Theorem tgrpset 41626

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 41625	. . . 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 6884	. . 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 6882	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
6	5	opeq2d 4843	. . . . . 6 ⊢ (𝑤 = 𝑊 → ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩)
7		eqidd 2763	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔))
8	5, 5, 7	mpoeq123dv 7492	. . . . . . 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 4843	. . . . . 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 4705	. . . . 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 2762	. . . . 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 5407	. . . . 5 ⊢ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))⟩} ∈ V
13	10, 11, 12	fvmpt 6990	. . . 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 4840	. . . . 5 ⊢ ⟨(Base‘ndx), 𝑇⟩ = ⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑊)⟩
16		eqid 2762	. . . . . . 7 ⊢ (𝑓 ∘ 𝑔) = (𝑓 ∘ 𝑔)
17	14, 14, 16	mpoeq123i 7493	. . . . . 6 ⊢ (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊), 𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ↦ (𝑓 ∘ 𝑔))
18	17	opeq2i 4840	. . . . 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 4702	. . . 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	eqtr4di 2815	. . 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 2817	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((TGrp‘𝐾)‘𝑊) = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
22	1, 21	eqtrid 2809	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐺 = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cpr 4589  ⟨cop 4593   ↦ cmpt 5190   ∘ ccom 5663  ‘cfv 6537   ∈ cmpo 7419  ndxcnx 17291  Basecbs 17307  +_gcplusg 17348  LHypclh 40865  LTrncltrn 40982  TGrpctgrp 41623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-oprab 7421  df-mpo 7422  df-tgrp 41624

This theorem is used by:  tgrpbase  41627  tgrpopr  41628  dvaabl  41905