MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  meetlem Structured version   Visualization version   GIF version

Theorem meetlem 17411
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 17410 . . . 4 (𝜑 → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
9 riotasbc 6898 . . . 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 17409 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1211sbceq1d 3657 . . 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 249 . 2 (𝜑[(𝑋 𝑌) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
14 ovex 6954 . . 3 (𝑋 𝑌) ∈ V
15 breq1 4889 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑋 ↔ (𝑋 𝑌) 𝑋))
16 breq1 4889 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑌 ↔ (𝑋 𝑌) 𝑌))
1715, 16anbi12d 624 . . . 4 (𝑥 = (𝑋 𝑌) → ((𝑥 𝑋𝑥 𝑌) ↔ ((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌)))
18 breq2 4890 . . . . . 6 (𝑥 = (𝑋 𝑌) → (𝑧 𝑥𝑧 (𝑋 𝑌)))
1918imbi2d 332 . . . . 5 (𝑥 = (𝑋 𝑌) → (((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2019ralbidv 3168 . . . 4 (𝑥 = (𝑋 𝑌) → (∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2117, 20anbi12d 624 . . 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 3687 . 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 210 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 386   = wceq 1601  wcel 2107  wral 3090  ∃!wreu 3092  [wsbc 3652  cop 4404   class class class wbr 4886  dom cdm 5355  cfv 6135  crio 6882  (class class class)co 6922  Basecbs 16255  lecple 16345  meetcmee 17331
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 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
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 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-glb 17361  df-meet 17363
This theorem is referenced by:  lemeet1  17412  lemeet2  17413  meetle  17414
  Copyright terms: Public domain W3C validator