Theorem lsatset 36141

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 3512	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6670	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lsatset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	syl6eqr 2874	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6		fveq2 6670	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
7		lsatset.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑊)
8	6, 7	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
9	8	sneqd 4579	. . . . . . 7 ⊢ (𝑤 = 𝑊 → {(0g‘𝑤)} = { 0 })
10	5, 9	difeq12d 4100	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) ∖ {(0g‘𝑤)}) = (𝑉 ∖ { 0 }))
11		fveq2 6670	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
12		lsatset.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
13	11, 12	syl6eqr 2874	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
14	13	fveq1d 6672	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LSpan‘𝑤)‘{𝑣}) = (𝑁‘{𝑣}))
15	10, 14	mpteq12dv 5151	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
16	15	rneqd 5808	. . . 4 ⊢ (𝑤 = 𝑊 → ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
17		df-lsatoms 36127	. . . 4 ⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
18	12	fvexi 6684	. . . . . . 7 ⊢ 𝑁 ∈ V
19	18	rnex 7617	. . . . . 6 ⊢ ran 𝑁 ∈ V
20		p0ex 5285	. . . . . 6 ⊢ {∅} ∈ V
21	19, 20	unex 7469	. . . . 5 ⊢ (ran 𝑁 ∪ {∅}) ∈ V
22		eqid 2821	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))
23		fvrn0 6698	. . . . . . . 8 ⊢ (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅})
24	23	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) → (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅}))
25	22, 24	fmpti 6876	. . . . . 6 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅})
26		frn 6520	. . . . . 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 5226	. . . 4 ⊢ ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ∈ V
29	16, 17, 28	fvmpt 6768	. . 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	syl5eq 2868	1 ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  {csn 4567   ↦ cmpt 5146  ran crn 5556  ⟶wf 6351  ‘cfv 6355  Basecbs 16483  0_gc0g 16713  LSpanclspn 19743  LSAtomsclsa 36125

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-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

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-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  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-fv 6363  df-lsatoms 36127

This theorem is referenced by:  islsat  36142  lsatlss  36147