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Theorem opprlem 20280

Description: Lemma for opprbas 20281 and oppradd 20282. (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 17136	. 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 20276	. . 3 ⊢ 𝑂 = (𝑅 sSet ⟨(._r‘ndx), tpos (._r‘𝑅)⟩)
8	7	fveq2i 6835	. 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 4574 ‘cfv 6490 (class class class)co 7358 tpos ctpos 8166 sSet csts 17091 Slot cslot 17109 ndxcnx 17121 Basecbs 17137 ._rcmulr 17179 opp_rcoppr 20274

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 5231 ax-nul 5241 ax-pr 5368 ax-un 7680

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 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-res 5634 df-iota 6446 df-fun 6492 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-tpos 8167 df-sets 17092 df-slot 17110 df-oppr 20275

This theorem is referenced by: opprbas 20281 oppradd 20282 opprmndb 42955 opprgrpb 42956 opprablb 42957

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