Theorem joinlem 18393

Description: Lemma for join properties. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)

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)
joinlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )

Assertion

Ref	Expression
joinlem	⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))

Distinct variable groups:   𝑧,𝐵   𝑧, ∨   𝑧,𝐾   𝑧,𝑋   𝑧,𝑌

Allowed substitution hints:   𝜑(𝑧)   ≤ (𝑧)   𝑉(𝑧)

Proof of Theorem joinlem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		joinval2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		joinval2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		joinval2.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		joinval2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		joinval2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		joinval2.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		joinlem.e	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
8	1, 2, 3, 4, 5, 6, 7	joineu 18392	. . . 4 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
9		riotasbc 7380	. . . 4 ⊢ (∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)) → [(℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))) / 𝑥]((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → [(℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))) / 𝑥]((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
11	1, 2, 3, 4, 5, 6	joinval2 18391	. . . 4 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
12	11	sbceq1d 3770	. . 3 ⊢ (𝜑 → ([(𝑋 ∨ 𝑌) / 𝑥]((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ [(℩𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))) / 𝑥]((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
13	10, 12	mpbird 257	. 2 ⊢ (𝜑 → [(𝑋 ∨ 𝑌) / 𝑥]((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
14		ovex 7438	. . 3 ⊢ (𝑋 ∨ 𝑌) ∈ V
15		breq2 5123	. . . . 5 ⊢ (𝑥 = (𝑋 ∨ 𝑌) → (𝑋 ≤ 𝑥 ↔ 𝑋 ≤ (𝑋 ∨ 𝑌)))
16		breq2 5123	. . . . 5 ⊢ (𝑥 = (𝑋 ∨ 𝑌) → (𝑌 ≤ 𝑥 ↔ 𝑌 ≤ (𝑋 ∨ 𝑌)))
17	15, 16	anbi12d 632	. . . 4 ⊢ (𝑥 = (𝑋 ∨ 𝑌) → ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ↔ (𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌))))
18		breq1 5122	. . . . . 6 ⊢ (𝑥 = (𝑋 ∨ 𝑌) → (𝑥 ≤ 𝑧 ↔ (𝑋 ∨ 𝑌) ≤ 𝑧))
19	18	imbi2d 340	. . . . 5 ⊢ (𝑥 = (𝑋 ∨ 𝑌) → (((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
20	19	ralbidv 3163	. . . 4 ⊢ (𝑥 = (𝑋 ∨ 𝑌) → (∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
21	17, 20	anbi12d 632	. . 3 ⊢ (𝑥 = (𝑋 ∨ 𝑌) → (((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧))))
22	14, 21	sbcie 3807	. 2 ⊢ ([(𝑋 ∨ 𝑌) / 𝑥]((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
23	13, 22	sylib 218	1 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃!wreu 3357  [wsbc 3765  ⟨cop 4607   class class class wbr 5119  dom cdm 5654  ‘cfv 6531  ℩crio 7361  (class class class)co 7405  Basecbs 17228  lecple 17278  joincjn 18323

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-lub 18356  df-join 18358

This theorem is referenced by:  lejoin1  18394  lejoin2  18395  joinle  18396