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Theorem meetval2 18154
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 𝐡 = (Baseβ€˜πΎ)
meetval2.l ≀ = (leβ€˜πΎ)
meetval2.m ∧ = (meetβ€˜πΎ)
meetval2.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
meetval2.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
meetval2.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
meetval2 (πœ‘ β†’ (𝑋 ∧ π‘Œ) = (β„©π‘₯ ∈ 𝐡 ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
Distinct variable groups:   π‘₯,𝑧,𝐡   π‘₯, ∧ ,𝑧   π‘₯,𝐾,𝑧   π‘₯,𝑋,𝑧   π‘₯,π‘Œ,𝑧
Allowed substitution hints:   πœ‘(π‘₯,𝑧)   ≀ (π‘₯,𝑧)   𝑉(π‘₯,𝑧)

Proof of Theorem meetval2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2736 . . 3 (glbβ€˜πΎ) = (glbβ€˜πΎ)
2 meetval2.m . . 3 ∧ = (meetβ€˜πΎ)
3 meetval2.k . . 3 (πœ‘ β†’ 𝐾 ∈ 𝑉)
4 meetval2.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
5 meetval2.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
61, 2, 3, 4, 5meetval 18150 . 2 (πœ‘ β†’ (𝑋 ∧ π‘Œ) = ((glbβ€˜πΎ)β€˜{𝑋, π‘Œ}))
7 meetval2.b . . 3 𝐡 = (Baseβ€˜πΎ)
8 meetval2.l . . 3 ≀ = (leβ€˜πΎ)
9 biid 262 . . 3 ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ (βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)))
104, 5prssd 4761 . . 3 (πœ‘ β†’ {𝑋, π‘Œ} βŠ† 𝐡)
117, 8, 1, 9, 3, 10glbval 18128 . 2 (πœ‘ β†’ ((glbβ€˜πΎ)β€˜{𝑋, π‘Œ}) = (β„©π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯))))
127, 8, 2, 3, 4, 5meetval2lem 18153 . . . 4 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯)) ↔ ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
1312riotabidv 7262 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (β„©π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯))) = (β„©π‘₯ ∈ 𝐡 ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
144, 5, 13syl2anc 585 . 2 (πœ‘ β†’ (β„©π‘₯ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}π‘₯ ≀ 𝑦 ∧ βˆ€π‘§ ∈ 𝐡 (βˆ€π‘¦ ∈ {𝑋, π‘Œ}𝑧 ≀ 𝑦 β†’ 𝑧 ≀ π‘₯))) = (β„©π‘₯ ∈ 𝐡 ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
156, 11, 143eqtrd 2780 1 (πœ‘ β†’ (𝑋 ∧ π‘Œ) = (β„©π‘₯ ∈ 𝐡 ((π‘₯ ≀ 𝑋 ∧ π‘₯ ≀ π‘Œ) ∧ βˆ€π‘§ ∈ 𝐡 ((𝑧 ≀ 𝑋 ∧ 𝑧 ≀ π‘Œ) β†’ 𝑧 ≀ π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  βˆ€wral 3062  {cpr 4567   class class class wbr 5081  β€˜cfv 6454  β„©crio 7259  (class class class)co 7303  Basecbs 16953  lecple 17010  glbcglb 18069  meetcmee 18071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5496  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-riota 7260  df-ov 7306  df-oprab 7307  df-glb 18106  df-meet 18108
This theorem is referenced by:  meetlem  18156
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