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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 17645 | . . 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 17604 | . 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋) |
11 | 6, 10 | eqbrtrd 5080 | 1 ⊢ (𝜑 → 0 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 dom cdm 5549 ‘cfv 6349 Basecbs 16477 lecple 16566 glbcglb 17547 0.cp0 17641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-glb 17579 df-p0 17643 |
This theorem is referenced by: op0le 36316 atl0le 36434 |
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