Theorem lsatset 39578

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 3474	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6863	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		lsatset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2814	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6		fveq2 6863	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = (0g‘𝑊))
7		lsatset.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑊)
8	6, 7	eqtr4di 2814	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (0g‘𝑤) = 0 )
9	8	sneqd 4593	. . . . . . 7 ⊢ (𝑤 = 𝑊 → {(0g‘𝑤)} = { 0 })
10	5, 9	difeq12d 4081	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((Base‘𝑤) ∖ {(0g‘𝑤)}) = (𝑉 ∖ { 0 }))
11		fveq2 6863	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
12		lsatset.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
13	11, 12	eqtr4di 2814	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
14	13	fveq1d 6865	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((LSpan‘𝑤)‘{𝑣}) = (𝑁‘{𝑣}))
15	10, 14	mpteq12dv 5186	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
16	15	rneqd 5912	. . . 4 ⊢ (𝑤 = 𝑊 → ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})) = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))
17		df-lsatoms 39564	. . . 4 ⊢ LSAtoms = (𝑤 ∈ V ↦ ran (𝑣 ∈ ((Base‘𝑤) ∖ {(0g‘𝑤)}) ↦ ((LSpan‘𝑤)‘{𝑣})))
18	12	fvexi 6877	. . . . . . 7 ⊢ 𝑁 ∈ V
19	18	rnex 7887	. . . . . 6 ⊢ ran 𝑁 ∈ V
20		snex 5395	. . . . . 6 ⊢ {∅} ∈ V
21	19, 20	unex 7723	. . . . 5 ⊢ (ran 𝑁 ∪ {∅}) ∈ V
22		eqid 2761	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) = (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣}))
23		fvrn0 6891	. . . . . . . 8 ⊢ (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅})
24	23	a1i 11	. . . . . . 7 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) → (𝑁‘{𝑣}) ∈ (ran 𝑁 ∪ {∅}))
25	22, 24	fmpti 7089	. . . . . 6 ⊢ (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})):(𝑉 ∖ { 0 })⟶(ran 𝑁 ∪ {∅})
26		frn 6695	. . . . . 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 5277	. . . 4 ⊢ ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})) ∈ V
29	16, 17, 28	fvmpt 6971	. . 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 2808	1 ⊢ (𝑊 ∈ 𝑋 → 𝐴 = ran (𝑣 ∈ (𝑉 ∖ { 0 }) ↦ (𝑁‘{𝑣})))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  ∅c0 4285  {csn 4581   ↦ cmpt 5180  ran crn 5646  ⟶wf 6513  ‘cfv 6517  Basecbs 17228  0_gc0g 17451  LSpanclspn 21018  LSAtomsclsa 39562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-lsatoms 39564

This theorem is referenced by:  islsat  39579  lsatlss  39584