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Theorem iseqeq3 9216
Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
iseqeq3 |- ( F = G -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) )
Proof of Theorem iseqeq3
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 904 . . . . . . . 8 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> F = G )
21fveq1d 5180 . . . . . . 7 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> ( F ` ( x + 1 ) ) = ( G ` ( x + 1 ) ) )
32oveq2d 5528 . . . . . 6 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( G ` ( x + 1 ) ) ) )
43opeq2d 3556 . . . . 5 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. )
54mpt2eq3dva 5569 . . . 4 |- ( F = G -> ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) )
6 fveq1 5177 . . . . 5 |- ( F = G -> ( F ` M ) = ( G ` M ) )
76opeq2d 3556 . . . 4 |- ( F = G -> <. M , ( F ` M ) >. = <. M , ( G ` M ) >. )
8 freceq1 5979 . . . . 5 |- ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( 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. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) )
9 freceq2 5980 . . . . 5 |- ( <. M , ( F ` M ) >. = <. M , ( G ` M ) >. -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
108, 9sylan9eq 2092 . . . 4 |- ( ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) /\ <. M , ( F ` M ) >. = <. M , ( G ` M ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
115, 7, 10syl2anc 391 . . 3 |- ( F = G -> frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
1211rneqd 4563 . 2 |- ( F = G -> 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. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. ) )
13 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 ) >. )
14 df-iseq 9212 . 2 |- seq M ( .+ , G , S ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. S |-> <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. ) , <. M , ( G ` M ) >. )
1512, 13, 143eqtr4g 2097 1 |- ( F = G -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ w3a 885   = wceq 1243   e. wcel 1393   <.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-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212
This theorem is referenced by:  expival  9257  sumeq1  9874
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