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Theorem ustex3sym 24140
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 24139 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑣 ∈ π‘ˆ) β†’ βˆƒπ‘€ ∈ π‘ˆ (◑𝑀 = 𝑀 ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣))
21ad4ant13 749 . . 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 769 . . . . . 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 783 . . . . . . . . 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 767 . . . . . . . . 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 24137 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑀 ∈ π‘ˆ) β†’ 𝑀 βŠ† (𝑀 ∘ 𝑀))
74, 5, 6syl2anc 582 . . . . . . . 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 771 . . . . . . . 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 5853 . . . . . . . . . 10 ((𝑀 ∘ 𝑀) βŠ† 𝑣 β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑀 ∘ 𝑣))
109adantl 480 . . . . . . . . 9 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑀 ∘ 𝑣))
11 sstr 3981 . . . . . . . . . 10 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ 𝑀 βŠ† 𝑣)
12 coss1 5852 . . . . . . . . . 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 3983 . . . . . . . 8 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑣 ∘ 𝑣))
157, 8, 14syl2anc 582 . . . . . . 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 774 . . . . . . 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 3983 . . . . . 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 510 . . . . 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 411 . . . 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 3158 . . 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 24133 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘£ ∈ π‘ˆ (𝑣 ∘ 𝑣) βŠ† 𝑉)
2321, 22r19.29a 3152 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 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   βŠ† wss 3939  β—‘ccnv 5671   ∘ ccom 5676  β€˜cfv 6543  UnifOncust 24122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-ust 24123
This theorem is referenced by:  utopreg  24175
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