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Mirrors > Home > MPE Home > Th. List > opprlemOLD | Structured version Visualization version GIF version |
Description: Obsolete version of opprlem 20062 as of 6-Nov-2024. Lemma for opprbas 20064 and oppradd 20066. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
opprbas.1 | β’ π = (opprβπ ) |
opprlemOLD.2 | β’ πΈ = Slot π |
opprlemOLD.3 | β’ π β β |
opprlemOLD.4 | β’ π < 3 |
Ref | Expression |
---|---|
opprlemOLD | β’ (πΈβπ ) = (πΈβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprlemOLD.2 | . . . 4 β’ πΈ = Slot π | |
2 | opprlemOLD.3 | . . . 4 β’ π β β | |
3 | 1, 2 | ndxid 17077 | . . 3 β’ πΈ = Slot (πΈβndx) |
4 | 2 | nnrei 12170 | . . . . 5 β’ π β β |
5 | opprlemOLD.4 | . . . . 5 β’ π < 3 | |
6 | 4, 5 | ltneii 11276 | . . . 4 β’ π β 3 |
7 | 1, 2 | ndxarg 17076 | . . . . 5 β’ (πΈβndx) = π |
8 | mulrndx 17182 | . . . . 5 β’ (.rβndx) = 3 | |
9 | 7, 8 | neeq12i 3007 | . . . 4 β’ ((πΈβndx) β (.rβndx) β π β 3) |
10 | 6, 9 | mpbir 230 | . . 3 β’ (πΈβndx) β (.rβndx) |
11 | 3, 10 | setsnid 17089 | . 2 β’ (πΈβπ ) = (πΈβ(π sSet β¨(.rβndx), tpos (.rβπ )β©)) |
12 | eqid 2733 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
13 | eqid 2733 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
14 | opprbas.1 | . . . 4 β’ π = (opprβπ ) | |
15 | 12, 13, 14 | opprval 20058 | . . 3 β’ π = (π sSet β¨(.rβndx), tpos (.rβπ )β©) |
16 | 15 | fveq2i 6849 | . 2 β’ (πΈβπ) = (πΈβ(π sSet β¨(.rβndx), tpos (.rβπ )β©)) |
17 | 11, 16 | eqtr4i 2764 | 1 β’ (πΈβπ ) = (πΈβπ) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 β wne 2940 β¨cop 4596 class class class wbr 5109 βcfv 6500 (class class class)co 7361 tpos ctpos 8160 < clt 11197 βcn 12161 3c3 12217 sSet csts 17043 Slot cslot 17061 ndxcnx 17073 Basecbs 17091 .rcmulr 17142 opprcoppr 20056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-i2m1 11127 ax-1ne0 11128 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-ltxr 11202 df-nn 12162 df-2 12224 df-3 12225 df-sets 17044 df-slot 17062 df-ndx 17074 df-mulr 17155 df-oppr 20057 |
This theorem is referenced by: opprbasOLD 20065 oppraddOLD 20067 |
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