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Theorem opprlem 20283

Description: Lemma for opprbas 20284 and oppradd 20285. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)

**Hypotheses**
Ref	Expression
opprbas.1	⊢ 𝑂 = (opp_r‘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 17140	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩))
4		eqid 2737	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2737	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (opp_r‘𝑅)
7	4, 5, 6	opprval 20279	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩)
8	7	fveq2i 6838	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩))
9	3, 8	eqtr4i 2763	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ≠ wne 2933 ⟨cop 4587 ‘cfv 6493 (class class class)co 7361 tpos ctpos 8170 sSet csts 17095 Slot cslot 17113 ndxcnx 17125 Basecbs 17141 ._rcmulr 17183 opp_rcoppr 20277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pr 5378 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3401 df-v 3443 df-sbc 3742 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-res 5637 df-iota 6449 df-fun 6495 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-tpos 8171 df-sets 17096 df-slot 17114 df-oppr 20278

This theorem is referenced by: opprbas 20284 oppradd 20285 opprmndb 42844 opprgrpb 42845 opprablb 42846

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