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| Mirrors > Home > MPE Home > Th. List > p0le | Structured version Visualization version GIF version | ||
| Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.) | 
| Ref | Expression | 
|---|---|
| p0le.b | ⊢ 𝐵 = (Base‘𝐾) | 
| p0le.g | ⊢ 𝐺 = (glb‘𝐾) | 
| p0le.l | ⊢ ≤ = (le‘𝐾) | 
| p0le.0 | ⊢ 0 = (0.‘𝐾) | 
| p0le.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) | 
| p0le.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| p0le.d | ⊢ (𝜑 → 𝐵 ∈ dom 𝐺) | 
| Ref | Expression | 
|---|---|
| p0le | ⊢ (𝜑 → 0 ≤ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | p0le.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
| 2 | p0le.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 3 | p0le.g | . . . 4 ⊢ 𝐺 = (glb‘𝐾) | |
| 4 | p0le.0 | . . . 4 ⊢ 0 = (0.‘𝐾) | |
| 5 | 2, 3, 4 | p0val 18473 | . . 3 ⊢ (𝐾 ∈ 𝑉 → 0 = (𝐺‘𝐵)) | 
| 6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → 0 = (𝐺‘𝐵)) | 
| 7 | p0le.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 8 | p0le.d | . . 3 ⊢ (𝜑 → 𝐵 ∈ dom 𝐺) | |
| 9 | p0le.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 10 | 2, 7, 3, 1, 8, 9 | glble 18418 | . 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋) | 
| 11 | 6, 10 | eqbrtrd 5164 | 1 ⊢ (𝜑 → 0 ≤ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 dom cdm 5684 ‘cfv 6560 Basecbs 17248 lecple 17305 glbcglb 18357 0.cp0 18469 | 
| 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 | 
| 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-glb 18393 df-p0 18471 | 
| This theorem is referenced by: op0le 39188 atl0le 39306 | 
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