Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > atlen0 | Structured version Visualization version GIF version |
Description: A lattice element is nonzero if an atom is under it. (Contributed by NM, 26-May-2012.) |
Ref | Expression |
---|---|
atlen0.b | ⊢ 𝐵 = (Base‘𝐾) |
atlen0.l | ⊢ ≤ = (le‘𝐾) |
atlen0.z | ⊢ 0 = (0.‘𝐾) |
atlen0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atlen0 | ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1187 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ AtLat) | |
2 | atlen0.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
3 | atlen0.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
4 | 2, 3 | atl0cl 36454 | . . . . 5 ⊢ (𝐾 ∈ AtLat → 0 ∈ 𝐵) |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ∈ 𝐵) |
6 | simpl2 1188 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ∈ 𝐵) | |
7 | 1, 5, 6 | 3jca 1124 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
8 | simpl3 1189 | . . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐴) | |
9 | atlen0.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 2, 9 | atbase 36440 | . . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ∈ 𝐵) |
12 | eqid 2821 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
13 | 3, 12, 9 | atcvr0 36439 | . . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
14 | 1, 8, 13 | syl2anc 586 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ( ⋖ ‘𝐾)𝑃) |
15 | eqid 2821 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
16 | 2, 15, 12 | cvrlt 36421 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) ∧ 0 ( ⋖ ‘𝐾)𝑃) → 0 (lt‘𝐾)𝑃) |
17 | 1, 5, 11, 14, 16 | syl31anc 1369 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑃) |
18 | simpr 487 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑃 ≤ 𝑋) | |
19 | atlpos 36452 | . . . . . 6 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
20 | 1, 19 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝐾 ∈ Poset) |
21 | atlen0.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
22 | 2, 21, 15 | pltletr 17581 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (( 0 (lt‘𝐾)𝑃 ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋)) |
23 | 20, 5, 11, 6, 22 | syl13anc 1368 | . . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → (( 0 (lt‘𝐾)𝑃 ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋)) |
24 | 17, 18, 23 | mp2and 697 | . . 3 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 (lt‘𝐾)𝑋) |
25 | 15 | pltne 17572 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑋 → 0 ≠ 𝑋)) |
26 | 7, 24, 25 | sylc 65 | . 2 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 0 ≠ 𝑋) |
27 | 26 | necomd 3071 | 1 ⊢ (((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ 𝑋) → 𝑋 ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 Posetcpo 17550 ltcplt 17551 0.cp0 17647 ⋖ ccvr 36413 Atomscatm 36414 AtLatcal 36415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-proset 17538 df-poset 17556 df-plt 17568 df-glb 17585 df-p0 17649 df-lat 17656 df-covers 36417 df-ats 36418 df-atl 36449 |
This theorem is referenced by: ps-2b 36633 2atm 36678 2llnm4 36721 dalem21 36845 dalem54 36877 trlval3 37338 cdlemc5 37346 |
Copyright terms: Public domain | W3C validator |