Theorem opprlem 20260

Description: Lemma for opprbas 20262 and oppradd 20264. (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 17163	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
4		eqid 2727	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2727	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
7	4, 5, 6	opprval 20256	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩)
8	7	fveq2i 6894	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
9	3, 8	eqtr4i 2758	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Syntax hints:   = wceq 1534   ≠ wne 2935  ⟨cop 4630  ‘cfv 6542  (class class class)co 7414  tpos ctpos 8222   sSet csts 17117  Slot cslot 17135  ndxcnx 17147  Basecbs 17165  ._rcmulr 17219  opp_rcoppr 20254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-tpos 8223  df-sets 17118  df-slot 17136  df-oppr 20255

This theorem is referenced by:  opprbas  20262  oppradd  20264