Theorem opprlem 20623

Description: Lemma for opprbas 20624 and oppradd 20625. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)

Hypotheses

Ref	Expression
opprbas.1	⊢ 𝑂 = (oppr‘𝑅)
opprlem.2	⊢ 𝐸 = Slot (𝐸‘ndx)
opprlem.3	⊢ (𝐸‘ndx) ≠ (.r‘ndx)

Assertion

Ref	Expression
opprlem	⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Proof of Theorem opprlem

Step	Hyp	Ref	Expression
1		opprlem.2	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
2		opprlem.3	. . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx)
3	1, 2	setsnid 17430	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
4		eqid 2763	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2763	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
7	4, 5, 6	opprval 20619	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩)
8	7	fveq2i 6919	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
9	3, 8	eqtr4i 2789	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2959  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451  tpos ctpos 8280   sSet csts 17385  Slot cslot 17403  ndxcnx 17415  Basecbs 17431  ._rcmulr 17473  opp_rcoppr 20617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-res 5670  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-tpos 8281  df-sets 17386  df-slot 17404  df-oppr 20618

This theorem is used by:  opprbas  20624  oppradd  20625  opprmndb  43844  opprgrpb  43845  opprablb  43846