Theorem islpln 40118

Description: The predicate "is a lattice plane". (Contributed by NM, 16-Jun-2012.)

Hypotheses

Ref	Expression
lplnset.b	⊢ 𝐵 = (Base‘𝐾)
lplnset.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
lplnset.n	⊢ 𝑁 = (LLines‘𝐾)
lplnset.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
islpln	⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)))

Distinct variable groups:   𝑦,𝑁   𝑦,𝐾   𝑦,𝑋

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝑃(𝑦)

Proof of Theorem islpln

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lplnset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		lplnset.c	. . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾)
3		lplnset.n	. . . 4 ⊢ 𝑁 = (LLines‘𝐾)
4		lplnset.p	. . . 4 ⊢ 𝑃 = (LPlanes‘𝐾)
5	1, 2, 3, 4	lplnset 40117	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})
6	5	eleq2d 2847	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥}))
7		breq2 5103	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝐶𝑥 ↔ 𝑦𝐶𝑋))
8	7	rexbidv 3185	. . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝑁 𝑦𝐶𝑥 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋))
9	8	elrab 3650	. 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋))
10	6, 9	bitrdi 289	1 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  {crab 3413   class class class wbr 5099  ‘cfv 6517  Basecbs 17228   ⋖ ccvr 39850  LLinesclln 40079  LPlanesclpl 40080

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 6473  df-fun 6519  df-fv 6525  df-lplanes 40087

This theorem is referenced by:  islpln4  40119  lplni  40120  lplnbase  40122  lplnnle2at  40129