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Theorem joinval2 17208
Description: Value of join for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 11-Sep-2018.)
Hypotheses
Ref Expression
joinval2.b 𝐵 = (Base‘𝐾)
joinval2.l = (le‘𝐾)
joinval2.j = (join‘𝐾)
joinval2.k (𝜑𝐾𝑉)
joinval2.x (𝜑𝑋𝐵)
joinval2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
joinval2 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
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 joinval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2802 . . 3 (lub‘𝐾) = (lub‘𝐾)
2 joinval2.j . . 3 = (join‘𝐾)
3 joinval2.k . . 3 (𝜑𝐾𝑉)
4 joinval2.x . . 3 (𝜑𝑋𝐵)
5 joinval2.y . . 3 (𝜑𝑌𝐵)
61, 2, 3, 4, 5joinval 17204 . 2 (𝜑 → (𝑋 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌}))
7 joinval2.b . . 3 𝐵 = (Base‘𝐾)
8 joinval2.l . . 3 = (le‘𝐾)
9 biid 252 . . 3 ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)))
10 prssi 4536 . . . 4 ((𝑋𝐵𝑌𝐵) → {𝑋, 𝑌} ⊆ 𝐵)
114, 5, 10syl2anc 575 . . 3 (𝜑 → {𝑋, 𝑌} ⊆ 𝐵)
127, 8, 1, 9, 3, 11lubval 17183 . 2 (𝜑 → ((lub‘𝐾)‘{𝑋, 𝑌}) = (𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧))))
137, 8, 2, 3, 4, 5joinval2lem 17207 . . . 4 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
1413riotabidv 6831 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧))) = (𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
154, 5, 14syl2anc 575 . 2 (𝜑 → (𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧))) = (𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
166, 12, 153eqtrd 2840 1 (𝜑 → (𝑋 𝑌) = (𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2155  wral 3092  wss 3763  {cpr 4366   class class class wbr 4837  cfv 6095  crio 6828  (class class class)co 6868  Basecbs 16062  lecple 16154  lubclub 17141  joincjn 17143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-riota 6829  df-ov 6871  df-oprab 6872  df-lub 17173  df-join 17175
This theorem is referenced by:  joinlem  17210
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