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Theorem lemeet1 17416
Description: A meet's first argument is less than or equal to the meet. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetval2.b 𝐵 = (Base‘𝐾)
meetval2.l = (le‘𝐾)
meetval2.m = (meet‘𝐾)
meetval2.k (𝜑𝐾𝑉)
meetval2.x (𝜑𝑋𝐵)
meetval2.y (𝜑𝑌𝐵)
meetlem.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
lemeet1 (𝜑 → (𝑋 𝑌) 𝑋)

Proof of Theorem lemeet1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 meetval2.b . . 3 𝐵 = (Base‘𝐾)
2 meetval2.l . . 3 = (le‘𝐾)
3 meetval2.m . . 3 = (meet‘𝐾)
4 meetval2.k . . 3 (𝜑𝐾𝑉)
5 meetval2.x . . 3 (𝜑𝑋𝐵)
6 meetval2.y . . 3 (𝜑𝑌𝐵)
7 meetlem.e . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
81, 2, 3, 4, 5, 6, 7meetlem 17415 . 2 (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
98simplld 758 1 (𝜑 → (𝑋 𝑌) 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  cop 4404   class class class wbr 4888  dom cdm 5357  cfv 6137  (class class class)co 6924  Basecbs 16259  lecple 16349  meetcmee 17335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-glb 17365  df-meet 17367
This theorem is referenced by:  meetle  17418  latmle1  17466
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