Theorem lshpset 37836

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 3492	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6888	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4		lshpset.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊)
5	3, 4	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6		fveq2 6888	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7		lshpset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
8	6, 7	eqtr4di 2790	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
9	8	neeq2d 3001	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠 ≠ 𝑉))
10		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11		lshpset.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑊)
12	10, 11	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
13	12	fveq1d 6890	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
14	13, 8	eqeq12d 2748	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
15	8, 14	rexeqbidv 3343	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
16	9, 15	anbi12d 631	. . . . 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 3449	. . . 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 37835	. . . 4 ⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
19	4	fvexi 6902	. . . . 5 ⊢ 𝑆 ∈ V
20	19	rabex 5331	. . . 4 ⊢ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
21	17, 18, 20	fvmpt 6995	. . 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	eqtrid 2784	1 ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∃wrex 3070  {crab 3432  Vcvv 3474   ∪ cun 3945  {csn 4627  ‘cfv 6540  Basecbs 17140  LSubSpclss 20534  LSpanclspn 20574  LSHypclsh 37833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-lshyp 37835

This theorem is referenced by:  islshp  37837