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Theorem tgptsmscls 22685
Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 22645, 0nsg 18259. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
Hypotheses
Ref Expression
tgptsmscls.b 𝐵 = (Base‘𝐺)
tgptsmscls.j 𝐽 = (TopOpen‘𝐺)
tgptsmscls.1 (𝜑𝐺 ∈ CMnd)
tgptsmscls.2 (𝜑𝐺 ∈ TopGrp)
tgptsmscls.a (𝜑𝐴𝑉)
tgptsmscls.f (𝜑𝐹:𝐴𝐵)
tgptsmscls.x (𝜑𝑋 ∈ (𝐺 tsums 𝐹))
Assertion
Ref Expression
tgptsmscls (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))

Proof of Theorem tgptsmscls
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tgptsmscls.2 . . . . . . . . . 10 (𝜑𝐺 ∈ TopGrp)
21adantr 481 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp)
3 tgpgrp 22614 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5 eqid 2818 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
650subg 18242 . . . . . . . . . 10 (𝐺 ∈ Grp → {(0g𝐺)} ∈ (SubGrp‘𝐺))
74, 6syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → {(0g𝐺)} ∈ (SubGrp‘𝐺))
8 tgptsmscls.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
98clssubg 22644 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ {(0g𝐺)} ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺))
102, 7, 9syl2anc 584 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺))
11 tgptsmscls.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
12 eqid 2818 . . . . . . . . 9 (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
1311, 12eqger 18268 . . . . . . . 8 (((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺) → (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) Er 𝐵)
1410, 13syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) Er 𝐵)
15 tgptsmscls.1 . . . . . . . . . 10 (𝜑𝐺 ∈ CMnd)
16 tgptps 22616 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
171, 16syl 17 . . . . . . . . . 10 (𝜑𝐺 ∈ TopSp)
18 tgptsmscls.a . . . . . . . . . 10 (𝜑𝐴𝑉)
19 tgptsmscls.f . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
2011, 15, 17, 18, 19tsmscl 22670 . . . . . . . . 9 (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
2120sselda 3964 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥𝐵)
22 tgptsmscls.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐺 tsums 𝐹))
2320, 22sseldd 3965 . . . . . . . . 9 (𝜑𝑋𝐵)
2423adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋𝐵)
25 eqid 2818 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
2615adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd)
2718adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴𝑉)
2819adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴𝐵)
2922adantr 481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹))
30 simpr 485 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹))
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 22684 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ (𝐺 tsums (𝐹f (-g𝐺)𝐹)))
3228ffvelrnda 6843 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (𝐹𝑘) ∈ 𝐵)
3328feqmptd 6726 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
3427, 32, 32, 33, 33offval2 7415 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹f (-g𝐺)𝐹) = (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))))
354adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → 𝐺 ∈ Grp)
3611, 5, 25grpsubid 18121 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3735, 32, 36syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3837mpteq2dva 5152 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))) = (𝑘𝐴 ↦ (0g𝐺)))
3934, 38eqtrd 2853 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹f (-g𝐺)𝐹) = (𝑘𝐴 ↦ (0g𝐺)))
4039oveq2d 7161 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹f (-g𝐺)𝐹)) = (𝐺 tsums (𝑘𝐴 ↦ (0g𝐺))))
412, 16syl 17 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopSp)
4211, 5grpidcl 18069 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
434, 42syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (0g𝐺) ∈ 𝐵)
4443adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (0g𝐺) ∈ 𝐵)
4544fmpttd 6871 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)):𝐴𝐵)
46 fconstmpt 5607 . . . . . . . . . . . 12 (𝐴 × {(0g𝐺)}) = (𝑘𝐴 ↦ (0g𝐺))
47 fvexd 6678 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐺) ∈ V)
4818, 47fczfsuppd 8839 . . . . . . . . . . . . 13 (𝜑 → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
4948adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
5046, 49eqbrtrrid 5093 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)) finSupp (0g𝐺))
5111, 5, 26, 41, 27, 45, 50, 8tsmsgsum 22674 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘𝐴 ↦ (0g𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}))
52 cmnmnd 18851 . . . . . . . . . . . . . 14 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5326, 52syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
545gsumz 17988 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5553, 27, 54syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5655sneqd 4569 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))} = {(0g𝐺)})
5756fveq2d 6667 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}) = ((cls‘𝐽)‘{(0g𝐺)}))
5840, 51, 573eqtrd 2857 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹f (-g𝐺)𝐹)) = ((cls‘𝐽)‘{(0g𝐺)}))
5931, 58eleqtrd 2912 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))
60 isabl 18839 . . . . . . . . . 10 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
614, 26, 60sylanbrc 583 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
6211subgss 18218 . . . . . . . . . 10 (((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺) → ((cls‘𝐽)‘{(0g𝐺)}) ⊆ 𝐵)
6310, 62syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g𝐺)}) ⊆ 𝐵)
6411, 25, 12eqgabl 18884 . . . . . . . . 9 ((𝐺 ∈ Abel ∧ ((cls‘𝐽)‘{(0g𝐺)}) ⊆ 𝐵) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋 ↔ (𝑥𝐵𝑋𝐵 ∧ (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))))
6561, 63, 64syl2anc 584 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋 ↔ (𝑥𝐵𝑋𝐵 ∧ (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))))
6621, 24, 59, 65mpbir3and 1334 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋)
6714, 66ersym 8290 . . . . . 6 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥)
6812releqg 18265 . . . . . . 7 Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
69 relelec 8323 . . . . . . 7 (Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) → (𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) ↔ 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥))
7068, 69ax-mp 5 . . . . . 6 (𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) ↔ 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥)
7167, 70sylibr 235 . . . . 5 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})))
72 eqid 2818 . . . . . . 7 ((cls‘𝐽)‘{(0g𝐺)}) = ((cls‘𝐽)‘{(0g𝐺)})
7311, 8, 5, 12, 72snclseqg 22651 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑋𝐵) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = ((cls‘𝐽)‘{𝑋}))
742, 24, 73syl2anc 584 . . . . 5 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = ((cls‘𝐽)‘{𝑋}))
7571, 74eleqtrd 2912 . . . 4 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
7675ex 413 . . 3 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
7776ssrdv 3970 . 2 (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
7811, 8, 15, 17, 18, 19, 22tsmscls 22673 . 2 (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
7977, 78eqssd 3981 1 (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  {csn 4557   class class class wbr 5057  cmpt 5137   × cxp 5546  Rel wrel 5553  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396   Er wer 8275  [cec 8276   finSupp cfsupp 8821  Basecbs 16471  TopOpenctopn 16683  0gc0g 16701   Σg cgsu 16702  Mndcmnd 17899  Grpcgrp 18041  -gcsg 18043  SubGrpcsubg 18211   ~QG cqg 18213  CMndccmn 18835  Abelcabl 18836  TopSpctps 21468  clsccl 21554  TopGrpctgp 22607   tsums ctsu 22661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-ec 8280  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-gsum 16704  df-topgen 16705  df-plusf 17839  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-subg 18214  df-eqg 18216  df-ghm 18294  df-cntz 18385  df-cmn 18837  df-abl 18838  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cn 21763  df-cnp 21764  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-tmd 22608  df-tgp 22609  df-tsms 22662
This theorem is referenced by:  tgptsmscld  22686
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