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Theorem iseqovex 10456
Description: Closure of a function used in proving sequence builder theorems. This can be thought of as a lemma for the small number of sequence builder theorems which need it. (Contributed by Jim Kingdon, 31-May-2020.)
Hypotheses
Ref Expression
iseqovex.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqovex.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
Assertion
Ref Expression
iseqovex  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  e.  S )
Distinct variable groups:    w, F, x, y, z    w,  .+ , x, y, z    w, S, x, y, z    ph, w, x, y, z    w, M, x, z
Allowed substitution hint:    M( y)

Proof of Theorem iseqovex
StepHypRef Expression
1 eqidd 2178 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) )
2 simprr 531 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  w  =  y )
3 simprl 529 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  z  =  x )
43oveq1d 5890 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  (
z  +  1 )  =  ( x  + 
1 ) )
54fveq2d 5520 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
62, 5oveq12d 5893 . . 3  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  =  x  /\  w  =  y ) )  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
7 simprl 529 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  x  e.  ( ZZ>= `  M )
)
8 simprr 531 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  y  e.  S )
9 iseqovex.pl . . . . . 6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
109caovclg 6027 . . . . 5  |-  ( (
ph  /\  ( z  e.  S  /\  w  e.  S ) )  -> 
( z  .+  w
)  e.  S )
1110adantlr 477 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( ZZ>= `  M )  /\  y  e.  S ) )  /\  ( z  e.  S  /\  w  e.  S
) )  ->  (
z  .+  w )  e.  S )
12 fveq2 5516 . . . . . 6  |-  ( z  =  ( x  + 
1 )  ->  ( F `  z )  =  ( F `  ( x  +  1
) ) )
1312eleq1d 2246 . . . . 5  |-  ( z  =  ( x  + 
1 )  ->  (
( F `  z
)  e.  S  <->  ( F `  ( x  +  1 ) )  e.  S
) )
14 iseqovex.f . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
1514ralrimiva 2550 . . . . . . 7  |-  ( ph  ->  A. x  e.  (
ZZ>= `  M ) ( F `  x )  e.  S )
16 fveq2 5516 . . . . . . . . 9  |-  ( x  =  z  ->  ( F `  x )  =  ( F `  z ) )
1716eleq1d 2246 . . . . . . . 8  |-  ( x  =  z  ->  (
( F `  x
)  e.  S  <->  ( F `  z )  e.  S
) )
1817cbvralv 2704 . . . . . . 7  |-  ( A. x  e.  ( ZZ>= `  M ) ( F `
 x )  e.  S  <->  A. z  e.  (
ZZ>= `  M ) ( F `  z )  e.  S )
1915, 18sylib 122 . . . . . 6  |-  ( ph  ->  A. z  e.  (
ZZ>= `  M ) ( F `  z )  e.  S )
2019adantr 276 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  A. z  e.  ( ZZ>= `  M )
( F `  z
)  e.  S )
21 peano2uz 9583 . . . . . 6  |-  ( x  e.  ( ZZ>= `  M
)  ->  ( x  +  1 )  e.  ( ZZ>= `  M )
)
227, 21syl 14 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x  +  1 )  e.  ( ZZ>= `  M
) )
2313, 20, 22rspcdva 2847 . . . 4  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  ( F `  ( x  +  1 ) )  e.  S )
2411, 8, 23caovcld 6028 . . 3  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  e.  S )
251, 6, 7, 8, 24ovmpod 6002 . 2  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
2625, 24eqeltrd 2254 1  |-  ( (
ph  /\  ( x  e.  ( ZZ>= `  M )  /\  y  e.  S
) )  ->  (
x ( z  e.  ( ZZ>= `  M ) ,  w  e.  S  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   A.wral 2455   ` cfv 5217  (class class class)co 5875    e. cmpo 5877   1c1 7812    + caddc 7814   ZZ>=cuz 9528
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529
This theorem is referenced by:  seq3val  10458  seq3-1  10460  seq3p1  10462
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