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Theorem iseqeq3 9759
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 941 . . . . . . . 8 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> F = G )
21fveq1d 5258 . . . . . . 7 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> ( F ` ( x + 1 ) ) = ( G ` ( x + 1 ) ) )
32oveq2d 5610 . . . . . 6 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> ( y .+ ( F ` ( x + 1 ) ) ) = ( y .+ ( G ` ( x + 1 ) ) ) )
43opeq2d 3606 . . . . 5 |- ( ( F = G /\ x e. ( ZZ>= ` M ) /\ y e. S ) -> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. = <. ( x + 1 ) , ( y .+ ( G ` ( x + 1 ) ) ) >. )
54mpt2eq3dva 5651 . . . 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 5255 . . . . 5 |- ( F = G -> ( F ` M ) = ( G ` M ) )
76opeq2d 3606 . . . 4 |- ( F = G -> <. M , ( F ` M ) >. = <. M , ( G ` M ) >. )
8 freceq1 6092 . . . . 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 6093 . . . . 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 2137 . . . 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 403 . . 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 4625 . 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 9755 . 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 9755 . 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 2142 1 |- ( F = G -> seq M ( .+ , F , S ) = seq M ( .+ , G , S ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ w3a 922   = wceq 1287   e. wcel 1436   <.cop 3428   ran crn 4405   ` cfv 4972  (class class class)co 5594   |-> cmpt2 5596  freccfrec 6090   1c1 7272   + caddc 7274   ZZ>=cuz 8928   seqcseq 9754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-cnv 4412  df-dm 4414  df-rn 4415  df-res 4416  df-iota 4937  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-recs 6005  df-frec 6091  df-iseq 9755
This theorem is referenced by:  expival  9808  sumeq1  10580  sumeq2d  10584  sumeq2  10585
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