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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubspi2N | Structured version Visualization version GIF version |
Description: Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubspset.l | ⊢ ≤ = (le‘𝐾) |
psubspset.j | ⊢ ∨ = (join‘𝐾) |
psubspset.a | ⊢ 𝐴 = (Atoms‘𝐾) |
psubspset.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
Ref | Expression |
---|---|
psubspi2N | ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7438 | . . . 4 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ 𝑟) = (𝑄 ∨ 𝑟)) | |
2 | 1 | breq2d 5160 | . . 3 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑟))) |
3 | oveq2 7439 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅)) | |
4 | 3 | breq2d 5160 | . . 3 ⊢ (𝑟 = 𝑅 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑅))) |
5 | 2, 4 | rspc2ev 3635 | . 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 39730 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋) |
11 | 5, 10 | sylan2 593 | 1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 lecple 17305 joincjn 18369 Atomscatm 39245 PSubSpcpsubsp 39479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-psubsp 39486 |
This theorem is referenced by: pclclN 39874 pclfinN 39883 pclfinclN 39933 |
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