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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leatb | Structured version Visualization version GIF version |
Description: A poset element less than or equal to an atom equals either zero or the atom. (atss 30129 analog.) (Contributed by NM, 17-Nov-2011.) |
Ref | Expression |
---|---|
leatom.b | ⊢ 𝐵 = (Base‘𝐾) |
leatom.l | ⊢ ≤ = (le‘𝐾) |
leatom.z | ⊢ 0 = (0.‘𝐾) |
leatom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
leatb | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leatom.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
2 | leatom.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
3 | leatom.z | . . . . . 6 ⊢ 0 = (0.‘𝐾) | |
4 | 1, 2, 3 | op0le 36482 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) |
5 | 4 | 3adant3 1129 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 0 ≤ 𝑋) |
6 | 5 | biantrurd 536 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ ( 0 ≤ 𝑋 ∧ 𝑋 ≤ 𝑃))) |
7 | opposet 36477 | . . . . . 6 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
8 | 7 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Poset) |
9 | 1, 3 | op0cl 36480 | . . . . . . 7 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
10 | leatom.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | 1, 10 | atbase 36585 | . . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) |
12 | id 22 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
13 | 9, 11, 12 | 3anim123i 1148 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
14 | 13 | 3com23 1123 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) |
15 | eqid 2798 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
16 | 3, 15, 10 | atcvr0 36584 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
17 | 16 | 3adant2 1128 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) |
18 | 1, 2, 15 | cvrnbtwn4 36575 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 0 ( ⋖ ‘𝐾)𝑃) → (( 0 ≤ 𝑋 ∧ 𝑋 ≤ 𝑃) ↔ ( 0 = 𝑋 ∨ 𝑋 = 𝑃))) |
19 | 8, 14, 17, 18 | syl3anc 1368 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (( 0 ≤ 𝑋 ∧ 𝑋 ≤ 𝑃) ↔ ( 0 = 𝑋 ∨ 𝑋 = 𝑃))) |
20 | eqcom 2805 | . . . . 5 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
21 | 20 | orbi1i 911 | . . . 4 ⊢ (( 0 = 𝑋 ∨ 𝑋 = 𝑃) ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃)) |
22 | 19, 21 | syl6bb 290 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (( 0 ≤ 𝑋 ∧ 𝑋 ≤ 𝑃) ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃))) |
23 | 6, 22 | bitrd 282 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃))) |
24 | orcom 867 | . 2 ⊢ ((𝑋 = 0 ∨ 𝑋 = 𝑃) ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 )) | |
25 | 23, 24 | syl6bb 290 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 lecple 16564 Posetcpo 17542 0.cp0 17639 OPcops 36468 ⋖ ccvr 36558 Atomscatm 36559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-proset 17530 df-poset 17548 df-plt 17560 df-glb 17577 df-p0 17641 df-oposet 36472 df-covers 36562 df-ats 36563 |
This theorem is referenced by: leat 36589 leat2 36590 meetat 36592 |
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