MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istgp2 Structured version   Visualization version   GIF version
Theorem istgp2 23586
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 23573 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
2 tgptps 23575 . . 3 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ TopSp)
3 tgpsubcn.2 . . . 4 𝐽 = (TopOpenβ€˜πΊ)
4 tgpsubcn.3 . . . 4 βˆ’ = (-gβ€˜πΊ)
53, 4tgpsubcn 23585 . . 3 (𝐺 ∈ TopGrp β†’ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
61, 2, 53jca 1128 . 2 (𝐺 ∈ TopGrp β†’ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
7 simp1 1136 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ Grp)
8 grpmnd 18822 . . . . 5 (𝐺 ∈ Grp β†’ 𝐺 ∈ Mnd)
983ad2ant1 1133 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ Mnd)
10 simp2 1137 . . . 4 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ TopSp)
11 eqid 2732 . . . . . . . 8 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
12 eqid 2732 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
13 eqid 2732 . . . . . . . 8 (invgβ€˜πΊ) = (invgβ€˜πΊ)
1473ad2ant1 1133 . . . . . . . 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 1137 . . . . . . . 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 1138 . . . . . . . 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 18892 . . . . . . 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 7482 . . . . . 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 2732 . . . . . . 7 (+π‘“β€˜πΊ) = (+π‘“β€˜πΊ)
2011, 12, 19plusffval 18563 . . . . . 6 (+π‘“β€˜πΊ) = (π‘₯ ∈ (Baseβ€˜πΊ), 𝑦 ∈ (Baseβ€˜πΊ) ↦ (π‘₯(+gβ€˜πΊ)𝑦))
2118, 20eqtr4di 2790 . . . . 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 22427 . . . . . . 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 23163 . . . . . 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 23164 . . . . . . 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 18867 . . . . . . . . . . 11 (𝐺 ∈ Grp β†’ (invgβ€˜πΊ):(Baseβ€˜πΊ)⟢(Baseβ€˜πΊ))
27263ad2ant1 1133 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (invgβ€˜πΊ):(Baseβ€˜πΊ)⟢(Baseβ€˜πΊ))
2827feqmptd 6957 . . . . . . . . 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 2732 . . . . . . . . . . . 12 (0gβ€˜πΊ) = (0gβ€˜πΊ)
3011, 4, 13, 29grpinvval2 18902 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((invgβ€˜πΊ)β€˜π‘₯) = ((0gβ€˜πΊ) βˆ’ π‘₯))
317, 30sylan 580 . . . . . . . . . 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 5247 . . . . . . . . 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 2772 . . . . . . . 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 18846 . . . . . . . . . . 11 (𝐺 ∈ Grp β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
35343ad2ant1 1133 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
3623, 23, 35cnmptc 23157 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ (0gβ€˜πΊ)) ∈ (𝐽 Cn 𝐽))
3723cnmptid 23156 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (π‘₯ ∈ (Baseβ€˜πΊ) ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
38 simp3 1138 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽))
3923, 36, 37, 38cnmpt12f 23161 . . . . . . . 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 2833 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ (invgβ€˜πΊ) ∈ (𝐽 Cn 𝐽))
4123, 23, 25, 40cnmpt21f 23167 . . . . . 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 23170 . . . . 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 2834 . . . 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 23569 . . . 4 (𝐺 ∈ TopMnd ↔ (𝐺 ∈ Mnd ∧ 𝐺 ∈ TopSp ∧ (+π‘“β€˜πΊ) ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)))
459, 10, 43, 44syl3anbrc 1343 . . 3 ((𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ βˆ’ ∈ ((𝐽 Γ—t 𝐽) Cn 𝐽)) β†’ 𝐺 ∈ TopMnd)
463, 13istgp 23572 . . 3 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invgβ€˜πΊ) ∈ (𝐽 Cn 𝐽)))
477, 45, 40, 46syl3anbrc 1343 . 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 1087   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5230  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17140  +gcplusg 17193  TopOpenctopn 17363  0gc0g 17381  +𝑓cplusf 18554  Mndcmnd 18621  Grpcgrp 18815  invgcminusg 18816  -gcsg 18817  TopOnctopon 22403  TopSpctps 22425   Cn ccn 22719   Γ—t ctx 23055  TopMndctmd 23565  TopGrpctgp 23566
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-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  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-iun 4998  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-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-0g 17383  df-topgen 17385  df-plusf 18556  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-top 22387  df-topon 22404  df-topsp 22426  df-bases 22440  df-cn 22722  df-cnp 22723  df-tx 23057  df-tmd 23567  df-tgp 23568
This theorem is referenced by:  distgp  23594  indistgp  23595  qustgplem  23616  ngptgp  24136  cnfldtgp  24376
  Copyright terms: Public domain W3C validator