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Theorem opprlem 20262

Description: Lemma for opprbas 20263 and oppradd 20264. (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 17154	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩))
4		eqid 2729	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2729	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (opp_r‘𝑅)
7	4, 5, 6	opprval 20258	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩)
8	7	fveq2i 6843	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩))
9	3, 8	eqtr4i 2755	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ≠ wne 2925 ⟨cop 4591 ‘cfv 6499 (class class class)co 7369 tpos ctpos 8181 sSet csts 17109 Slot cslot 17127 ndxcnx 17139 Basecbs 17155 ._rcmulr 17197 opp_rcoppr 20256

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-res 5643 df-iota 6452 df-fun 6501 df-fv 6507 df-ov 7372 df-oprab 7373 df-mpo 7374 df-tpos 8182 df-sets 17110 df-slot 17128 df-oppr 20257

This theorem is referenced by: opprbas 20263 oppradd 20264 opprmndb 42492 opprgrpb 42493 opprablb 42494

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