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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 32366 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 39188 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 0 ≤ 𝑋) | 
| 5 | 4 | 3adant3 1132 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 0 ≤ 𝑋) | 
| 6 | 5 | biantrurd 532 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ ( 0 ≤ 𝑋 ∧ 𝑋 ≤ 𝑃))) | 
| 7 | opposet 39183 | . . . . . 6 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
| 8 | 7 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 𝐾 ∈ Poset) | 
| 9 | 1, 3 | op0cl 39186 | . . . . . . 7 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) | 
| 10 | leatom.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 11 | 1, 10 | atbase 39291 | . . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵) | 
| 12 | id 22 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
| 13 | 9, 11, 12 | 3anim123i 1151 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) | 
| 14 | 13 | 3com23 1126 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → ( 0 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) | 
| 15 | eqid 2736 | . . . . . . 7 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
| 16 | 3, 15, 10 | atcvr0 39290 | . . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) | 
| 17 | 16 | 3adant2 1131 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → 0 ( ⋖ ‘𝐾)𝑃) | 
| 18 | 1, 2, 15 | cvrnbtwn4 39281 | . . . . 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 1372 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (( 0 ≤ 𝑋 ∧ 𝑋 ≤ 𝑃) ↔ ( 0 = 𝑋 ∨ 𝑋 = 𝑃))) | 
| 20 | eqcom 2743 | . . . . 5 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
| 21 | 20 | orbi1i 913 | . . . 4 ⊢ (( 0 = 𝑋 ∨ 𝑋 = 𝑃) ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃)) | 
| 22 | 19, 21 | bitrdi 287 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (( 0 ≤ 𝑋 ∧ 𝑋 ≤ 𝑃) ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃))) | 
| 23 | 6, 22 | bitrd 279 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 0 ∨ 𝑋 = 𝑃))) | 
| 24 | orcom 870 | . 2 ⊢ ((𝑋 = 0 ∨ 𝑋 = 𝑃) ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 )) | |
| 25 | 23, 24 | bitrdi 287 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) → (𝑋 ≤ 𝑃 ↔ (𝑋 = 𝑃 ∨ 𝑋 = 0 ))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 ‘cfv 6560 Basecbs 17248 lecple 17305 Posetcpo 18354 0.cp0 18469 OPcops 39174 ⋖ ccvr 39264 Atomscatm 39265 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-proset 18341 df-poset 18360 df-plt 18376 df-glb 18393 df-p0 18471 df-oposet 39178 df-covers 39268 df-ats 39269 | 
| This theorem is referenced by: leat 39295 leat2 39296 meetat 39298 | 
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