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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 5155 | . . 3 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑟))) |
| 3 | oveq2 7439 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅)) | |
| 4 | 3 | breq2d 5155 | . . 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 39749 | . 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 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 lecple 17304 joincjn 18357 Atomscatm 39264 PSubSpcpsubsp 39498 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-psubsp 39505 |
| This theorem is referenced by: pclclN 39893 pclfinN 39902 pclfinclN 39952 |
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