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Theorem meetval2 17635
 Description: Value of meet for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 11-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)
Assertion
Ref Expression
meetval2 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
Distinct variable groups:   𝑥,𝑧,𝐵   𝑥, ,𝑧   𝑥,𝐾,𝑧   𝑥,𝑋,𝑧   𝑥,𝑌,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧)   (𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem meetval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2823 . . 3 (glb‘𝐾) = (glb‘𝐾)
2 meetval2.m . . 3 = (meet‘𝐾)
3 meetval2.k . . 3 (𝜑𝐾𝑉)
4 meetval2.x . . 3 (𝜑𝑋𝐵)
5 meetval2.y . . 3 (𝜑𝑌𝐵)
61, 2, 3, 4, 5meetval 17631 . 2 (𝜑 → (𝑋 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
7 meetval2.b . . 3 𝐵 = (Base‘𝐾)
8 meetval2.l . . 3 = (le‘𝐾)
9 biid 263 . . 3 ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)))
104, 5prssd 4757 . . 3 (𝜑 → {𝑋, 𝑌} ⊆ 𝐵)
117, 8, 1, 9, 3, 10glbval 17609 . 2 (𝜑 → ((glb‘𝐾)‘{𝑋, 𝑌}) = (𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥))))
127, 8, 2, 3, 4, 5meetval2lem 17634 . . . 4 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1312riotabidv 7118 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥))) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
144, 5, 13syl2anc 586 . 2 (𝜑 → (𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥))) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
156, 11, 143eqtrd 2862 1 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {cpr 4571   class class class wbr 5068  ‘cfv 6357  ℩crio 7115  (class class class)co 7158  Basecbs 16485  lecple 16574  glbcglb 17555  meetcmee 17557 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 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463 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 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-glb 17587  df-meet 17589 This theorem is referenced by:  meetlem  17637
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