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Theorem opprlem 20258

Description: Lemma for opprbas 20259 and oppradd 20260. (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 17185	. 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 = (opp_r‘𝑅)
7	4, 5, 6	opprval 20254	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩)
8	7	fveq2i 6864	. 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩))
9	3, 8	eqtr4i 2756	1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ≠ wne 2926 ⟨cop 4598 ‘cfv 6514 (class class class)co 7390 tpos ctpos 8207 sSet csts 17140 Slot cslot 17158 ndxcnx 17170 Basecbs 17186 ._rcmulr 17228 opp_rcoppr 20252

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 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-res 5653 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-tpos 8208 df-sets 17141 df-slot 17159 df-oppr 20253

This theorem is referenced by: opprbas 20259 oppradd 20260 opprmndb 42506 opprgrpb 42507 opprablb 42508

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