Theorem lshpset 35559

Description: The set of all hyperplanes of a left module or left vector space. The vector 𝑣 is called a generating vector for the hyperplane. (Contributed by NM, 29-Jun-2014.)

Hypotheses

Ref	Expression
lshpset.v	⊢ 𝑉 = (Base‘𝑊)
lshpset.n	⊢ 𝑁 = (LSpan‘𝑊)
lshpset.s	⊢ 𝑆 = (LSubSp‘𝑊)
lshpset.h	⊢ 𝐻 = (LSHyp‘𝑊)

Assertion

Ref	Expression
lshpset	⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})

Distinct variable groups:   𝑆,𝑠   𝑣,𝑉   𝑣,𝑠,𝑊

Allowed substitution hints:   𝑆(𝑣)   𝐻(𝑣,𝑠)   𝑁(𝑣,𝑠)   𝑉(𝑠)   𝑋(𝑣,𝑠)

Proof of Theorem lshpset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpset.h	. 2 ⊢ 𝐻 = (LSHyp‘𝑊)
2		elex 3427	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6493	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4		lshpset.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊)
5	3, 4	syl6eqr 2826	. . . . 5 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6		fveq2 6493	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7		lshpset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
8	6, 7	syl6eqr 2826	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
9	8	neeq2d 3021	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠 ≠ 𝑉))
10		fveq2 6493	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11		lshpset.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑊)
12	10, 11	syl6eqr 2826	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
13	12	fveq1d 6495	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
14	13, 8	eqeq12d 2787	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
15	8, 14	rexeqbidv 3336	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
16	9, 15	anbi12d 621	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤)) ↔ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)))
17	5, 16	rabeqbidv 3402	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))} = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
18		df-lshyp 35558	. . . 4 ⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
19	4	fvexi 6507	. . . . 5 ⊢ 𝑆 ∈ V
20	19	rabex 5085	. . . 4 ⊢ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
21	17, 18, 20	fvmpt 6589	. . 3 ⊢ (𝑊 ∈ V → (LSHyp‘𝑊) = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
22	2, 21	syl 17	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSHyp‘𝑊) = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
23	1, 22	syl5eq 2820	1 ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ≠ wne 2961  ∃wrex 3083  {crab 3086  Vcvv 3409   ∪ cun 3821  {csn 4435  ‘cfv 6182  Basecbs 16333  LSubSpclss 19419  LSpanclspn 19459  LSHypclsh 35556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-iota 6146  df-fun 6184  df-fv 6190  df-lshyp 35558

This theorem is referenced by:  islshp  35560