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

Theorem istmd 23266
Description: The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
Hypotheses
Ref Expression
istmd.1 𝐹 = (+π‘“β€˜πΊ)
istmd.2 𝐽 = (TopOpenβ€˜πΊ)
Assertion
Ref Expression
istmd (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))

Proof of Theorem istmd
Dummy variables 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3908 . . 3 (𝐺 ∈ (Mnd ∩ TopSp) ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp))
21anbi1i 625 . 2 ((𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
3 fvexd 6815 . . . 4 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) ∈ V)
4 simpl 484 . . . . . . 7 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑓 = 𝐺)
54fveq2d 6804 . . . . . 6 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (+π‘“β€˜π‘“) = (+π‘“β€˜πΊ))
6 istmd.1 . . . . . 6 𝐹 = (+π‘“β€˜πΊ)
75, 6eqtr4di 2794 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (+π‘“β€˜π‘“) = 𝐹)
8 id 22 . . . . . . . 8 (𝑗 = (TopOpenβ€˜π‘“) β†’ 𝑗 = (TopOpenβ€˜π‘“))
9 fveq2 6800 . . . . . . . . 9 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = (TopOpenβ€˜πΊ))
10 istmd.2 . . . . . . . . 9 𝐽 = (TopOpenβ€˜πΊ)
119, 10eqtr4di 2794 . . . . . . . 8 (𝑓 = 𝐺 β†’ (TopOpenβ€˜π‘“) = 𝐽)
128, 11sylan9eqr 2798 . . . . . . 7 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ 𝑗 = 𝐽)
1312, 12oveq12d 7321 . . . . . 6 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ (𝑗 Γ—t 𝑗) = (𝐽 Γ—t 𝐽))
1413, 12oveq12d 7321 . . . . 5 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ ((𝑗 Γ—t 𝑗) Cn 𝑗) = ((𝐽 Γ—t 𝐽) Cn 𝐽))
157, 14eleq12d 2831 . . . 4 ((𝑓 = 𝐺 ∧ 𝑗 = (TopOpenβ€˜π‘“)) β†’ ((+π‘“β€˜π‘“) ∈ ((𝑗 Γ—t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
163, 15sbcied 3766 . . 3 (𝑓 = 𝐺 β†’ ([(TopOpenβ€˜π‘“) / 𝑗](+π‘“β€˜π‘“) ∈ ((𝑗 Γ—t 𝑗) Cn 𝑗) ↔ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
17 df-tmd 23264 . . 3 TopMnd = {𝑓 ∈ (Mnd ∩ TopSp) ∣ [(TopOpenβ€˜π‘“) / 𝑗](+π‘“β€˜π‘“) ∈ ((𝑗 Γ—t 𝑗) Cn 𝑗)}
1816, 17elrab2 3632 . 2 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ (Mnd ∩ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
19 df-3an 1089 . 2 ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ↔ ((𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp) ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
202, 18, 193bitr4i 304 1 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ 𝐹 ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104  Vcvv 3437  [wsbc 3721   ∩ cin 3891  β€˜cfv 6454  (class class class)co 7303  TopOpenctopn 17173  +𝑓cplusf 18364  Mndcmnd 18426  TopSpctps 22122   Cn ccn 22416   Γ—t ctx 22752  TopMndctmd 23262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-nul 5239
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-iota 6406  df-fv 6462  df-ov 7306  df-tmd 23264
This theorem is referenced by:  tmdmnd  23267  tmdtps  23268  tmdcn  23275  istgp2  23283  oppgtmd  23289  efmndtmd  23293  submtmd  23296  prdstmdd  23316  nrgtrg  23895  mhmhmeotmd  31918  xrge0tmdALT  31937
  Copyright terms: Public domain W3C validator