12		opeq1 3904	. . . . . . . . . . . . . . . . 17 |- ( z = 1o -> <. z , 1o >. = <. 1o , 1o >. )
13	12	eceq1d 6843	. . . . . . . . . . . . . . . 16 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
14		df-1nqqs 7718	. . . . . . . . . . . . . . . 16 |- 1Q = [ <. 1o , 1o >. ] ~Q
15	13, 14	eqtr4di 2289	. . . . . . . . . . . . . . 15 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = 1Q )
16	15	breq2d 4142	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( l <Q [ <. z , 1o >. ] ~Q <-> l <Q 1Q ) )
17	16	abbidv 2358	. . . . . . . . . . . . 13 |- ( z = 1o -> { l | l <Q [ <. z , 1o >. ] ~Q } = { l | l <Q 1Q } )
18	15	breq1d 4140	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( [ <. z , 1o >. ] ~Q <Q u <-> 1Q <Q u ) )
19	18	abbidv 2358	. . . . . . . . . . . . 13 |- ( z = 1o -> { u | [ <. z , 1o >. ] ~Q <Q u } = { u | 1Q <Q u } )
20	17, 19	opeq12d 3912	. . . . . . . . . . . 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 7834	. . . . . . . . . . . 12 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
22	20, 21	eqtr4di 2289	. . . . . . . . . . 11 |- ( z = 1o -> <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. = 1P )
23	22	oveq1d 6100	. . . . . . . . . 10 |- ( z = 1o -> ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( 1P +P. 1P ) )
24	23	opeq1d 3910	. . . . . . . . 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 6843	. . . . . . . 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 8099	. . . . . . . 8 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
27	25, 26	eqtr4di 2289	. . . . . . 7 |- ( z = 1o -> [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R )
28	27	opeq1d 3910	. . . . . 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 8187	. . . . . 6 |- 1 = <. 1R , 0R >.
30	28, 29	eqtr4di 2289	. . . . 5 |- ( z = 1o -> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 )
31	30	eqeq1d 2247	. . . 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 2929	. . 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 430	. 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 533	. . . . . . 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 7694	. . . . . . 7 |- ( ( z e. N. /\ 1o e. N. ) -> ( z +N 1o ) e. N. )
36	34, 10, 35	sylancl 417	. . . . . 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 8214	. . . . . . . 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 6100	. . . . . . 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 2273	. . . . . 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 3904	. . . . . . . . . . . . . . . 16 |- ( v = ( z +N 1o ) -> <. v , 1o >. = <. ( z +N 1o ) , 1o >. )
43	42	eceq1d 6843	. . . . . . . . . . . . . . 15 |- ( v = ( z +N 1o ) -> [ <. v , 1o >. ] ~Q = [ <. ( z +N 1o ) , 1o >. ] ~Q )
44	43	breq2d 4142	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q ) )
45	44	abbidv 2358	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } )
46	43	breq1d 4140	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u ) )
47	46	abbidv 2358	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } )
48	45, 47	opeq12d 3912	. . . . . . . . . . . 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 6100	. . . . . . . . . . 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 3910	. . . . . . . . . 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 6843	. . . . . . . . 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 3910	. . . . . . . 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 2247	. . . . . . 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 2929	. . . . . 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 415	. . . . 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 2668	. . 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 3904	. . . . . . . . . . . . 13 |- ( v = z -> <. v , 1o >. = <. z , 1o >. )
59	58	eceq1d 6843	. . . . . . . . . . . 12 |- ( v = z -> [ <. v , 1o >. ] ~Q = [ <. z , 1o >. ] ~Q )
60	59	breq2d 4142	. . . . . . . . . . 11 |- ( v = z -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. z , 1o >. ] ~Q ) )
61	60	abbidv 2358	. . . . . . . . . 10 |- ( v = z -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. z , 1o >. ] ~Q } )
62	59	breq1d 4140	. . . . . . . . . . 11 |- ( v = z -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. z , 1o >. ] ~Q <Q u ) )
63	62	abbidv 2358	. . . . . . . . . 10 |- ( v = z -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. z , 1o >. ] ~Q <Q u } )
64	61, 63	opeq12d 3912	. . . . . . . . 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 6100	. . . . . . . 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 3910	. . . . . . 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 6843	. . . . . 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 3910	. . . . 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 2247	. . . 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 2787	. . 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 8260	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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   <.cop 3712   |^|cint 3970   class class class wbr 4130  (class class class)co 6085   1oc1o 6680   [cec 6805   N.cnpi 7639   +N cpli 7640   ~Q ceq 7646   1Qc1q 7648   <Q cltq 7652   1Pc1p 7659   +P. cpp 7660   ~R cer 7663   0Rc0r 7665   1Rc1r 7666   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-r 8189  df-add 8190

This theorem is used by:  axcaucvglemres  8266