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Mirrors > Home > ILE Home > Th. List > opprsllem | GIF version |
Description: Lemma for opprbasg 13252 and oppraddg 13253. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.) |
Ref | Expression |
---|---|
opprbas.1 | β’ π = (opprβπ ) |
opprsllem.2 | β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) |
opprlem.3 | β’ (πΈβndx) β (.rβndx) |
Ref | Expression |
---|---|
opprsllem | β’ (π β π β (πΈβπ ) = (πΈβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulrslid 12592 | . . . . 5 β’ (.r = Slot (.rβndx) β§ (.rβndx) β β) | |
2 | 1 | slotex 12491 | . . . 4 β’ (π β π β (.rβπ ) β V) |
3 | tposexg 6261 | . . . 4 β’ ((.rβπ ) β V β tpos (.rβπ ) β V) | |
4 | 2, 3 | syl 14 | . . 3 β’ (π β π β tpos (.rβπ ) β V) |
5 | opprsllem.2 | . . . 4 β’ (πΈ = Slot (πΈβndx) β§ (πΈβndx) β β) | |
6 | opprlem.3 | . . . 4 β’ (πΈβndx) β (.rβndx) | |
7 | 1 | simpri 113 | . . . 4 β’ (.rβndx) β β |
8 | 5, 6, 7 | setsslnid 12516 | . . 3 β’ ((π β π β§ tpos (.rβπ ) β V) β (πΈβπ ) = (πΈβ(π sSet β¨(.rβndx), tpos (.rβπ )β©))) |
9 | 4, 8 | mpdan 421 | . 2 β’ (π β π β (πΈβπ ) = (πΈβ(π sSet β¨(.rβndx), tpos (.rβπ )β©))) |
10 | eqid 2177 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
11 | eqid 2177 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
12 | opprbas.1 | . . . 4 β’ π = (opprβπ ) | |
13 | 10, 11, 12 | opprvalg 13246 | . . 3 β’ (π β π β π = (π sSet β¨(.rβndx), tpos (.rβπ )β©)) |
14 | 13 | fveq2d 5521 | . 2 β’ (π β π β (πΈβπ) = (πΈβ(π sSet β¨(.rβndx), tpos (.rβπ )β©))) |
15 | 9, 14 | eqtr4d 2213 | 1 β’ (π β π β (πΈβπ ) = (πΈβπ)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 β wne 2347 Vcvv 2739 β¨cop 3597 βcfv 5218 (class class class)co 5877 tpos ctpos 6247 βcn 8921 ndxcnx 12461 sSet csts 12462 Slot cslot 12463 Basecbs 12464 .rcmulr 12539 opprcoppr 13244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-tpos 6248 df-inn 8922 df-2 8980 df-3 8981 df-ndx 12467 df-slot 12468 df-sets 12471 df-mulr 12552 df-oppr 13245 |
This theorem is referenced by: opprbasg 13252 oppraddg 13253 |
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