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Theorem meetlem 16965
 Description: Lemma for meet properties. (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
meetlem (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
Distinct variable groups:   𝑧,𝐵   𝑧,   𝑧,𝐾   𝑧,𝑋   𝑧,𝑌
Allowed substitution hints:   𝜑(𝑧)   (𝑧)   𝑉(𝑧)

Proof of Theorem meetlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 meetval2.b . . . . 5 𝐵 = (Base‘𝐾)
2 meetval2.l . . . . 5 = (le‘𝐾)
3 meetval2.m . . . . 5 = (meet‘𝐾)
4 meetval2.k . . . . 5 (𝜑𝐾𝑉)
5 meetval2.x . . . . 5 (𝜑𝑋𝐵)
6 meetval2.y . . . . 5 (𝜑𝑌𝐵)
7 meetlem.e . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
81, 2, 3, 4, 5, 6, 7meeteu 16964 . . . 4 (𝜑 → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
9 riotasbc 6591 . . . 4 (∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)) → [(𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
108, 9syl 17 . . 3 (𝜑[(𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
111, 2, 3, 4, 5, 6meetval2 16963 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1211sbceq1d 3427 . . 3 (𝜑 → ([(𝑋 𝑌) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)) ↔ [(𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1310, 12mpbird 247 . 2 (𝜑[(𝑋 𝑌) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
14 ovex 6643 . . 3 (𝑋 𝑌) ∈ V
15 breq1 4626 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑋 ↔ (𝑋 𝑌) 𝑋))
16 breq1 4626 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑌 ↔ (𝑋 𝑌) 𝑌))
1715, 16anbi12d 746 . . . 4 (𝑥 = (𝑋 𝑌) → ((𝑥 𝑋𝑥 𝑌) ↔ ((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌)))
18 breq2 4627 . . . . . 6 (𝑥 = (𝑋 𝑌) → (𝑧 𝑥𝑧 (𝑋 𝑌)))
1918imbi2d 330 . . . . 5 (𝑥 = (𝑋 𝑌) → (((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2019ralbidv 2982 . . . 4 (𝑥 = (𝑋 𝑌) → (∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2117, 20anbi12d 746 . . 3 (𝑥 = (𝑋 𝑌) → (((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)) ↔ (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌)))))
2214, 21sbcie 3457 . 2 ([(𝑋 𝑌) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)) ↔ (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2313, 22sylib 208 1 (𝜑 → (((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2908  ∃!wreu 2910  [wsbc 3422  ⟨cop 4161   class class class wbr 4623  dom cdm 5084  ‘cfv 5857  ℩crio 6575  (class class class)co 6615  Basecbs 15800  lecple 15888  meetcmee 16885 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-glb 16915  df-meet 16917 This theorem is referenced by:  lemeet1  16966  lemeet2  16967  meetle  16968
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