Theorem p0le 18415

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 18413	. . 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 18358	. 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋)
11	6, 10	eqbrtrd 5165	1 ⊢ (𝜑 → 0 ≤ 𝑋)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   class class class wbr 5143  dom cdm 5673  ‘cfv 6543  Basecbs 17174  lecple 17234  glbcglb 18296  0.cp0 18409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-glb 18333  df-p0 18411

This theorem is referenced by:  op0le  38653  atl0le  38771