Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > erngfmul | Structured version Visualization version GIF version |
Description: Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) |
Ref | Expression |
---|---|
erngset.h | β’ π» = (LHypβπΎ) |
erngset.t | β’ π = ((LTrnβπΎ)βπ) |
erngset.e | β’ πΈ = ((TEndoβπΎ)βπ) |
erngset.d | β’ π· = ((EDRingβπΎ)βπ) |
erng.m | β’ Β· = (.rβπ·) |
Ref | Expression |
---|---|
erngfmul | β’ ((πΎ β π β§ π β π») β Β· = (π β πΈ, π‘ β πΈ β¦ (π β π‘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erngset.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | erngset.t | . . . 4 β’ π = ((LTrnβπΎ)βπ) | |
3 | erngset.e | . . . 4 β’ πΈ = ((TEndoβπΎ)βπ) | |
4 | erngset.d | . . . 4 β’ π· = ((EDRingβπΎ)βπ) | |
5 | 1, 2, 3, 4 | erngset 39159 | . . 3 β’ ((πΎ β π β§ π β π») β π· = {β¨(Baseβndx), πΈβ©, β¨(+gβndx), (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ))))β©, β¨(.rβndx), (π β πΈ, π‘ β πΈ β¦ (π β π‘))β©}) |
6 | 5 | fveq2d 6842 | . 2 β’ ((πΎ β π β§ π β π») β (.rβπ·) = (.rβ{β¨(Baseβndx), πΈβ©, β¨(+gβndx), (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ))))β©, β¨(.rβndx), (π β πΈ, π‘ β πΈ β¦ (π β π‘))β©})) |
7 | erng.m | . 2 β’ Β· = (.rβπ·) | |
8 | 3 | fvexi 6852 | . . . 4 β’ πΈ β V |
9 | 8, 8 | mpoex 8001 | . . 3 β’ (π β πΈ, π‘ β πΈ β¦ (π β π‘)) β V |
10 | eqid 2738 | . . . 4 β’ {β¨(Baseβndx), πΈβ©, β¨(+gβndx), (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ))))β©, β¨(.rβndx), (π β πΈ, π‘ β πΈ β¦ (π β π‘))β©} = {β¨(Baseβndx), πΈβ©, β¨(+gβndx), (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ))))β©, β¨(.rβndx), (π β πΈ, π‘ β πΈ β¦ (π β π‘))β©} | |
11 | 10 | rngmulr 17117 | . . 3 β’ ((π β πΈ, π‘ β πΈ β¦ (π β π‘)) β V β (π β πΈ, π‘ β πΈ β¦ (π β π‘)) = (.rβ{β¨(Baseβndx), πΈβ©, β¨(+gβndx), (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ))))β©, β¨(.rβndx), (π β πΈ, π‘ β πΈ β¦ (π β π‘))β©})) |
12 | 9, 11 | ax-mp 5 | . 2 β’ (π β πΈ, π‘ β πΈ β¦ (π β π‘)) = (.rβ{β¨(Baseβndx), πΈβ©, β¨(+gβndx), (π β πΈ, π‘ β πΈ β¦ (π β π β¦ ((π βπ) β (π‘βπ))))β©, β¨(.rβndx), (π β πΈ, π‘ β πΈ β¦ (π β π‘))β©}) |
13 | 6, 7, 12 | 3eqtr4g 2803 | 1 β’ ((πΎ β π β§ π β π») β Β· = (π β πΈ, π‘ β πΈ β¦ (π β π‘))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3444 {ctp 4589 β¨cop 4591 β¦ cmpt 5187 β ccom 5635 βcfv 6492 β cmpo 7352 ndxcnx 17000 Basecbs 17018 +gcplusg 17068 .rcmulr 17069 LHypclh 38343 LTrncltrn 38460 TEndoctendo 39111 EDRingcedring 39112 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-nn 12088 df-2 12150 df-3 12151 df-n0 12348 df-z 12434 df-uz 12697 df-fz 13354 df-struct 16954 df-slot 16989 df-ndx 17001 df-base 17019 df-plusg 17081 df-mulr 17082 df-edring 39116 |
This theorem is referenced by: erngmul 39165 erngdvlem3 39349 erngdvlem4 39350 dvafmulr 39370 dvhfmulr 39444 |
Copyright terms: Public domain | W3C validator |