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Theorem opprlem 20339
Description: Lemma for opprbas 20341 and oppradd 20343. (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 17245 . 2 (𝐸𝑅) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑅)⟩))
4 eqid 2737 . . . 4 (Base‘𝑅) = (Base‘𝑅)
5 eqid 2737 . . . 4 (.r𝑅) = (.r𝑅)
6 opprbas.1 . . . 4 𝑂 = (oppr𝑅)
74, 5, 6opprval 20335 . . 3 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑅)⟩)
87fveq2i 6909 . 2 (𝐸𝑂) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r𝑅)⟩))
93, 8eqtr4i 2768 1 (𝐸𝑅) = (𝐸𝑂)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wne 2940  cop 4632  cfv 6561  (class class class)co 7431  tpos ctpos 8250   sSet csts 17200  Slot cslot 17218  ndxcnx 17230  Basecbs 17247  .rcmulr 17298  opprcoppr 20333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-tpos 8251  df-sets 17201  df-slot 17219  df-oppr 20334
This theorem is referenced by:  opprbas  20341  oppradd  20343  opprmndb  42521  opprgrpb  42522  opprablb  42523
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