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Theorem joinval2 17131
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 2724 . . 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 17127 . 2 (𝜑 → (𝑋 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌}))
7 joinval2.b . . 3 𝐵 = (Base‘𝐾)
8 joinval2.l . . 3 = (le‘𝐾)
9 biid 251 . . 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 4461 . . . 4 ((𝑋𝐵𝑌𝐵) → {𝑋, 𝑌} ⊆ 𝐵)
114, 5, 10syl2anc 696 . . 3 (𝜑 → {𝑋, 𝑌} ⊆ 𝐵)
127, 8, 1, 9, 3, 11lubval 17106 . 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 17130 . . . 4 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
1413riotabidv 6728 . . 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 696 . 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 2762 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 383   = wceq 1596  wcel 2103  wral 3014  wss 3680  {cpr 4287   class class class wbr 4760  cfv 6001  crio 6725  (class class class)co 6765  Basecbs 15980  lecple 16071  lubclub 17064  joincjn 17066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-lub 17096  df-join 17098
This theorem is referenced by:  joinlem  17133
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