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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgrpfset | Structured version Visualization version GIF version |
Description: The translation group maps for a lattice 𝐾. (Contributed by NM, 5-Jun-2013.) |
Ref | Expression |
---|---|
tgrpset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
tgrpfset | ⊢ (𝐾 ∈ 𝑉 → (TGrp‘𝐾) = (𝑤 ∈ 𝐻 ↦ {〈(Base‘ndx), ((LTrn‘𝐾)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3512 | . 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) | |
2 | fveq2 6670 | . . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) | |
3 | tgrpset.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | 2, 3 | syl6eqr 2874 | . . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) |
5 | fveq2 6670 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾)) | |
6 | 5 | fveq1d 6672 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤)) |
7 | 6 | opeq2d 4810 | . . . . 5 ⊢ (𝑘 = 𝐾 → 〈(Base‘ndx), ((LTrn‘𝑘)‘𝑤)〉 = 〈(Base‘ndx), ((LTrn‘𝐾)‘𝑤)〉) |
8 | eqidd 2822 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔)) | |
9 | 6, 6, 8 | mpoeq123dv 7229 | . . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))) |
10 | 9 | opeq2d 4810 | . . . . 5 ⊢ (𝑘 = 𝐾 → 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉 = 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉) |
11 | 7, 10 | preq12d 4677 | . . . 4 ⊢ (𝑘 = 𝐾 → {〈(Base‘ndx), ((LTrn‘𝑘)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉} = {〈(Base‘ndx), ((LTrn‘𝐾)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉}) |
12 | 4, 11 | mpteq12dv 5151 | . . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {〈(Base‘ndx), ((LTrn‘𝑘)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉}) = (𝑤 ∈ 𝐻 ↦ {〈(Base‘ndx), ((LTrn‘𝐾)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉})) |
13 | df-tgrp 37894 | . . 3 ⊢ TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {〈(Base‘ndx), ((LTrn‘𝑘)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉})) | |
14 | 12, 13, 3 | mptfvmpt 6990 | . 2 ⊢ (𝐾 ∈ V → (TGrp‘𝐾) = (𝑤 ∈ 𝐻 ↦ {〈(Base‘ndx), ((LTrn‘𝐾)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉})) |
15 | 1, 14 | syl 17 | 1 ⊢ (𝐾 ∈ 𝑉 → (TGrp‘𝐾) = (𝑤 ∈ 𝐻 ↦ {〈(Base‘ndx), ((LTrn‘𝐾)‘𝑤)〉, 〈(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {cpr 4569 〈cop 4573 ↦ cmpt 5146 ∘ ccom 5559 ‘cfv 6355 ∈ cmpo 7158 ndxcnx 16480 Basecbs 16483 +gcplusg 16565 LHypclh 37135 LTrncltrn 37252 TGrpctgrp 37893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-oprab 7160 df-mpo 7161 df-tgrp 37894 |
This theorem is referenced by: tgrpset 37896 |
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