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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngbase | Structured version Visualization version GIF version |
Description: The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom 𝑊). TODO: the .t hypothesis isn't used. (Also look at others.) (Contributed by NM, 9-Jun-2013.) |
Ref | Expression |
---|---|
erngset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
erngset.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
erngset.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
erngset.d | ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) |
erng.c | ⊢ 𝐶 = (Base‘𝐷) |
Ref | Expression |
---|---|
erngbase | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erngset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | erngset.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | erngset.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | erngset.d | . . . 4 ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | erngset 38810 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐷 = {〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))〉}) |
6 | 5 | fveq2d 6775 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (Base‘𝐷) = (Base‘{〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))〉})) |
7 | erng.c | . 2 ⊢ 𝐶 = (Base‘𝐷) | |
8 | 3 | fvexi 6785 | . . 3 ⊢ 𝐸 ∈ V |
9 | eqid 2740 | . . . 4 ⊢ {〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))〉} = {〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))〉} | |
10 | 9 | rngbase 17007 | . . 3 ⊢ (𝐸 ∈ V → 𝐸 = (Base‘{〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))〉})) |
11 | 8, 10 | ax-mp 5 | . 2 ⊢ 𝐸 = (Base‘{〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))〉}) |
12 | 6, 7, 11 | 3eqtr4g 2805 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 {ctp 4571 〈cop 4573 ↦ cmpt 5162 ∘ ccom 5594 ‘cfv 6432 ∈ cmpo 7273 ndxcnx 16892 Basecbs 16910 +gcplusg 16960 .rcmulr 16961 LHypclh 37994 LTrncltrn 38111 TEndoctendo 38762 EDRingcedring 38763 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-edring 38767 |
This theorem is referenced by: erng1lem 38997 erngdvlem1 38998 erngdvlem2N 38999 erngdvlem3 39000 erngdvlem4 39001 erng0g 39004 erng1r 39005 dvabase 39017 dvhbase 39093 |
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