Theorem p0le 18459

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 18457	. . 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 18402	. 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋)
11	6, 10	eqbrtrd 5122	1 ⊢ (𝜑 → 0 ≤ 𝑋)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142   class class class wbr 5100  dom cdm 5647  ‘cfv 6521  Basecbs 17245  lecple 17293  glbcglb 18342  0.cp0 18453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-glb 18377  df-p0 18455

This theorem is referenced by:  op0le  39810  atl0le  39928