Theorem nntopi 8042

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 2217	. . 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 2509	. 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 2217	. . 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 2509	. 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 2217	. . 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 2509	. 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 2217	. . 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 2509	. 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 7463	. . 3 |- 1o e. N.
11		eqid 2207	. . 3 |- 1 = 1
12		opeq1 3833	. . . . . . . . . . . . . . . . 17 |- ( z = 1o -> <. z , 1o >. = <. 1o , 1o >. )
13	12	eceq1d 6679	. . . . . . . . . . . . . . . 16 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = [ <. 1o , 1o >. ] ~Q )
14		df-1nqqs 7499	. . . . . . . . . . . . . . . 16 |- 1Q = [ <. 1o , 1o >. ] ~Q
15	13, 14	eqtr4di 2258	. . . . . . . . . . . . . . 15 |- ( z = 1o -> [ <. z , 1o >. ] ~Q = 1Q )
16	15	breq2d 4071	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( l <Q [ <. z , 1o >. ] ~Q <-> l <Q 1Q ) )
17	16	abbidv 2325	. . . . . . . . . . . . 13 |- ( z = 1o -> { l | l <Q [ <. z , 1o >. ] ~Q } = { l | l <Q 1Q } )
18	15	breq1d 4069	. . . . . . . . . . . . . 14 |- ( z = 1o -> ( [ <. z , 1o >. ] ~Q <Q u <-> 1Q <Q u ) )
19	18	abbidv 2325	. . . . . . . . . . . . 13 |- ( z = 1o -> { u | [ <. z , 1o >. ] ~Q <Q u } = { u | 1Q <Q u } )
20	17, 19	opeq12d 3841	. . . . . . . . . . . 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 7615	. . . . . . . . . . . 12 |- 1P = <. { l | l <Q 1Q } , { u | 1Q <Q u } >.
22	20, 21	eqtr4di 2258	. . . . . . . . . . 11 |- ( z = 1o -> <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. = 1P )
23	22	oveq1d 5982	. . . . . . . . . 10 |- ( z = 1o -> ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( 1P +P. 1P ) )
24	23	opeq1d 3839	. . . . . . . . 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 6679	. . . . . . . 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 7880	. . . . . . . 8 |- 1R = [ <. ( 1P +P. 1P ) , 1P >. ] ~R
27	25, 26	eqtr4di 2258	. . . . . . 7 |- ( z = 1o -> [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = 1R )
28	27	opeq1d 3839	. . . . . 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 7968	. . . . . 6 |- 1 = <. 1R , 0R >.
30	28, 29	eqtr4di 2258	. . . . 5 |- ( z = 1o -> <. [ <. ( <. { l | l <Q [ <. z , 1o >. ] ~Q } , { u | [ <. z , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = 1 )
31	30	eqeq1d 2216	. . . 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 2884	. . 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 7475	. . . . . . 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 7995	. . . . . . . 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 5982	. . . . . . 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 2242	. . . . . 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 3833	. . . . . . . . . . . . . . . 16 |- ( v = ( z +N 1o ) -> <. v , 1o >. = <. ( z +N 1o ) , 1o >. )
43	42	eceq1d 6679	. . . . . . . . . . . . . . 15 |- ( v = ( z +N 1o ) -> [ <. v , 1o >. ] ~Q = [ <. ( z +N 1o ) , 1o >. ] ~Q )
44	43	breq2d 4071	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q ) )
45	44	abbidv 2325	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. ( z +N 1o ) , 1o >. ] ~Q } )
46	43	breq1d 4069	. . . . . . . . . . . . . 14 |- ( v = ( z +N 1o ) -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u ) )
47	46	abbidv 2325	. . . . . . . . . . . . 13 |- ( v = ( z +N 1o ) -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. ( z +N 1o ) , 1o >. ] ~Q <Q u } )
48	45, 47	opeq12d 3841	. . . . . . . . . . . 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 5982	. . . . . . . . . . 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 3839	. . . . . . . . . 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 6679	. . . . . . . . 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 3839	. . . . . . . 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 2216	. . . . . . 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 2884	. . . . . 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 2625	. . 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 3833	. . . . . . . . . . . . 13 |- ( v = z -> <. v , 1o >. = <. z , 1o >. )
59	58	eceq1d 6679	. . . . . . . . . . . 12 |- ( v = z -> [ <. v , 1o >. ] ~Q = [ <. z , 1o >. ] ~Q )
60	59	breq2d 4071	. . . . . . . . . . 11 |- ( v = z -> ( l <Q [ <. v , 1o >. ] ~Q <-> l <Q [ <. z , 1o >. ] ~Q ) )
61	60	abbidv 2325	. . . . . . . . . 10 |- ( v = z -> { l | l <Q [ <. v , 1o >. ] ~Q } = { l | l <Q [ <. z , 1o >. ] ~Q } )
62	59	breq1d 4069	. . . . . . . . . . 11 |- ( v = z -> ( [ <. v , 1o >. ] ~Q <Q u <-> [ <. z , 1o >. ] ~Q <Q u ) )
63	62	abbidv 2325	. . . . . . . . . 10 |- ( v = z -> { u | [ <. v , 1o >. ] ~Q <Q u } = { u | [ <. z , 1o >. ] ~Q <Q u } )
64	61, 63	opeq12d 3841	. . . . . . . . 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 5982	. . . . . . . 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 3839	. . . . . . 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 6679	. . . . . 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 3839	. . . . 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 2216	. . . 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 2743	. . 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 8041	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 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   <.cop 3646   |^|cint 3899   class class class wbr 4059  (class class class)co 5967   1oc1o 6518   [cec 6641   N.cnpi 7420   +N cpli 7421   ~Q ceq 7427   1Qc1q 7429   <Q cltq 7433   1Pc1p 7440   +P. cpp 7441   ~R cer 7444   0Rc0r 7446   1Rc1r 7447   1c1 7961   + caddc 7963

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-2o 6526  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-rq 7500  df-ltnqqs 7501  df-enq0 7572  df-nq0 7573  df-0nq0 7574  df-plq0 7575  df-mq0 7576  df-inp 7614  df-i1p 7615  df-iplp 7616  df-enr 7874  df-nr 7875  df-plr 7876  df-0r 7879  df-1r 7880  df-c 7966  df-1 7968  df-r 7970  df-add 7971

This theorem is referenced by:  axcaucvglemres  8047