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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 7165 | . . . 4 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ 𝑟) = (𝑄 ∨ 𝑟)) | |
2 | 1 | breq2d 5080 | . . 3 ⊢ (𝑞 = 𝑄 → (𝑃 ≤ (𝑞 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑟))) |
3 | oveq2 7166 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅)) | |
4 | 3 | breq2d 5080 | . . 3 ⊢ (𝑟 = 𝑅 → (𝑃 ≤ (𝑄 ∨ 𝑟) ↔ 𝑃 ≤ (𝑄 ∨ 𝑅))) |
5 | 2, 4 | rspc2ev 3637 | . 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 36885 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 lecple 16574 joincjn 17556 Atomscatm 36401 PSubSpcpsubsp 36634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-psubsp 36641 |
This theorem is referenced by: pclclN 37029 pclfinN 37038 pclfinclN 37088 |
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