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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tgrpbase | Structured version Visualization version GIF version |
Description: The base set of the translation group is the set of all translations (for a fiducial co-atom 𝑊). (Contributed by NM, 5-Jun-2013.) |
Ref | Expression |
---|---|
tgrpset.h | ⊢ 𝐻 = (LHyp‘𝐾) |
tgrpset.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
tgrpset.g | ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊) |
tgrp.c | ⊢ 𝐶 = (Base‘𝐺) |
Ref | Expression |
---|---|
tgrpbase | ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgrpset.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | tgrpset.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | tgrpset.g | . . . 4 ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | tgrpset 40727 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐺 = {〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉}) |
5 | 4 | fveq2d 6910 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (Base‘𝐺) = (Base‘{〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉})) |
6 | tgrp.c | . 2 ⊢ 𝐶 = (Base‘𝐺) | |
7 | 2 | fvexi 6920 | . . 3 ⊢ 𝑇 ∈ V |
8 | eqid 2734 | . . . 4 ⊢ {〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉} = {〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉} | |
9 | 8 | grpbase 17331 | . . 3 ⊢ (𝑇 ∈ V → 𝑇 = (Base‘{〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉})) |
10 | 7, 9 | ax-mp 5 | . 2 ⊢ 𝑇 = (Base‘{〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉}) |
11 | 5, 6, 10 | 3eqtr4g 2799 | 1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 {cpr 4632 〈cop 4636 ∘ ccom 5692 ‘cfv 6562 ∈ cmpo 7432 ndxcnx 17226 Basecbs 17244 +gcplusg 17297 LHypclh 39966 LTrncltrn 40083 TGrpctgrp 40724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-tgrp 40725 |
This theorem is referenced by: tgrpgrplem 40731 tgrpabl 40733 |
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