MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmphtr Structured version   Visualization version   GIF version

Theorem hmphtr 23775
Description: "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmphtr ((𝐽𝐾𝐾𝐿) → 𝐽𝐿)

Proof of Theorem hmphtr
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 23768 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 hmph 23768 . 2 (𝐾𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3 n0 4346 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4 n0 4346 . . 3 ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5 exdistrv 1952 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6 hmeoco 23764 . . . . . 6 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔𝑓) ∈ (𝐽Homeo𝐿))
7 hmphi 23769 . . . . . 6 ((𝑔𝑓) ∈ (𝐽Homeo𝐿) → 𝐽𝐿)
86, 7syl 17 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
98exlimivv 1928 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
105, 9sylbir 234 . . 3 ((∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
113, 4, 10syl2anb 596 . 2 (((𝐽Homeo𝐾) ≠ ∅ ∧ (𝐾Homeo𝐿) ≠ ∅) → 𝐽𝐿)
121, 2, 11syl2anb 596 1 ((𝐽𝐾𝐾𝐿) → 𝐽𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wex 1774  wcel 2099  wne 2930  c0 4322   class class class wbr 5145  ccom 5678  (class class class)co 7416  Homeochmeo 23745  chmph 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-1o 8488  df-map 8849  df-top 22884  df-topon 22901  df-cn 23219  df-hmeo 23747  df-hmph 23748
This theorem is referenced by:  hmpher  23776  xrhmph  24960
  Copyright terms: Public domain W3C validator