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Theorem tgptsmscls 23583
Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 23543, 0nsg 19021. (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 23511 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5 eqid 2731 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
650subg 19003 . . . . . . . . . 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 23542 . . . . . . . . 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 2731 . . . . . . . . 9 (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
1311, 12eqger 19030 . . . . . . . 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 23513 . . . . . . . . . . 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 23568 . . . . . . . . 9 (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
2120sselda 3978 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥𝐵)
22 tgptsmscls.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐺 tsums 𝐹))
2320, 22sseldd 3979 . . . . . . . . 9 (𝜑𝑋𝐵)
2423adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋𝐵)
25 eqid 2731 . . . . . . . . . 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 23582 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ (𝐺 tsums (𝐹f (-g𝐺)𝐹)))
3228ffvelcdmda 7071 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (𝐹𝑘) ∈ 𝐵)
3328feqmptd 6946 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
3427, 32, 32, 33, 33offval2 7673 . . . . . . . . . . . 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 18881 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3735, 32, 36syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3837mpteq2dva 5241 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))) = (𝑘𝐴 ↦ (0g𝐺)))
3934, 38eqtrd 2771 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹f (-g𝐺)𝐹) = (𝑘𝐴 ↦ (0g𝐺)))
4039oveq2d 7409 . . . . . . . . . 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 18825 . . . . . . . . . . . . . 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 7099 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)):𝐴𝐵)
46 fconstmpt 5730 . . . . . . . . . . . 12 (𝐴 × {(0g𝐺)}) = (𝑘𝐴 ↦ (0g𝐺))
47 fvexd 6893 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐺) ∈ V)
4818, 47fczfsuppd 9364 . . . . . . . . . . . . 13 (𝜑 → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
4948adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
5046, 49eqbrtrrid 5177 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)) finSupp (0g𝐺))
5111, 5, 26, 41, 27, 45, 50, 8tsmsgsum 23572 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘𝐴 ↦ (0g𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}))
52 cmnmnd 19629 . . . . . . . . . . . . . 14 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5326, 52syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
545gsumz 18692 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5553, 27, 54syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5655sneqd 4634 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))} = {(0g𝐺)})
5756fveq2d 6882 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}) = ((cls‘𝐽)‘{(0g𝐺)}))
5840, 51, 573eqtrd 2775 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹f (-g𝐺)𝐹)) = ((cls‘𝐽)‘{(0g𝐺)}))
5931, 58eleqtrd 2834 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))
60 isabl 19616 . . . . . . . . . 10 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
614, 26, 60sylanbrc 583 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
6211subgss 18979 . . . . . . . . . 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 19663 . . . . . . . . 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 1342 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋)
6714, 66ersym 8698 . . . . . 6 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥)
6812releqg 19027 . . . . . . 7 Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
69 relelec 8731 . . . . . . 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 233 . . . . 5 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})))
72 eqid 2731 . . . . . . 7 ((cls‘𝐽)‘{(0g𝐺)}) = ((cls‘𝐽)‘{(0g𝐺)})
7311, 8, 5, 12, 72snclseqg 23549 . . . . . 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 2834 . . . 4 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
7675ex 413 . . 3 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
7776ssrdv 3984 . 2 (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
7811, 8, 15, 17, 18, 19, 22tsmscls 23571 . 2 (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
7977, 78eqssd 3995 1 (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3473  wss 3944  {csn 4622   class class class wbr 5141  cmpt 5224   × cxp 5667  Rel wrel 5674  wf 6528  cfv 6532  (class class class)co 7393  f cof 7651   Er wer 8683  [cec 8684   finSupp cfsupp 9344  Basecbs 17126  TopOpenctopn 17349  0gc0g 17367   Σg cgsu 17368  Mndcmnd 18602  Grpcgrp 18794  -gcsg 18796  SubGrpcsubg 18972   ~QG cqg 18974  CMndccmn 19612  Abelcabl 19613  TopSpctps 22363  clsccl 22451  TopGrpctgp 23504   tsums ctsu 23559
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-om 7839  df-1st 7957  df-2nd 7958  df-supp 8129  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-ec 8688  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-fsupp 9345  df-oi 9487  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-n0 12455  df-z 12541  df-uz 12805  df-fz 13467  df-fzo 13610  df-seq 13949  df-hash 14273  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-0g 17369  df-gsum 17370  df-topgen 17371  df-plusf 18542  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-mhm 18647  df-submnd 18648  df-grp 18797  df-minusg 18798  df-sbg 18799  df-subg 18975  df-eqg 18977  df-ghm 19056  df-cntz 19147  df-cmn 19614  df-abl 19615  df-fbas 20875  df-fg 20876  df-top 22325  df-topon 22342  df-topsp 22364  df-bases 22378  df-cld 22452  df-ntr 22453  df-cls 22454  df-nei 22531  df-cn 22660  df-cnp 22661  df-tx 22995  df-hmeo 23188  df-fil 23279  df-fm 23371  df-flim 23372  df-flf 23373  df-tmd 23505  df-tgp 23506  df-tsms 23560
This theorem is referenced by:  tgptsmscld  23584
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