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Theorem ustex3sym 23585
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 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† 𝑉))
Distinct variable groups:   𝑀,π‘ˆ   𝑀,𝑉   𝑀,𝑋

Proof of Theorem ustex3sym
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 ustex2sym 23584 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑣 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣))
21ad4ant13 750 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣))
3 simprl 770 . . . . . 6 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ ◑𝑀 = 𝑀)
4 simp-5l 784 . . . . . . . . 9 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ π‘ˆ ∈ (UnifOnβ€˜π‘‹))
5 simplr 768 . . . . . . . . 9 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ 𝑀 ∈ π‘ˆ)
6 ustssco 23582 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑀 ∈ π‘ˆ) β†’ 𝑀 βŠ† (𝑀 ∘ 𝑀))
74, 5, 6syl2anc 585 . . . . . . . 8 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ 𝑀 βŠ† (𝑀 ∘ 𝑀))
8 simprr 772 . . . . . . . 8 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ (𝑀 ∘ 𝑀) βŠ† 𝑣)
9 coss2 5813 . . . . . . . . . 10 ((𝑀 ∘ 𝑀) βŠ† 𝑣 β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑀 ∘ 𝑣))
109adantl 483 . . . . . . . . 9 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑀 ∘ 𝑣))
11 sstr 3953 . . . . . . . . . 10 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ 𝑀 βŠ† 𝑣)
12 coss1 5812 . . . . . . . . . 10 (𝑀 βŠ† 𝑣 β†’ (𝑀 ∘ 𝑣) βŠ† (𝑣 ∘ 𝑣))
1311, 12syl 17 . . . . . . . . 9 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (𝑀 ∘ 𝑣) βŠ† (𝑣 ∘ 𝑣))
1410, 13sstrd 3955 . . . . . . . 8 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑣 ∘ 𝑣))
157, 8, 14syl2anc 585 . . . . . . 7 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑣 ∘ 𝑣))
16 simpllr 775 . . . . . . 7 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ (𝑣 ∘ 𝑣) βŠ† 𝑉)
1715, 16sstrd 3955 . . . . . 6 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† 𝑉)
183, 17jca 513 . . . . 5 ((((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) ∧ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣)) β†’ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† 𝑉))
1918ex 414 . . . 4 (((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) ∧ 𝑀 ∈ π‘ˆ) β†’ ((◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† 𝑉)))
2019reximdva 3162 . . 3 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) β†’ (βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† 𝑉)))
212, 20mpd 15 . 2 ((((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) ∧ 𝑣 ∈ π‘ˆ) ∧ (𝑣 ∘ 𝑣) βŠ† 𝑉) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† 𝑉))
22 ustexhalf 23578 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘£ ∈ π‘ˆ (𝑣 ∘ 𝑣) βŠ† 𝑉)
2321, 22r19.29a 3156 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† 𝑉))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   βŠ† wss 3911  β—‘ccnv 5633   ∘ ccom 5638  β€˜cfv 6497  UnifOncust 23567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-ust 23568
This theorem is referenced by:  utopreg  23620
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