MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istgp2 Structured version   Visualization version   GIF version
Theorem istgp2 23595
Description: A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tgpsubcn.2 𝐽 = (TopOpenβ€˜πΊ)
tgpsubcn.3 βˆ’ = (-gβ€˜πΊ)
Assertion
Ref Expression
istgp2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
Proof of Theorem istgp2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgpgrp 23582 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
2 tgptps 23584 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopSp)
3 tgpsubcn.2 . . . 4 𝐽 = (TopOpenβ€˜πΊ)
4 tgpsubcn.3 . . . 4 βˆ’ = (-gβ€˜πΊ)
53, 4tgpsubcn 23594 . . 3 (𝐺 ∈ TopGrp β†’ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
61, 2, 53jca 1129 . 2 (𝐺 ∈ TopGrp β†’ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
7 simp1 1137 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ Grp)
8 grpmnd 18826 . . . . 5 (𝐺 ∈ Grp β†’ 𝐺 ∈ Mnd)
983ad2ant1 1134 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ Mnd)
10 simp2 1138 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ TopSp)
11 eqid 2733 . . . . . . . 8 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
12 eqid 2733 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
13 eqid 2733 . . . . . . . 8 (invgβ€˜πΊ) = (invgβ€˜πΊ)
1473ad2ant1 1134 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ (Baseβ€˜πΊ)) β†’ 𝐺 ∈ Grp)
15 simp2 1138 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ (Baseβ€˜πΊ)) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
16 simp3 1139 . . . . . . . 8 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ (Baseβ€˜πΊ)) β†’ 𝑦 ∈ (Baseβ€˜πΊ))
1711, 12, 4, 13, 14, 15, 16grpsubinv 18896 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ 𝑦 ∈ (Baseβ€˜πΊ)) β†’ (π‘₯ βˆ’ ((invgβ€˜πΊ)β€˜π‘¦)) = (π‘₯(+gβ€˜πΊ)𝑦))
1817mpoeq3dva 7486 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯ βˆ’ ((invgβ€˜πΊ)β€˜π‘¦))) = (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦)))
19 eqid 2733 . . . . . . 7 (+π‘“β€˜πΊ) = (+π‘“β€˜πΊ)
2011, 12, 19plusffval 18567 . . . . . 6 (+π‘“β€˜πΊ) = (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦))
2118, 20eqtr4di 2791 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯ βˆ’ ((invgβ€˜πΊ)β€˜π‘¦))) = (+π‘“β€˜πΊ))
2211, 3istps 22436 . . . . . . 7 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
2310, 22sylib 217 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜(Baseβ€˜πΊ)))
2423, 23cnmpt1st 23172 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
2523, 23cnmpt2nd 23173 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
2611, 13grpinvf 18871 . . . . . . . . . . 11 (𝐺 ∈ Grp β†’ (invgβ€˜πΊ):(Baseβ€˜πΊ)⟢(Baseβ€˜πΊ))
27263ad2ant1 1134 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (invgβ€˜πΊ):(Baseβ€˜πΊ)⟢(Baseβ€˜πΊ))
2827feqmptd 6961 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (invgβ€˜πΊ) = (π‘₯ ∈ (Baseβ€˜πΊ) ↦ ((invgβ€˜πΊ)β€˜π‘₯)))
29 eqid 2733 . . . . . . . . . . . 12 (0gβ€˜πΊ) = (0gβ€˜πΊ)
3011, 4, 13, 29grpinvval2 18906 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((invgβ€˜πΊ)β€˜π‘₯) = ((0gβ€˜πΊ) βˆ’ π‘₯))
317, 30sylan 581 . . . . . . . . . 10 (((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((invgβ€˜πΊ)β€˜π‘₯) = ((0gβ€˜πΊ) βˆ’ π‘₯))
3231mpteq2dva 5249 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ ((invgβ€˜πΊ)β€˜π‘₯)) = (π‘₯ ∈ (Baseβ€˜πΊ) ↦ ((0gβ€˜πΊ) βˆ’ π‘₯)))
3328, 32eqtrd 2773 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (invgβ€˜πΊ) = (π‘₯ ∈ (Baseβ€˜πΊ) ↦ ((0gβ€˜πΊ) βˆ’ π‘₯)))
3411, 29grpidcl 18850 . . . . . . . . . . 11 (𝐺 ∈ Grp β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
35343ad2ant1 1134 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
3623, 23, 35cnmptc 23166 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ (0gβ€˜πΊ)) ∈ (𝐽 Cn 𝐽))
3723cnmptid 23165 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
38 simp3 1139 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
3923, 36, 37, 38cnmpt12f 23170 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ ((0gβ€˜πΊ) βˆ’ π‘₯)) ∈ (𝐽 Cn 𝐽))
4033, 39eqeltrd 2834 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (invgβ€˜πΊ) ∈ (𝐽 Cn 𝐽))
4123, 23, 25, 40cnmpt21f 23176 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ ((invgβ€˜πΊ)β€˜π‘¦)) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
4223, 23, 24, 41, 38cnmpt22f 23179 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯ βˆ’ ((invgβ€˜πΊ)β€˜π‘¦))) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
4321, 42eqeltrrd 2835 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (+π‘“β€˜πΊ) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
4419, 3istmd 23578 . . . 4 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+π‘“β€˜πΊ) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
459, 10, 43, 44syl3anbrc 1344 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ TopMnd)
463, 13istgp 23581 . . 3 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invgβ€˜πΊ) ∈ (𝐽 Cn 𝐽)))
477, 45, 40, 46syl3anbrc 1344 . 2 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ TopGrp)
486, 47impbii 208 1 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5232  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  Basecbs 17144  +gcplusg 17197  TopOpenctopn 17367  0gc0g 17385  +𝑓cplusf 18558  Mndcmnd 18625  Grpcgrp 18819  invgcminusg 18820  -gcsg 18821  TopOnctopon 22412  TopSpctps 22434   Cn ccn 22728   Γ—t ctx 23064  TopMndctmd 23574  TopGrpctgp 23575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-0g 17387  df-topgen 17389  df-plusf 18560  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cn 22731  df-cnp 22732  df-tx 23066  df-tmd 23576  df-tgp 23577
This theorem is referenced by:  distgp  23603  indistgp  23604  qustgplem  23625  ngptgp  24145  cnfldtgp  24385
  Copyright terms: Public domain W3C validator