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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 17427 | . . 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 17386 | . 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋) |
11 | 6, 10 | eqbrtrd 4908 | 1 ⊢ (𝜑 → 0 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 class class class wbr 4886 dom cdm 5355 ‘cfv 6135 Basecbs 16255 lecple 16345 glbcglb 17329 0.cp0 17423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-glb 17361 df-p0 17425 |
This theorem is referenced by: op0le 35342 atl0le 35460 |
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