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| Description: Lemma for opprbas 20341 and oppradd 20343. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.) | 
| Ref | Expression | 
|---|---|
| opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) | 
| opprlem.2 | ⊢ 𝐸 = Slot (𝐸‘ndx) | 
| opprlem.3 | ⊢ (𝐸‘ndx) ≠ (.r‘ndx) | 
| Ref | Expression | 
|---|---|
| opprlem | ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | opprlem.2 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
| 2 | opprlem.3 | . . 3 ⊢ (𝐸‘ndx) ≠ (.r‘ndx) | |
| 3 | 1, 2 | setsnid 17245 | . 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 = (oppr‘𝑅) | |
| 7 | 4, 5, 6 | opprval 20335 | . . 3 ⊢ 𝑂 = (𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉) | 
| 8 | 7 | fveq2i 6909 | . 2 ⊢ (𝐸‘𝑂) = (𝐸‘(𝑅 sSet 〈(.r‘ndx), tpos (.r‘𝑅)〉)) | 
| 9 | 3, 8 | eqtr4i 2768 | 1 ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ≠ wne 2940 〈cop 4632 ‘cfv 6561 (class class class)co 7431 tpos ctpos 8250 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 Basecbs 17247 .rcmulr 17298 opprcoppr 20333 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-tpos 8251 df-sets 17201 df-slot 17219 df-oppr 20334 | 
| This theorem is referenced by: opprbas 20341 oppradd 20343 opprmndb 42521 opprgrpb 42522 opprablb 42523 | 
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