Theorem meetval2 18028

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	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
meetval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
meetval2	⊢ (𝜑 → (𝑋 ∧ 𝑌) = (℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 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.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
2		meetval2.m	. . 3 ⊢ ∧ = (meet‘𝐾)
3		meetval2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
4		meetval2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		meetval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	1, 2, 3, 4, 5	meetval 18024	. 2 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
7		meetval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
8		meetval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
9		biid 260	. . 3 ⊢ ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
10	4, 5	prssd 4752	. . 3 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ 𝐵)
11	7, 8, 1, 9, 3, 10	glbval 18002	. 2 ⊢ (𝜑 → ((glb‘𝐾)‘{𝑋, 𝑌}) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
12	7, 8, 2, 3, 4, 5	meetval2lem 18027	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
13	12	riotabidv 7214	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
14	4, 5, 13	syl2anc 583	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
15	6, 11, 14	3eqtrd 2782	1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = (℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {cpr 4560   class class class wbr 5070  ‘cfv 6418  ℩crio 7211  (class class class)co 7255  Basecbs 16840  lecple 16895  glbcglb 17943  meetcmee 17945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-glb 17980  df-meet 17982

This theorem is referenced by:  meetlem  18030