| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elpaddatiN | Structured version Visualization version GIF version | ||
| Description: Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| paddfval.l | ⊢ ≤ = (le‘𝐾) |
| paddfval.j | ⊢ ∨ = (join‘𝐾) |
| paddfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| paddfval.p | ⊢ + = (+𝑃‘𝐾) |
| Ref | Expression |
|---|---|
| elpaddatiN | ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑅 ∈ (𝑋 + {𝑄}))) → ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | paddfval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 2 | paddfval.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
| 3 | paddfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 4 | paddfval.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
| 5 | 1, 2, 3, 4 | elpaddat 39806 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑅 ∈ (𝑋 + {𝑄}) ↔ (𝑅 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)))) |
| 6 | simpr 484 | . . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)) → ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)) | |
| 7 | 5, 6 | biimtrdi 253 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑅 ∈ (𝑋 + {𝑄}) → ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄))) |
| 8 | 7 | impr 454 | 1 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑅 ∈ (𝑋 + {𝑄}))) → ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ⊆ wss 3951 ∅c0 4333 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 lecple 17304 joincjn 18357 Latclat 18476 Atomscatm 39264 +𝑃cpadd 39797 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-iun 4993 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-lub 18391 df-join 18393 df-lat 18477 df-ats 39268 df-padd 39798 |
| This theorem is referenced by: osumcllem7N 39964 pexmidlem4N 39975 |
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