Theorem p0le 18487

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.)

Hypotheses

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)

Assertion

Ref	Expression
p0le	⊢ (𝜑 → 0 ≤ 𝑋)

Proof of Theorem p0le

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 18485	. . 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 18430	. 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋)
11	6, 10	eqbrtrd 5170	1 ⊢ (𝜑 → 0 ≤ 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  dom cdm 5689  ‘cfv 6563  Basecbs 17245  lecple 17305  glbcglb 18368  0.cp0 18481

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-glb 18405  df-p0 18483

This theorem is referenced by:  op0le  39168  atl0le  39286