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Theorem ialgrlemconst 11735
Description: Lemma for ialgr0 11736. Closure of a constant function, in a form suitable for theorems such as seq3-1 10245 or seqf 10246. (Contributed by Jim Kingdon, 22-Jul-2021.)
Hypotheses
Ref Expression
ialgrlemconst.z 𝑍 = (ℤ𝑀)
ialgrlemconst.a (𝜑𝐴𝑆)
Assertion
Ref Expression
ialgrlemconst ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)

Proof of Theorem ialgrlemconst
StepHypRef Expression
1 ialgrlemconst.a . . 3 (𝜑𝐴𝑆)
2 ialgrlemconst.z . . . . 5 𝑍 = (ℤ𝑀)
32eleq2i 2206 . . . 4 (𝑥𝑍𝑥 ∈ (ℤ𝑀))
43biimpri 132 . . 3 (𝑥 ∈ (ℤ𝑀) → 𝑥𝑍)
5 fvconst2g 5634 . . 3 ((𝐴𝑆𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
61, 4, 5syl2an 287 . 2 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
71adantr 274 . 2 ((𝜑𝑥 ∈ (ℤ𝑀)) → 𝐴𝑆)
86, 7eqeltrd 2216 1 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  {csn 3527   × cxp 4537  cfv 5123  cuz 9338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131
This theorem is referenced by:  ialgr0  11736  algrp1  11738
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