Theorem rmxfval 37468

Description: Value of the X sequence. Not used after rmxyval 37480 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)

Assertion

Ref	Expression
rmxfval	|- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm N ) = ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )

Distinct variable groups:   A, b   N, b

Proof of Theorem rmxfval

Dummy variables n a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . . . . . . . 11 |- ( a = A -> ( a ^ 2 ) = ( A ^ 2 ) )
2	1	oveq1d 6665	. . . . . . . . . 10 |- ( a = A -> ( ( a ^ 2 ) - 1 ) = ( ( A ^ 2 ) - 1 ) )
3	2	fveq2d 6195	. . . . . . . . 9 |- ( a = A -> ( sqr ` ( ( a ^ 2 ) - 1 ) ) = ( sqr ` ( ( A ^ 2 ) - 1 ) ) )
4	3	oveq1d 6665	. . . . . . . 8 |- ( a = A -> ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) = ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) )
5	4	oveq2d 6666	. . . . . . 7 |- ( a = A -> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) = ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) )
6	5	mpteq2dv 4745	. . . . . 6 |- ( a = A -> ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) )
7	6	cnveqd 5298	. . . . 5 |- ( a = A -> `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) )
8	7	adantr 481	. . . 4 |- ( ( a = A /\ n = N ) -> `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) = `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) )
9		id 22	. . . . . 6 |- ( a = A -> a = A )
10	9, 3	oveq12d 6668	. . . . 5 |- ( a = A -> ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) = ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) )
11		id 22	. . . . 5 |- ( n = N -> n = N )
12	10, 11	oveqan12d 6669	. . . 4 |- ( ( a = A /\ n = N ) -> ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) = ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) )
13	8, 12	fveq12d 6197	. . 3 |- ( ( a = A /\ n = N ) -> ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) = ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) )
14	13	fveq2d 6195	. 2 |- ( ( a = A /\ n = N ) -> ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) = ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )
15		df-rmx 37466	. 2 |- Xrm = ( a e. ( ZZ>= ` 2 ) , n e. ZZ |-> ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( a ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( a + ( sqr ` ( ( a ^ 2 ) - 1 ) ) ) ^ n ) ) ) )
16		fvex 6201	. 2 |- ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) e. _V
17	14, 15, 16	ovmpt2a 6791	1 |- ( ( A e. ( ZZ>= ` 2 ) /\ N e. ZZ ) -> ( A Xrm N ) = ( 1st ` ( `' ( b e. ( NN0 X. ZZ ) |-> ( ( 1st ` b ) + ( ( sqr ` ( ( A ^ 2 ) - 1 ) ) x. ( 2nd ` b ) ) ) ) ` ( ( A + ( sqr ` ( ( A ^ 2 ) - 1 ) ) ) ^ N ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ^cexp 12860   sqrcsqrt 13973   X_rm crmx 37464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rmx 37466

This theorem is referenced by:  rmxyval  37480