Theorem opprlem 20230

Description: Lemma for opprbas 20232 and oppradd 20234. (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 17146	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
4		eqid 2730	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2730	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
7	4, 5, 6	opprval 20226	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩)
8	7	fveq2i 6893	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
9	3, 8	eqtr4i 2761	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Syntax hints:   = wceq 1539   ≠ wne 2938  ⟨cop 4633  ‘cfv 6542  (class class class)co 7411  tpos ctpos 8212   sSet csts 17100  Slot cslot 17118  ndxcnx 17130  Basecbs 17148  ._rcmulr 17202  opp_rcoppr 20224

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-tpos 8213  df-sets 17101  df-slot 17119  df-oppr 20225

This theorem is referenced by:  opprbas  20232  oppradd  20234