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Theorem zlmlem 20940
Description: Lemma for zlmbas 20942 and zlmplusg 20944. (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 17089 . . . 4 (𝐸𝐺) = (𝐸‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩))
4 zlmlem.4 . . . . 5 (𝐸‘ndx) ≠ ( ·𝑠 ‘ndx)
51, 4setsnid 17089 . . . 4 (𝐸‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩))
63, 5eqtri 2761 . . 3 (𝐸𝐺) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩))
7 zlmbas.w . . . . 5 𝑊 = (ℤMod‘𝐺)
8 eqid 2733 . . . . 5 (.g𝐺) = (.g𝐺)
97, 8zlmval 20939 . . . 4 (𝐺 ∈ V → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩))
109fveq2d 6850 . . 3 (𝐺 ∈ V → (𝐸𝑊) = (𝐸‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g𝐺)⟩)))
116, 10eqtr4id 2792 . 2 (𝐺 ∈ V → (𝐸𝐺) = (𝐸𝑊))
121str0 17069 . . . 4 ∅ = (𝐸‘∅)
1312eqcomi 2742 . . 3 (𝐸‘∅) = ∅
1413, 7fveqprc 17071 . 2 𝐺 ∈ V → (𝐸𝐺) = (𝐸𝑊))
1511, 14pm2.61i 182 1 (𝐸𝐺) = (𝐸𝑊)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  wne 2940  Vcvv 3447  c0 4286  cop 4596  cfv 6500  (class class class)co 7361   sSet csts 17043  Slot cslot 17061  ndxcnx 17073  Scalarcsca 17144   ·𝑠 cvsca 17145  .gcmg 18880  ringczring 20892  ℤModczlm 20924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-res 5649  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-sets 17044  df-slot 17062  df-zlm 20928
This theorem is referenced by:  zlmbas  20942  zlmplusg  20944  zlmmulr  20946
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