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Theorem zlmlem 21552
Description: Lemma for zlmbas 21554 and zlmplusg 21556. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 3-Nov-2024.)
Hypotheses
Ref Expression
zlmbas.w 𝑊 = (ℤMod‘𝐺)
zlmlem.2 𝐸 = Slot (𝐸‘ndx)
zlmlem.3 (𝐸‘ndx) ≠ (Scalar‘ndx)
zlmlem.4 (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
Assertion
Ref Expression
zlmlem (𝐸𝐺) = (𝐸𝑊)

Proof of Theorem zlmlem
StepHypRef Expression
1 zlmlem.2 . . . . 5 𝐸 = Slot (𝐸‘ndx)
2 zlmlem.3 . . . . 5 (𝐸‘ndx) ≠ (Scalar‘ndx)
31, 2setsnid 17258 . . . 4 (𝐸𝐺) = (𝐸‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4 zlmlem.4 . . . . 5 (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
51, 4setsnid 17258 . . . 4 (𝐸‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩))
63, 5eqtri 2768 . . 3 (𝐸𝐺) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩))
7 zlmbas.w . . . . 5 𝑊 = (ℤMod‘𝐺)
8 eqid 2740 . . . . 5 (.g𝐺) = (.g𝐺)
97, 8zlmval 21551 . . . 4 (𝐺 ∈ V → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩))
109fveq2d 6926 . . 3 (𝐺 ∈ V → (𝐸𝑊) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩)))
116, 10eqtr4id 2799 . 2 (𝐺 ∈ V → (𝐸𝐺) = (𝐸𝑊))
121str0 17238 . . . 4 ∅ = (𝐸‘∅)
1312eqcomi 2749 . . 3 (𝐸‘∅) = ∅
1413, 7fveqprc 17240 . 2 𝐺 ∈ V → (𝐸𝐺) = (𝐸𝑊))
1511, 14pm2.61i 182 1 (𝐸𝐺) = (𝐸𝑊)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2108  wne 2946  Vcvv 3488  c0 4352  cop 4654  cfv 6575  (class class class)co 7450   sSet csts 17212  Slot cslot 17230  ndxcnx 17242  Scalarcsca 17316   ·𝑠 cvsca 17317  .gcmg 19109  ringczring 21482  ℤModczlm 21536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6527  df-fun 6577  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-sets 17213  df-slot 17231  df-zlm 21540
This theorem is referenced by:  zlmbas  21554  zlmplusg  21556  zlmmulr  21558
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