Theorem tgrpset 37883

Description: The translation group for a fiducial co-atom 𝑊. (Contributed by NM, 5-Jun-2013.)

Hypotheses

Ref	Expression
tgrpset.h	⊢ 𝐻 = (LHyp‘𝐾)
tgrpset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tgrpset.g	⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)

Assertion

Ref	Expression
tgrpset	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐺 = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})

Distinct variable groups:   𝑓,𝑔,𝐾   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔

Allowed substitution hints:   𝐺(𝑓,𝑔)   𝐻(𝑓,𝑔)   𝑉(𝑓,𝑔)

Proof of Theorem tgrpset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgrpset.g	. 2 ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)
2		tgrpset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3	2	tgrpfset 37882	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (TGrp‘𝐾) = (𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
4	3	fveq1d 6674	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((TGrp‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩})‘𝑊))
5		fveq2 6672	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
6	5	opeq2d 4812	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩)
7		eqidd 2824	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
8	5, 5, 7	mpoeq123dv 7231	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔)))
9	8	opeq2d 4812	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))⟩)
10	6, 9	preq12d 4679	. . . . 5 ⊢ (𝑤 = 𝑊 → {⟨(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))⟩})
11		eqid 2823	. . . . 5 ⊢ (𝑤 ∈ 𝐻 ↦ {⟨(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))⟩})
12		prex 5335	. . . . 5 ⊢ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))⟩} ∈ V
13	10, 11, 12	fvmpt 6770	. . . 4 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩})‘𝑊) = {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))⟩})
14		tgrpset.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
15	14	opeq2i 4809	. . . . 5 ⊢ ⟨(Base‘ndx), 𝑇⟩ = ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩
16		eqid 2823	. . . . . . 7 ⊢ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔)
17	14, 14, 16	mpoeq123i 7232	. . . . . 6 ⊢ (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))
18	17	opeq2i 4809	. . . . 5 ⊢ ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩ = ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))⟩
19	15, 18	preq12i 4676	. . . 4 ⊢ {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩} = {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))⟩}
20	13, 19	syl6eqr 2876	. . 3 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩})‘𝑊) = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
21	4, 20	sylan9eq 2878	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((TGrp‘𝐾)‘𝑊) = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
22	1, 21	syl5eq 2870	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐺 = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cpr 4571  ⟨cop 4575   ↦ cmpt 5148   ∘ ccom 5561  ‘cfv 6357   ∈ cmpo 7160  ndxcnx 16482  Basecbs 16485  +_gcplusg 16567  LHypclh 37122  LTrncltrn 37239  TGrpctgrp 37880

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 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-oprab 7162  df-mpo 7163  df-tgrp 37881

This theorem is referenced by:  tgrpbase  37884  tgrpopr  37885  dvaabl  38162