Theorem glble 18294

Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
glbprop.b	⊢ 𝐵 = (Base‘𝐾)
glbprop.l	⊢ ≤ = (le‘𝐾)
glbprop.u	⊢ 𝑈 = (glb‘𝐾)
glbprop.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
glbprop.s	⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
glble.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
glble	⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋)

Proof of Theorem glble

Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5099	. 2 ⊢ (𝑦 = 𝑋 → ((𝑈‘𝑆) ≤ 𝑦 ↔ (𝑈‘𝑆) ≤ 𝑋))
2		glbprop.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		glbprop.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		glbprop.u	. . . 4 ⊢ 𝑈 = (glb‘𝐾)
5		glbprop.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6		glbprop.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
7	2, 3, 4, 5, 6	glbprop 18293	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))
8	7	simpld 494	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦)
9		glble.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	1, 8, 9	rspcdva 3580	1 ⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5095  dom cdm 5623  ‘cfv 6486  Basecbs 17138  lecple 17186  glbcglb 18234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-glb 18269

This theorem is referenced by:  p0le  18351  clatglble  18441  glbsscl  48946