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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 39787 | . . 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 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 lecple 17305 joincjn 18369 Latclat 18489 Atomscatm 39245 +𝑃cpadd 39778 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
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-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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-iun 4998 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-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-lub 18404 df-join 18406 df-lat 18490 df-ats 39249 df-padd 39779 |
This theorem is referenced by: osumcllem7N 39945 pexmidlem4N 39956 |
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