Theorem iseqeq4 9217

Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)

Assertion

Ref	Expression
iseqeq4	|- ( S = T -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )

Proof of Theorem iseqeq4

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2040	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
2		mpt2eq12 5565	. . . . 5 |- ( ( ( ZZ>= ` M ) = ( ZZ>= ` M ) /\ S = T ) -> ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
3	1, 2	mpan 400	. . . 4 |- ( S = T -> ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
4		freceq1 5979	. . . 4 |- ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
5	3, 4	syl 14	. . 3 |- ( S = T -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
6	5	rneqd 4563	. 2 |- ( S = T -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
7		df-iseq 9212	. 2 |- seq M ( .+ , F , S ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
8		df-iseq 9212	. 2 |- seq M ( .+ , F , T ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. T |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
9	6, 7, 8	3eqtr4g 2097	1 |- ( S = T -> seq M ( .+ , F , S ) = seq M ( .+ , F , T ) )

Syntax hints:   -> wi 4   = wceq 1243   <.cop 3378   ran crn 4346   ` cfv 4902  (class class class)co 5512   |-> cmpt2 5514  freccfrec 5977   1c1 6890   + caddc 6892   ZZ>=cuz 8473   seqcseq 9211

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212

This theorem is referenced by: (None)