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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atle | Structured version Visualization version GIF version |
Description: Any nonzero element has an atom under it. (Contributed by NM, 28-Jun-2012.) |
Ref | Expression |
---|---|
atle.b | ⊢ 𝐵 = (Base‘𝐾) |
atle.l | ⊢ ≤ = (le‘𝐾) |
atle.z | ⊢ 0 = (0.‘𝐾) |
atle.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atle | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑝 ∈ 𝐴 𝑝 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝐾 ∈ HL) | |
2 | hlop 38964 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
3 | 2 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝐾 ∈ OP) |
4 | atle.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | atle.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
6 | 4, 5 | op0cl 38786 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
7 | 3, 6 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 ∈ 𝐵) |
8 | simp2 1134 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐵) | |
9 | simp3 1135 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ≠ 0 ) | |
10 | eqid 2725 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
11 | 4, 10, 5 | opltn0 38792 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 0 (lt‘𝐾)𝑋 ↔ 𝑋 ≠ 0 )) |
12 | 3, 8, 11 | syl2anc 582 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ( 0 (lt‘𝐾)𝑋 ↔ 𝑋 ≠ 0 )) |
13 | 9, 12 | mpbird 256 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 (lt‘𝐾)𝑋) |
14 | atle.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
15 | eqid 2725 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
16 | atle.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
17 | 4, 14, 10, 15, 16 | hlrelat 39005 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 0 (lt‘𝐾)𝑋) → ∃𝑝 ∈ 𝐴 ( 0 (lt‘𝐾)( 0 (join‘𝐾)𝑝) ∧ ( 0 (join‘𝐾)𝑝) ≤ 𝑋)) |
18 | 1, 7, 8, 13, 17 | syl31anc 1370 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑝 ∈ 𝐴 ( 0 (lt‘𝐾)( 0 (join‘𝐾)𝑝) ∧ ( 0 (join‘𝐾)𝑝) ≤ 𝑋)) |
19 | simpl1 1188 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ HL) | |
20 | hlol 38963 | . . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | |
21 | 19, 20 | syl 17 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ OL) |
22 | 4, 16 | atbase 38891 | . . . . . . . 8 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵) |
23 | 22 | adantl 480 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵) |
24 | 4, 15, 5 | olj02 38828 | . . . . . . 7 ⊢ ((𝐾 ∈ OL ∧ 𝑝 ∈ 𝐵) → ( 0 (join‘𝐾)𝑝) = 𝑝) |
25 | 21, 23, 24 | syl2anc 582 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ 𝑝 ∈ 𝐴) → ( 0 (join‘𝐾)𝑝) = 𝑝) |
26 | 25 | breq1d 5159 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ 𝑝 ∈ 𝐴) → (( 0 (join‘𝐾)𝑝) ≤ 𝑋 ↔ 𝑝 ≤ 𝑋)) |
27 | 26 | biimpd 228 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ 𝑝 ∈ 𝐴) → (( 0 (join‘𝐾)𝑝) ≤ 𝑋 → 𝑝 ≤ 𝑋)) |
28 | 27 | adantld 489 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ 𝑝 ∈ 𝐴) → (( 0 (lt‘𝐾)( 0 (join‘𝐾)𝑝) ∧ ( 0 (join‘𝐾)𝑝) ≤ 𝑋) → 𝑝 ≤ 𝑋)) |
29 | 28 | reximdva 3157 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (∃𝑝 ∈ 𝐴 ( 0 (lt‘𝐾)( 0 (join‘𝐾)𝑝) ∧ ( 0 (join‘𝐾)𝑝) ≤ 𝑋) → ∃𝑝 ∈ 𝐴 𝑝 ≤ 𝑋)) |
30 | 18, 29 | mpd 15 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → ∃𝑝 ∈ 𝐴 𝑝 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 lecple 17243 ltcplt 18303 joincjn 18306 0.cp0 18418 OPcops 38774 OLcol 38776 Atomscatm 38865 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: 1cvratex 39076 llnle 39121 lhpexle 39608 |
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