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Mirrors > Home > MPE Home > Th. List > Mathboxes > atcvrneN | Structured version Visualization version GIF version |
Description: Inequality derived from atom condition. (Contributed by NM, 7-Feb-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atcvrne.j | ⊢ ∨ = (join‘𝐾) |
atcvrne.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
atcvrne.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atcvrneN | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlatl 38962 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
2 | 1 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ AtLat) |
3 | simp21 1203 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ 𝐴) | |
4 | eqid 2725 | . . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
5 | atcvrne.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 4, 5 | atn0 38910 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ (0.‘𝐾)) |
7 | 2, 3, 6 | syl2anc 582 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ≠ (0.‘𝐾)) |
8 | simp1 1133 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝐾 ∈ HL) | |
9 | eqid 2725 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | 9, 5 | atbase 38891 | . . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
11 | 3, 10 | syl 17 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃 ∈ (Base‘𝐾)) |
12 | simp22 1204 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ∈ 𝐴) | |
13 | simp23 1205 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑅 ∈ 𝐴) | |
14 | simp3 1135 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑃𝐶(𝑄 ∨ 𝑅)) | |
15 | atcvrne.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
16 | atcvrne.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
17 | 9, 15, 4, 16, 5 | atcvrj0 39031 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅)) |
18 | 8, 11, 12, 13, 14, 17 | syl131anc 1380 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 = (0.‘𝐾) ↔ 𝑄 = 𝑅)) |
19 | 18 | necon3bid 2974 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → (𝑃 ≠ (0.‘𝐾) ↔ 𝑄 ≠ 𝑅)) |
20 | 7, 19 | mpbid 231 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃𝐶(𝑄 ∨ 𝑅)) → 𝑄 ≠ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 joincjn 18306 0.cp0 18418 ⋖ ccvr 38864 Atomscatm 38865 AtLatcal 38866 HLchlt 38952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-lat 18427 df-clat 18494 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 |
This theorem is referenced by: atleneN 39037 |
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