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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 37587 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑅 ∈ (𝑋 + {𝑄}) ↔ (𝑅 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)))) |
6 | simpr 488 | . . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)) → ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)) | |
7 | 5, 6 | syl6bi 256 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑅 ∈ (𝑋 + {𝑄}) → ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄))) |
8 | 7 | impr 458 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2942 ∃wrex 3064 ⊆ wss 3882 ∅c0 4253 {csn 4557 class class class wbr 5069 ‘cfv 6400 (class class class)co 7234 lecple 16839 joincjn 17848 Latclat 17967 Atomscatm 37046 +𝑃cpadd 37578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3711 df-csb 3828 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-1st 7782 df-2nd 7783 df-lub 17882 df-join 17884 df-lat 17968 df-ats 37050 df-padd 37579 |
This theorem is referenced by: osumcllem7N 37745 pexmidlem4N 37756 |
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