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Theorem ialgrlemconst 12037
Description: Lemma for ialgr0 12038. Closure of a constant function, in a form suitable for theorems such as seq3-1 10457 or seqf 10458. (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 2244 . . . 4 (𝑥𝑍𝑥 ∈ (ℤ𝑀))
43biimpri 133 . . 3 (𝑥 ∈ (ℤ𝑀) → 𝑥𝑍)
5 fvconst2g 5730 . . 3 ((𝐴𝑆𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
61, 4, 5syl2an 289 . 2 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
71adantr 276 . 2 ((𝜑𝑥 ∈ (ℤ𝑀)) → 𝐴𝑆)
86, 7eqeltrd 2254 1 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  {csn 3592   × cxp 4624  cfv 5216  cuz 9526
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-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-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224
This theorem is referenced by:  ialgr0  12038  algrp1  12040  mulgnn0z  12963  mulgnndir  12965  mulgpropdg  12978
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