Theorem psubspi2N 40194

Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
psubspset.l	⊢ ≤ = (le‘𝐾)
psubspset.j	⊢ ∨ = (join‘𝐾)
psubspset.a	⊢ 𝐴 = (Atoms‘𝐾)
psubspset.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
psubspi2N	⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃 ∈ 𝑋)

Proof of Theorem psubspi2N

Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7374	. . . 4 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ 𝑟) = (𝑄 ∨ 𝑟))
2	1	breq2d 5098	. . 3 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑟)))
3		oveq2 7375	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
4	3	breq2d 5098	. . 3 ⊢ (𝑟 = 𝑅 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑅)))
5	2, 4	rspc2ev 3578	. 2 ⊢ ((𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅)) → ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟))
6		psubspset.l	. . 3 ⊢ ≤ = (le‘𝐾)
7		psubspset.j	. . 3 ⊢ ∨ = (join‘𝐾)
8		psubspset.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
9		psubspset.s	. . 3 ⊢ 𝑆 = (PSubSp‘𝐾)
10	6, 7, 8, 9	psubspi 40193	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋)
11	5, 10	sylan2 594	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃 ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   class class class wbr 5086  ‘cfv 6499  (class class class)co 7367  lecple 17227  joincjn 18277  Atomscatm 39709  PSubSpcpsubsp 39942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5308  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6455  df-fun 6501  df-fv 6507  df-ov 7370  df-psubsp 39949

This theorem is referenced by:  pclclN  40337  pclfinN  40346  pclfinclN  40396