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Theorem ialgrlemconst 10558
Description: Lemma for ialgr0 10559. Closure of a constant function, in a form suitable for theorems such as iseq1 9522 or iseqfcl 9524. (Contributed by Jim Kingdon, 22-Jul-2021.)
Hypotheses
Ref Expression
ialgrlemconst.z  |-  Z  =  ( ZZ>= `  M )
ialgrlemconst.a  |-  ( ph  ->  A  e.  S )
Assertion
Ref Expression
ialgrlemconst  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( Z  X.  { A }
) `  x )  e.  S )

Proof of Theorem ialgrlemconst
StepHypRef Expression
1 ialgrlemconst.a . . 3  |-  ( ph  ->  A  e.  S )
2 ialgrlemconst.z . . . . 5  |-  Z  =  ( ZZ>= `  M )
32eleq2i 2146 . . . 4  |-  ( x  e.  Z  <->  x  e.  ( ZZ>= `  M )
)
43biimpri 131 . . 3  |-  ( x  e.  ( ZZ>= `  M
)  ->  x  e.  Z )
5 fvconst2g 5401 . . 3  |-  ( ( A  e.  S  /\  x  e.  Z )  ->  ( ( Z  X.  { A } ) `  x )  =  A )
61, 4, 5syl2an 283 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( Z  X.  { A }
) `  x )  =  A )
71adantr 270 . 2  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  A  e.  S )
86, 7eqeltrd 2156 1  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( Z  X.  { A }
) `  x )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {csn 3400    X. cxp 4363   ` cfv 4926   ZZ>=cuz 8689
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-fv 4934
This theorem is referenced by:  ialgr0  10559  ialgrf  10560  ialgrp1  10561
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