Theorem opprlem 19782

Description: Lemma for opprbas 19784 and oppradd 19786. (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 16838	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
4		eqid 2738	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2738	. . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (oppr‘𝑅)
7	4, 5, 6	opprval 19778	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩)
8	7	fveq2i 6759	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(.r‘ndx), tpos (.r‘𝑅)⟩))
9	3, 8	eqtr4i 2769	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Syntax hints:   = wceq 1539   ≠ wne 2942  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  tpos ctpos 8012   sSet csts 16792  Slot cslot 16810  ndxcnx 16822  Basecbs 16840  ._rcmulr 16889  opp_rcoppr 19776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-tpos 8013  df-sets 16793  df-slot 16811  df-oppr 19777

This theorem is referenced by:  opprbas  19784  oppradd  19786