Theorem joinval2lem 18552

Description: Lemma for joinval2 18553 and joineu 18554. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 18554 into joinlem 18555?

Hypotheses

Ref	Expression
joinval2.b	⊢ 𝐵 = (Base‘𝐾)
joinval2.l	⊢ ≤ = (le‘𝐾)
joinval2.j	⊢ ∨ = (join‘𝐾)
joinval2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joinval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
joinval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
joinval2lem	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

Distinct variable groups:   𝑥,𝑧,𝐵   𝑥, ∨ ,𝑧   𝑥,𝑦,𝐾,𝑧   𝑦, ≤   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝐵(𝑦)   ∨ (𝑦)   ≤ (𝑥, 𝑧)   𝑉(𝑥, 𝑦, 𝑧)

Proof of Theorem joinval2lem

Step	Hyp	Ref	Expression
1		breq1 5106	. . 3 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥))
2		breq1 5106	. . 3 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑥 ↔ 𝑌 ≤ 𝑥))
3	1, 2	ralprg 4657	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ↔ (𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥)))
4		breq1 5106	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧))
5		breq1 5106	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧))
6	4, 5	ralprg 4657	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 ↔ (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧)))
7	6	imbi1d 344	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
8	7	ralbidv 3186	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
9	3, 8	anbi12d 644	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {cpr 4586   class class class wbr 5103  ‘cfv 6538  Basecbs 17387  lecple 17435  joincjn 18485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104

This theorem is used by:  joinval2  18553  joineu  18554  joindm3  50076