Theorem islshp 39471

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 39470	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
6	5	eleq2d 2825	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ 𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)}))
7		neeq1 2996	. . . . 5 ⊢ (𝑠 = 𝑈 → (𝑠 ≠ 𝑉 ↔ 𝑈 ≠ 𝑉))
8		uneq1 4091	. . . . . . 7 ⊢ (𝑠 = 𝑈 → (𝑠 ∪ {𝑣}) = (𝑈 ∪ {𝑣}))
9	8	fveqeq2d 6835	. . . . . 6 ⊢ (𝑠 = 𝑈 → ((𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
10	9	rexbidv 3163	. . . . 5 ⊢ (𝑠 = 𝑈 → (∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉 ↔ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉))
11	7, 10	anbi12d 638	. . . 4 ⊢ (𝑠 = 𝑈 → ((𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉) ↔ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
12	11	elrab 3629	. . 3 ⊢ (𝑈 ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ↔ (𝑈 ∈ 𝑆 ∧ (𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))
13		3anass 1100	. . 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 279	. 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 288	1 ⊢ (𝑊 ∈ 𝑋 → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ 𝑆 ∧ 𝑈 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑈 ∪ {𝑣})) = 𝑉)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391   ∪ cun 3881  {csn 4555  ‘cfv 6485  Basecbs 17170  LSubSpclss 20921  LSpanclspn 20961  LSHypclsh 39467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-lshyp 39469

This theorem is referenced by:  islshpsm  39472  lshplss  39473  lshpne  39474  lshpnel2N  39477  lkrshp  39597  lshpset2N  39611  dochsatshp  41943