Theorem glble 18549

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 5106	. 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 18548	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))
8	7	simpld 500	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦)
9		glble.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	1, 8, 9	rspcdva 3577	1 ⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102  dom cdm 5650  ‘cfv 6538  Basecbs 17392  lecple 17440  glbcglb 18489

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-glb 18524

This theorem is used by:  p0le  18606  clatglble  18696  glbsscl  50168