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Theorem lejoin2 17452
Description: A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)
Hypotheses
Ref Expression
joinval2.b 𝐵 = (Base‘𝐾)
joinval2.l = (le‘𝐾)
joinval2.j = (join‘𝐾)
joinval2.k (𝜑𝐾𝑉)
joinval2.x (𝜑𝑋𝐵)
joinval2.y (𝜑𝑌𝐵)
joinlem.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
lejoin2 (𝜑𝑌 (𝑋 𝑌))

Proof of Theorem lejoin2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 joinval2.b . . 3 𝐵 = (Base‘𝐾)
2 joinval2.l . . 3 = (le‘𝐾)
3 joinval2.j . . 3 = (join‘𝐾)
4 joinval2.k . . 3 (𝜑𝐾𝑉)
5 joinval2.x . . 3 (𝜑𝑋𝐵)
6 joinval2.y . . 3 (𝜑𝑌𝐵)
7 joinlem.e . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
81, 2, 3, 4, 5, 6, 7joinlem 17450 . 2 (𝜑 → ((𝑋 (𝑋 𝑌) ∧ 𝑌 (𝑋 𝑌)) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → (𝑋 𝑌) 𝑧)))
98simplrd 766 1 (𝜑𝑌 (𝑋 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  wral 3105  cop 4478   class class class wbr 4962  dom cdm 5443  cfv 6225  (class class class)co 7016  Basecbs 16312  lecple 16401  joincjn 17383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-lub 17413  df-join 17415
This theorem is referenced by:  joinle  17453  latlej2  17500
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