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Theorem ialgrlemconst 12046
Description: Lemma for ialgr0 12047. Closure of a constant function, in a form suitable for theorems such as seq3-1 10463 or seqf 10464. (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 5733 . . 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 3594   × cxp 4626  cfv 5218  cuz 9531
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 4123  ax-pow 4176  ax-pr 4211
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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226
This theorem is referenced by:  ialgr0  12047  algrp1  12049  mulgnn0z  13020  mulgnndir  13022  mulgpropdg  13035
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