Theorem p0le 18499

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 18497	. . 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 18442	. 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋)
11	6, 10	eqbrtrd 5188	1 ⊢ (𝜑 → 0 ≤ 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  Basecbs 17258  lecple 17318  glbcglb 18380  0.cp0 18493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-glb 18417  df-p0 18495

This theorem is referenced by:  op0le  39142  atl0le  39260