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Theorem hmeogrp 25537
 Description: Homeomorphisms on a topology is a group for composition. This means from Felix Klein's point of view that a set equipped with a topology is a geometry, namely the so-called rubber sheet geometry. (Contributed by FL, 3-Feb-2008.) (Proof shortened by Mario Carneiro, 31-May-2014.)
Hypothesis
Ref Expression
hmeogrp.1
Assertion
Ref Expression
hmeogrp
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem hmeogrp
StepHypRef Expression
1 hmeogrp.1 . . . 4
2 oveq12 5867 . . . . . 6
32anidms 626 . . . . 5
4 mpt2eq12 5908 . . . . 5
53, 3, 4syl2anc 642 . . . 4
61, 5syl5eq 2327 . . 3
76eleq1d 2349 . 2
8 eqid 2283 . . 3
9 sn0top 16736 . . . 4
109elimel 3617 . . 3
118, 10hmeogrpi 25536 . 2
127, 11dedth 3606 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1623   wcel 1684  c0 3455  cif 3565  csn 3640   ccom 4693  (class class class)co 5858   cmpt2 5860  ctop 16631   chmeo 17444  cgr 20853 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446  df-grpo 20858
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