Theorem ustex3sym 22969

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 22968	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣))
2	1	ad4ant13 751	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣))
3		simprl 771	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → ◡𝑤 = 𝑤)
4		simp-5l 785	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → 𝑈 ∈ (UnifOn‘𝑋))
5		simplr 769	. . . . . . . . 9 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → 𝑤 ∈ 𝑈)
6		ustssco 22966	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑤 ∘ 𝑤))
7	4, 5, 6	syl2anc 587	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → 𝑤 ⊆ (𝑤 ∘ 𝑤))
8		simprr 773	. . . . . . . 8 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ (𝑣 ∘ 𝑣) ⊆ 𝑉) ∧ 𝑤 ∈ 𝑈) ∧ (◡𝑤 = 𝑤 ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣)) → (𝑤 ∘ 𝑤) ⊆ 𝑣)
9		coss2 5699	. . . . . . . . . 10 ⊢ ((𝑤 ∘ 𝑤) ⊆ 𝑣 → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ (𝑤 ∘ 𝑣))
10	9	adantl 485	. . . . . . . . 9 ⊢ ((𝑤 ⊆ (𝑤 ∘ 𝑤) ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ (𝑤 ∘ 𝑣))
11		sstr 3885	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ (𝑤 ∘ 𝑤) ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → 𝑤 ⊆ 𝑣)
12		coss1 5698	. . . . . . . . . 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 3887	. . . . . . . 8 ⊢ ((𝑤 ⊆ (𝑤 ∘ 𝑤) ∧ (𝑤 ∘ 𝑤) ⊆ 𝑣) → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ (𝑣 ∘ 𝑣))
15	7, 8, 14	syl2anc 587	. . . . . . 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 776	. . . . . . 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 3887	. . . . . 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 515	. . . . 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 416	. . . 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 3184	. . 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 22962	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑣 ∈ 𝑈 (𝑣 ∘ 𝑣) ⊆ 𝑉)
23	21, 22	r19.29a 3199	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑉))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ∃wrex 3054   ⊆ wss 3843  ^◡ccnv 5524   ∘ ccom 5529  ‘cfv 6339  UnifOncust 22951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-iota 6297  df-fun 6341  df-fv 6347  df-ust 22952

This theorem is referenced by:  utopreg  23004