Theorem islshp 39556

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 39555	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
6	5	eleq2d 2847	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ 𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}))
7		neeq1 3018	. . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ≠ 𝑉 ↔ 𝑈 ≠ 𝑉))
8		uneq1 4114	. . . . . . 7 ⊢ (𝑠 = 𝑈 → (𝑠 ∪ {𝑣}) = (𝑈 ∪ {𝑣}))
9	8	fveqeq2d 6869	. . . . . 6 ⊢ (𝑠 = 𝑈 → ((𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
10	9	rexbidv 3185	. . . . 5 ⊢ (𝑠 = 𝑈 → (∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
11	7, 10	anbi12d 641	. . . 4 ⊢ (𝑠 = 𝑈 → ((𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉) ↔ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
12	11	elrab 3650	. . 3 ⊢ (𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈 ∈ 𝑆 ∧ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
13		3anass 1105	. . 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 280	. 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 289	1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  {crab 3413   ∪ cun 3902  {csn 4581  ‘cfv 6515  Basecbs 17226  LSubSpclss 20976  LSpanclspn 21016  LSHypclsh 39552

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

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-pw 4556  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-iota 6471  df-fun 6517  df-fv 6523  df-lshyp 39554

This theorem is referenced by:  islshpsm  39557  lshplss  39558  lshpne  39559  lshpnel2N  39562  lkrshp  39682  lshpset2N  39696  dochsatshp  42028