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Theorem opprlem 20365

Description: Lemma for opprbas 20367 and oppradd 20369. (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 17256	. 2 ⊢ (𝐸‘𝑅) = (𝐸‘(𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩))
4		eqid 2740	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
5		eqid 2740	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
6		opprbas.1	. . . 4 ⊢ 𝑂 = (opp_r‘𝑅)
7	4, 5, 6	opprval 20361	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩)
8	7	fveq2i 6923	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩))
9	3, 8	eqtr4i 2771	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ≠ wne 2946 ⟨cop 4654 ‘cfv 6573 (class class class)co 7448 tpos ctpos 8266 sSet csts 17210 Slot cslot 17228 ndxcnx 17240 Basecbs 17258 ._rcmulr 17312 opp_rcoppr 20359

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-res 5712 df-iota 6525 df-fun 6575 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-tpos 8267 df-sets 17211 df-slot 17229 df-oppr 20360

This theorem is referenced by: opprbas 20367 oppradd 20369 opprmndb 42466 opprgrpb 42467 opprablb 42468

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