Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > erngbase-rN | 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). (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
erngset.h-r | ⊢ 𝐻 = (LHyp‘𝐾) |
erngset.t-r | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
erngset.e-r | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
erngset.d-r | ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊) |
erng.c-r | ⊢ 𝐶 = (Base‘𝐷) |
Ref | Expression |
---|---|
erngbase-rN | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erngset.h-r | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | erngset.t-r | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | erngset.e-r | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | erngset.d-r | . . . 4 ⊢ 𝐷 = ((EDRingR‘𝐾)‘𝑊) | |
5 | 1, 2, 3, 4 | erngset-rN 38516 | . . 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 ↦ (𝑡 ∘ 𝑠))〉}) |
6 | 5 | fveq2d 6710 | . 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 ↦ (𝑡 ∘ 𝑠))〉})) |
7 | erng.c-r | . 2 ⊢ 𝐶 = (Base‘𝐷) | |
8 | 3 | fvexi 6720 | . . 3 ⊢ 𝐸 ∈ V |
9 | eqid 2734 | . . . 4 ⊢ {〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))〉} = {〈(Base‘ndx), 𝐸〉, 〈(+g‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx), (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑡 ∘ 𝑠))〉} | |
10 | 9 | rngbase 16822 | . . 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 ↦ (𝑡 ∘ 𝑠))〉})) |
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 ↦ (𝑡 ∘ 𝑠))〉}) |
12 | 6, 7, 11 | 3eqtr4g 2799 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3401 {ctp 4535 〈cop 4537 ↦ cmpt 5124 ∘ ccom 5544 ‘cfv 6369 ∈ cmpo 7204 ndxcnx 16681 Basecbs 16684 +gcplusg 16767 .rcmulr 16768 LHypclh 37692 LTrncltrn 37809 TEndoctendo 38460 EDRingRcedring-rN 38462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-plusg 16780 df-mulr 16781 df-edring-rN 38464 |
This theorem is referenced by: erngdvlem1-rN 38704 erngdvlem2-rN 38705 erngdvlem3-rN 38706 erngdvlem4-rN 38707 |
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