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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpaddat | Structured version Visualization version GIF version |
Description: Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.) |
Ref | Expression |
---|---|
paddfval.l | ⊢ ≤ = (le‘𝐾) |
paddfval.j | ⊢ ∨ = (join‘𝐾) |
paddfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
paddfval.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
elpaddat | ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1057 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝐾 ∈ Lat) | |
2 | simpl2 1058 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝑋 ⊆ 𝐴) | |
3 | simpl3 1059 | . . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝑄 ∈ 𝐴) | |
4 | 3 | snssd 4281 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → {𝑄} ⊆ 𝐴) |
5 | simpr 476 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → 𝑋 ≠ ∅) | |
6 | snnzg 4251 | . . . 4 ⊢ (𝑄 ∈ 𝐴 → {𝑄} ≠ ∅) | |
7 | 3, 6 | syl 17 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → {𝑄} ≠ ∅) |
8 | paddfval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
9 | paddfval.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
10 | paddfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | paddfval.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
12 | 8, 9, 10, 11 | elpaddn0 34104 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ {𝑄} ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ {𝑄} ≠ ∅)) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟)))) |
13 | 1, 2, 4, 5, 7, 12 | syl32anc 1326 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟)))) |
14 | oveq2 6557 | . . . . . . 7 ⊢ (𝑟 = 𝑄 → (𝑝 ∨ 𝑟) = (𝑝 ∨ 𝑄)) | |
15 | 14 | breq2d 4595 | . . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑆 ≤ (𝑝 ∨ 𝑟) ↔ 𝑆 ≤ (𝑝 ∨ 𝑄))) |
16 | 15 | rexsng 4166 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → (∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟) ↔ 𝑆 ≤ (𝑝 ∨ 𝑄))) |
17 | 3, 16 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟) ↔ 𝑆 ≤ (𝑝 ∨ 𝑄))) |
18 | 17 | rexbidv 3034 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟) ↔ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄))) |
19 | 18 | anbi2d 736 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → ((𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 ∃𝑟 ∈ {𝑄}𝑆 ≤ (𝑝 ∨ 𝑟)) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄)))) |
20 | 13, 19 | bitrd 267 | 1 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 Latclat 16868 Atomscatm 33568 +𝑃cpadd 34099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-lub 16797 df-join 16799 df-lat 16869 df-ats 33572 df-padd 34100 |
This theorem is referenced by: elpaddatiN 34109 elpadd2at 34110 pclfinclN 34254 |
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