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Theorem opprlem 20062
Description: Lemma for opprbas 20064 and oppradd 20066. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
Hypotheses
Ref Expression
opprbas.1 𝑂 = (opprβ€˜π‘…)
opprlem.2 𝐸 = Slot (πΈβ€˜ndx)
opprlem.3 (πΈβ€˜ndx) β‰  (.rβ€˜ndx)
Assertion
Ref Expression
opprlem (πΈβ€˜π‘…) = (πΈβ€˜π‘‚)

Proof of Theorem opprlem
StepHypRef Expression
1 opprlem.2 . . 3 𝐸 = Slot (πΈβ€˜ndx)
2 opprlem.3 . . 3 (πΈβ€˜ndx) β‰  (.rβ€˜ndx)
31, 2setsnid 17089 . 2 (πΈβ€˜π‘…) = (πΈβ€˜(𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘…)⟩))
4 eqid 2733 . . . 4 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
5 eqid 2733 . . . 4 (.rβ€˜π‘…) = (.rβ€˜π‘…)
6 opprbas.1 . . . 4 𝑂 = (opprβ€˜π‘…)
74, 5, 6opprval 20058 . . 3 𝑂 = (𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘…)⟩)
87fveq2i 6849 . 2 (πΈβ€˜π‘‚) = (πΈβ€˜(𝑅 sSet ⟨(.rβ€˜ndx), tpos (.rβ€˜π‘…)⟩))
93, 8eqtr4i 2764 1 (πΈβ€˜π‘…) = (πΈβ€˜π‘‚)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   β‰  wne 2940  βŸ¨cop 4596  β€˜cfv 6500  (class class class)co 7361  tpos ctpos 8160   sSet csts 17043  Slot cslot 17061  ndxcnx 17073  Basecbs 17091  .rcmulr 17142  opprcoppr 20056
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-tpos 8161  df-sets 17044  df-slot 17062  df-oppr 20057
This theorem is referenced by:  opprbas  20064  oppradd  20066
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