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Theorem ustex3sym 23713
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 23712 . . . 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 23710 . . . . . . . . 9 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑀 ∈ π‘ˆ) β†’ 𝑀 βŠ† (𝑀 ∘ 𝑀))
74, 5, 6syl2anc 584 . . . . . . . 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 5854 . . . . . . . . . 10 ((𝑀 ∘ 𝑀) βŠ† 𝑣 β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑀 ∘ 𝑣))
109adantl 482 . . . . . . . . 9 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑀 ∘ 𝑣))
11 sstr 3989 . . . . . . . . . 10 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ 𝑀 βŠ† 𝑣)
12 coss1 5853 . . . . . . . . . 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 3991 . . . . . . . 8 ((𝑀 βŠ† (𝑀 ∘ 𝑀) ∧ (𝑀 ∘ 𝑀) βŠ† 𝑣) β†’ (𝑀 ∘ (𝑀 ∘ 𝑀)) βŠ† (𝑣 ∘ 𝑣))
157, 8, 14syl2anc 584 . . . . . . 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 3991 . . . . . 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 512 . . . . 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 413 . . . 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 3168 . . 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 23706 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘£ ∈ π‘ˆ (𝑣 ∘ 𝑣) βŠ† 𝑉)
2321, 22r19.29a 3162 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3947  β—‘ccnv 5674   ∘ ccom 5679  β€˜cfv 6540  UnifOncust 23695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-ust 23696
This theorem is referenced by:  utopreg  23748
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