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Theorem opprlem 20356
Description: Lemma for opprbas 20358 and oppradd 20360. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
Hypotheses
Ref Expression
opprbas.1 𝑂 = (oppr𝑅)
opprlem.2 𝐸 = Slot (𝐸‘ndx)
opprlem.3 (𝐸‘ndx) ≠ (.r‘ndx)
Assertion
Ref Expression
opprlem (𝐸𝑅) = (𝐸𝑂)

Proof of Theorem opprlem
StepHypRef Expression
1 opprlem.2 . . 3 𝐸 = Slot (𝐸‘ndx)
2 opprlem.3 . . 3 (𝐸‘ndx) ≠ (.r‘ndx)
31, 2setsnid 17243 . 2 (𝐸𝑅) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑅)⟩))
4 eqid 2735 . . . 4 (Base‘𝑅) = (Base‘𝑅)
5 eqid 2735 . . . 4 (.r𝑅) = (.r𝑅)
6 opprbas.1 . . . 4 𝑂 = (oppr𝑅)
74, 5, 6opprval 20352 . . 3 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑅)⟩)
87fveq2i 6910 . 2 (𝐸𝑂) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑅)⟩))
93, 8eqtr4i 2766 1 (𝐸𝑅) = (𝐸𝑂)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wne 2938  cop 4637  cfv 6563  (class class class)co 7431  tpos ctpos 8249   sSet csts 17197  Slot cslot 17215  ndxcnx 17227  Basecbs 17245  .rcmulr 17299  opprcoppr 20350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-tpos 8250  df-sets 17198  df-slot 17216  df-oppr 20351
This theorem is referenced by:  opprbas  20358  oppradd  20360  opprmndb  42498  opprgrpb  42499  opprablb  42500
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