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Theorem meetlem 18417
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 18416 . . . 4 (𝜑 → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
9 riotasbc 7398 . . . 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 18415 . . . 4 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1211sbceq1d 3780 . . 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 256 . 2 (𝜑[(𝑋 𝑌) / 𝑥]((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
14 ovex 7456 . . 3 (𝑋 𝑌) ∈ V
15 breq1 5155 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑋 ↔ (𝑋 𝑌) 𝑋))
16 breq1 5155 . . . . 5 (𝑥 = (𝑋 𝑌) → (𝑥 𝑌 ↔ (𝑋 𝑌) 𝑌))
1715, 16anbi12d 630 . . . 4 (𝑥 = (𝑋 𝑌) → ((𝑥 𝑋𝑥 𝑌) ↔ ((𝑋 𝑌) 𝑋 ∧ (𝑋 𝑌) 𝑌)))
18 breq2 5156 . . . . . 6 (𝑥 = (𝑋 𝑌) → (𝑧 𝑥𝑧 (𝑋 𝑌)))
1918imbi2d 339 . . . . 5 (𝑥 = (𝑋 𝑌) → (((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2019ralbidv 3167 . . . 4 (𝑥 = (𝑋 𝑌) → (∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥) ↔ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 (𝑋 𝑌))))
2117, 20anbi12d 630 . . 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 3819 . 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 217 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 394   = wceq 1533  wcel 2098  wral 3050  ∃!wreu 3361  [wsbc 3775  cop 4638   class class class wbr 5152  dom cdm 5681  cfv 6553  crio 7378  (class class class)co 7423  Basecbs 17208  lecple 17268  meetcmee 18332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-glb 18367  df-meet 18369
This theorem is referenced by:  lemeet1  18418  lemeet2  18419  meetle  18420
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