Theorem islshp 39642

Description: The predicate "is a hyperplane" (of a left module or left vector space). (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
islshp	⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))

Distinct variable groups:   𝑣,𝑉   𝑣,𝑊   𝑣,𝑈

Allowed substitution hints:   𝑆(𝑣)   𝐻(𝑣)   𝑁(𝑣)   𝑋(𝑣)

Proof of Theorem islshp

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpset.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		lshpset.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑊)
3		lshpset.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
4		lshpset.h	. . . 4 ⊢ 𝐻 = (LSHyp‘𝑊)
5	1, 2, 3, 4	lshpset 39641	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
6	5	eleq2d 2855	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ 𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}))
7		neeq1 3026	. . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ≠ 𝑉 ↔ 𝑈 ≠ 𝑉))
8		uneq1 4123	. . . . . . 7 ⊢ (𝑠 = 𝑈 → (𝑠 ∪ {𝑣}) = (𝑈 ∪ {𝑣}))
9	8	fveqeq2d 6890	. . . . . 6 ⊢ (𝑠 = 𝑈 → ((𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
10	9	rexbidv 3195	. . . . 5 ⊢ (𝑠 = 𝑈 → (∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
11	7, 10	anbi12d 643	. . . 4 ⊢ (𝑠 = 𝑈 → ((𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉) ↔ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
12	11	elrab 3659	. . 3 ⊢ (𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈 ∈ 𝑆 ∧ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
13		3anass 1109	. . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉) ↔ (𝑈 ∈ 𝑆 ∧ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
14	12, 13	bitr4i 281	. 2 ⊢ (𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
15	6, 14	bitrdi 290	1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  {crab 3423   ∪ cun 3911  {csn 4594  ‘cfv 6537  Basecbs 17268  LSubSpclss 21029  LSpanclspn 21069  LSHypclsh 39638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-lshyp 39640

This theorem is referenced by:  islshpsm  39643  lshplss  39644  lshpne  39645  lshpnel2N  39648  lkrshp  39768  lshpset2N  39782  dochsatshp  42114