Theorem lsatset 38989

Description: The set of all 1-dim subspaces (atoms) of a left module or left vector space. (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
lsatset.v	⊢ 𝑉 = (Base‘𝑊)
lsatset.n	⊢ 𝑁 = (LSpan‘𝑊)
lsatset.z	⊢ 0 = (0g‘𝑊)
lsatset.a	⊢ 𝐴 = (LSAtoms‘𝑊)

Assertion

Ref	Expression
lsatset	⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))

Distinct variable groups:   𝑣,𝑁   𝑣,𝑉   𝑣,𝑊   𝑣, 0   𝑣,𝑋

Allowed substitution hint:   𝐴(𝑣)

Proof of Theorem lsatset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsatset.a	. 2 ⊢ 𝐴 = (LSAtoms‘𝑊)
2		elex 3457	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6822	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lsatset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6		fveq2 6822	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
7		lsatset.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑊)
8	6, 7	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
9	8	sneqd 4589	. . . . . . 7 ⊢ (𝑤 = 𝑊 → {(0g‘𝑤)} = { 0 })
10	5, 9	difeq12d 4078	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) ∖ {(0g‘𝑤)}) = (𝑉 ∖ { 0 }))
11		fveq2 6822	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
12		lsatset.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
13	11, 12	eqtr4di 2782	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
14	13	fveq1d 6824	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LSpan‘𝑤)‘{𝑣}) = (𝑁‘{𝑣}))
15	10, 14	mpteq12dv 5179	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
16	15	rneqd 5880	. . . 4 ⊢ (𝑤 = 𝑊 → ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
17		df-lsatoms 38975	. . . 4 ⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
18	12	fvexi 6836	. . . . . . 7 ⊢ 𝑁 ∈ V
19	18	rnex 7843	. . . . . 6 ⊢ ran 𝑁 ∈ V
20		p0ex 5323	. . . . . 6 ⊢ {∅} ∈ V
21	19, 20	unex 7680	. . . . 5 ⊢ (ran 𝑁 ∪ {∅}) ∈ V
22		eqid 2729	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))
23		fvrn0 6850	. . . . . . . 8 ⊢ (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅})
24	23	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) → (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅}))
25	22, 24	fmpti 7046	. . . . . 6 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅})
26		frn 6659	. . . . . 6 ⊢ ((𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅}) → ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ⊆ (ran 𝑁 ∪ {∅}))
27	25, 26	ax-mp 5	. . . . 5 ⊢ ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ⊆ (ran 𝑁 ∪ {∅})
28	21, 27	ssexi 5261	. . . 4 ⊢ ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ∈ V
29	16, 17, 28	fvmpt 6930	. . 3 ⊢ (𝑊 ∈ V → (LSAtoms‘𝑊) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
30	2, 29	syl 17	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSAtoms‘𝑊) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
31	1, 30	eqtrid 2776	1 ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3436   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4284  {csn 4577   ↦ cmpt 5173  ran crn 5620  ⟶wf 6478  ‘cfv 6482  Basecbs 17120  0_gc0g 17343  LSpanclspn 20874  LSAtomsclsa 38973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-lsatoms 38975

This theorem is referenced by:  islsat  38990  lsatlss  38995