Theorem ustex3sym 24135

Description: In an uniform structure, for any entourage 𝑉, there exists a symmetrical entourage smaller than a third of 𝑉. (Contributed by Thierry Arnoux, 16-Jan-2018.)

Assertion

Ref	Expression
ustex3sym	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉))

Distinct variable groups:   𝑤,𝑈   𝑤,𝑉   𝑤,𝑋

Proof of Theorem ustex3sym

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ustex2sym 24134	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣))
2	1	ad4ant13 750	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣))
3		simprl 770	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → ◡𝑤 = 𝑤)
4		simp-5l 784	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → 𝑈 ∈ (UnifOn‘𝑋))
5		simplr 768	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → 𝑤 ∈ 𝑈)
6		ustssco 24132	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑤 ∘ 𝑤))
7	4, 5, 6	syl2anc 583	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → 𝑤 ⊆ (𝑤 ∘ 𝑤))
8		simprr 772	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
9		coss2 5859	. . . . . . . . . 10 ⊢ ((𝑤 ∘ 𝑤) ⊆ 𝑣 → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ (𝑤 ∘ 𝑣))
10	9	adantl 481	. . . . . . . . 9 ⊢ ((𝑤 ⊆ (𝑤 ∘ 𝑤) ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ (𝑤 ∘ 𝑣))
11		sstr 3988	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ (𝑤 ∘ 𝑤) ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → 𝑤 ⊆ 𝑣)
12		coss1 5858	. . . . . . . . . 10 ⊢ (𝑤 ⊆ 𝑣 → (𝑤 ∘ 𝑣) ⊆ (𝑣 ∘ 𝑣))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ ((𝑤 ⊆ (𝑤 ∘ 𝑤) ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → (𝑤 ∘ 𝑣) ⊆ (𝑣 ∘ 𝑣))
14	10, 13	sstrd 3990	. . . . . . . 8 ⊢ ((𝑤 ⊆ (𝑤 ∘ 𝑤) ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ (𝑣 ∘ 𝑣))
15	7, 8, 14	syl2anc 583	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ (𝑣 ∘ 𝑣))
16		simpllr 775	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → (𝑣 ∘ 𝑣) ⊆ 𝑉)
17	15, 16	sstrd 3990	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉)
18	3, 17	jca 511	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉))
19	18	ex 412	. . . 4 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) → ((◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉)))
20	19	reximdva 3165	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → (∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉)))
21	2, 20	mpd 15	. 2 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉))
22		ustexhalf 24128	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑣 ∈ 𝑈 (𝑣 ∘ 𝑣) ⊆ 𝑉)
23	21, 22	r19.29a 3159	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3067   ⊆ wss 3947  ^◡ccnv 5677   ∘ ccom 5682  ‘cfv 6548  UnifOncust 24117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6500  df-fun 6550  df-fv 6556  df-ust 24118

This theorem is referenced by:  utopreg  24170