Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrnnid | Structured version Visualization version GIF version |
Description: If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom 𝑊 and not equal to its translation. (Contributed by NM, 24-May-2012.) |
Ref | Expression |
---|---|
ltrneq.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrneq.l | ⊢ ≤ = (le‘𝐾) |
ltrneq.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ltrneq.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrneq.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrnnid | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralinexa 2980 | . . . . 5 ⊢ (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → ¬ (𝐹‘𝑝) ≠ 𝑝) ↔ ¬ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) | |
2 | nne 2786 | . . . . . . . 8 ⊢ (¬ (𝐹‘𝑝) ≠ 𝑝 ↔ (𝐹‘𝑝) = 𝑝) | |
3 | 2 | biimpi 205 | . . . . . . 7 ⊢ (¬ (𝐹‘𝑝) ≠ 𝑝 → (𝐹‘𝑝) = 𝑝) |
4 | 3 | imim2i 16 | . . . . . 6 ⊢ ((¬ 𝑝 ≤ 𝑊 → ¬ (𝐹‘𝑝) ≠ 𝑝) → (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) |
5 | 4 | ralimi 2936 | . . . . 5 ⊢ (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → ¬ (𝐹‘𝑝) ≠ 𝑝) → ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) |
6 | 1, 5 | sylbir 224 | . . . 4 ⊢ (¬ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝) → ∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝)) |
7 | ltrneq.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
8 | ltrneq.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
9 | ltrneq.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | ltrneq.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | ltrneq.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
12 | 7, 8, 9, 10, 11 | ltrnid 34439 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∀𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 → (𝐹‘𝑝) = 𝑝) ↔ 𝐹 = ( I ↾ 𝐵))) |
13 | 6, 12 | syl5ib 233 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (¬ ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝) → 𝐹 = ( I ↾ 𝐵))) |
14 | 13 | necon1ad 2799 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐹 ≠ ( I ↾ 𝐵) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝))) |
15 | 14 | 3impia 1253 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵)) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑊 ∧ (𝐹‘𝑝) ≠ 𝑝)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 I cid 4948 ↾ cres 5040 ‘cfv 5804 Basecbs 15695 lecple 15775 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 |
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-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-laut 34293 df-ldil 34408 df-ltrn 34409 |
This theorem is referenced by: trlnidat 34478 |
Copyright terms: Public domain | W3C validator |