Theorem lshpset 39614

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 3478	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6871	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
4		lshpset.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑊)
5	3, 4	eqtr4di 2818	. . . . 5 ⊢ (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝑆)
6		fveq2 6871	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
7		lshpset.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
8	6, 7	eqtr4di 2818	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
9	8	neeq2d 3020	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑠 ≠ (Base‘𝑤) ↔ 𝑠 ≠ 𝑉))
10		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = (LSpan‘𝑊))
11		lshpset.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑊)
12	10, 11	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (LSpan‘𝑤) = 𝑁)
13	12	fveq1d 6873	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (𝑁‘(𝑠 ∪ {𝑣})))
14	13, 8	eqeq12d 2781	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
15	8, 14	rexeqbidv 3340	. . . . . 6 ⊢ (𝑤 = 𝑊 → (∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤) ↔ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉))
16	9, 15	anbi12d 643	. . . . 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 3435	. . . 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 39613	. . . 4 ⊢ LSHyp = (𝑤 ∈ V ↦ {𝑠 ∈ (LSubSp‘𝑤) ∣ (𝑠 ≠ (Base‘𝑤) ∧ ∃𝑣 ∈ (Base‘𝑤)((LSpan‘𝑤)‘(𝑠 ∪ {𝑣})) = (Base‘𝑤))})
19	4	fvexi 6885	. . . . 5 ⊢ 𝑆 ∈ V
20	19	rabex 5300	. . . 4 ⊢ {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)} ∈ V
21	17, 18, 20	fvmpt 6979	. . 3 ⊢ (𝑊 ∈ V → (LSHyp‘𝑊) = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
22	2, 21	syl 18	. 2 ⊢ (𝑊 ∈ 𝑋 → (LSHyp‘𝑊) = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})
23	1, 22	eqtrid 2812	1 ⊢ (𝑊 ∈ 𝑋 → 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 ≠ 𝑉 ∧ ∃𝑣 ∈ 𝑉 (𝑁‘(𝑠 ∪ {𝑣})) = 𝑉)})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  {crab 3417  Vcvv 3457   ∪ cun 3905  {csn 4585  ‘cfv 6525  Basecbs 17259  LSubSpclss 21021  LSpanclspn 21061  LSHypclsh 39611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-lshyp 39613

This theorem is referenced by:  islshp  39615