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Theorem meetlem 17614
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 17613 . . . 4 (𝜑 → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
9 riotasbc 7109 . . . 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 17612 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1211sbceq1d 3757 . . 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 259 . 2 (𝜑[(𝑋 𝑌) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
14 ovex 7166 . . 3 (𝑋 𝑌) ∈ V
15 breq1 5045 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑋 ↔ (𝑋 𝑌) 𝑋))
16 breq1 5045 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑌 ↔ (𝑋 𝑌) 𝑌))
1715, 16anbi12d 632 . . . 4 (𝑥 = (𝑋 𝑌) → ((𝑥 𝑋𝑥 𝑌) ↔ ((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌)))
18 breq2 5046 . . . . . 6 (𝑥 = (𝑋 𝑌) → (𝑧 𝑥𝑧 (𝑋 𝑌)))
1918imbi2d 343 . . . . 5 (𝑥 = (𝑋 𝑌) → (((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2019ralbidv 3184 . . . 4 (𝑥 = (𝑋 𝑌) → (∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2117, 20anbi12d 632 . . 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 3792 . 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 220 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 398   = wceq 1537  wcel 2114  wral 3125  ∃!wreu 3127  [wsbc 3752  cop 4549   class class class wbr 5042  dom cdm 5531  cfv 6331  crio 7090  (class class class)co 7133  Basecbs 16462  lecple 16551  meetcmee 17534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-glb 17564  df-meet 17566
This theorem is referenced by:  lemeet1  17615  lemeet2  17616  meetle  17617
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