Theorem opprlem 20579

Description: Lemma for opprbas 20580 and oppradd 20581. (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 17393	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
4		eqid 2760	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2760	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
7	4, 5, 6	opprval 20575	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩)
8	7	fveq2i 6891	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
9	3, 8	eqtr4i 2786	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

This proof depends on syntax axioms:   = wceq 1569   ≠ wne 2955  ⟨cop 4834  ‘cfv 6541  (class class class)co 7422  tpos ctpos 8247   sSet csts 17348  Slot cslot 17366  ndxcnx 17378  Basecbs 17394  ._rcmulr 17436  opp_rcoppr 20573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6496  df-fun 6543  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-tpos 8248  df-sets 17349  df-slot 17367  df-oppr 20574

This theorem is used by:  opprbas  20580  oppradd  20581  opprmndb  43685  opprgrpb  43686  opprablb  43687