Theorem p0le 18384

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 18382	. . 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 18327	. 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋)
11	6, 10	eqbrtrd 5094	1 ⊢ (𝜑 → 0 ≤ 𝑋)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  dom cdm 5618  ‘cfv 6485  Basecbs 17170  lecple 17218  glbcglb 18267  0.cp0 18378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-glb 18302  df-p0 18380

This theorem is referenced by:  op0le  39678  atl0le  39796