Theorem meetlem 18296

Description: Lemma for meet properties. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-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)
meetlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )

Assertion

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

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

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

Proof of Theorem meetlem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		meetval2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		meetval2.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		meetval2.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
4		meetval2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		meetval2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		meetval2.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		meetlem.e	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
8	1, 2, 3, 4, 5, 6, 7	meeteu 18295	. . . 4 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))
9		riotasbc 7316	. . . 4 ⊢ (∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)) → [(℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) / 𝑥]((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → [(℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) / 𝑥]((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))
11	1, 2, 3, 4, 5, 6	meetval2 18294	. . . 4 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = (℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
12	11	sbceq1d 3741	. . 3 ⊢ (𝜑 → ([(𝑋 ∧ 𝑌) / 𝑥]((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)) ↔ [(℩𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) / 𝑥]((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
13	10, 12	mpbird 257	. 2 ⊢ (𝜑 → [(𝑋 ∧ 𝑌) / 𝑥]((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))
14		ovex 7374	. . 3 ⊢ (𝑋 ∧ 𝑌) ∈ V
15		breq1 5089	. . . . 5 ⊢ (𝑥 = (𝑋 ∧ 𝑌) → (𝑥 ≤ 𝑋 ↔ (𝑋 ∧ 𝑌) ≤ 𝑋))
16		breq1 5089	. . . . 5 ⊢ (𝑥 = (𝑋 ∧ 𝑌) → (𝑥 ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) ≤ 𝑌))
17	15, 16	anbi12d 632	. . . 4 ⊢ (𝑥 = (𝑋 ∧ 𝑌) → ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ↔ ((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌)))
18		breq2 5090	. . . . . 6 ⊢ (𝑥 = (𝑋 ∧ 𝑌) → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ (𝑋 ∧ 𝑌)))
19	18	imbi2d 340	. . . . 5 ⊢ (𝑥 = (𝑋 ∧ 𝑌) → (((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥) ↔ ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
20	19	ralbidv 3155	. . . 4 ⊢ (𝑥 = (𝑋 ∧ 𝑌) → (∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
21	17, 20	anbi12d 632	. . 3 ⊢ (𝑥 = (𝑋 ∧ 𝑌) → (((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)) ↔ (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)))))
22	14, 21	sbcie 3778	. 2 ⊢ ([(𝑋 ∧ 𝑌) / 𝑥]((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)) ↔ (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))
23	13, 22	sylib 218	1 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃!wreu 3344  [wsbc 3736  ⟨cop 4577   class class class wbr 5086  dom cdm 5611  ‘cfv 6476  ℩crio 7297  (class class class)co 7341  Basecbs 17115  lecple 17163  meetcmee 18213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-glb 18246  df-meet 18248

This theorem is referenced by:  lemeet1  18297  lemeet2  18298  meetle  18299