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Theorem tgptsmscls 24174
Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 24134, 0nsg 19200. (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 480 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp)
3 tgpgrp 24102 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5 eqid 2735 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
650subg 19182 . . . . . . . . . 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 24133 . . . . . . . . 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 2735 . . . . . . . . 9 (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
1311, 12eqger 19209 . . . . . . . 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 24104 . . . . . . . . . . 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 24159 . . . . . . . . 9 (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
2120sselda 3995 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥𝐵)
22 tgptsmscls.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐺 tsums 𝐹))
2320, 22sseldd 3996 . . . . . . . . 9 (𝜑𝑋𝐵)
2423adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋𝐵)
25 eqid 2735 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
2615adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd)
2718adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴𝑉)
2819adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴𝐵)
2922adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹))
30 simpr 484 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹))
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 24173 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ (𝐺 tsums (𝐹f (-g𝐺)𝐹)))
3228ffvelcdmda 7104 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (𝐹𝑘) ∈ 𝐵)
3328feqmptd 6977 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
3427, 32, 32, 33, 33offval2 7717 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹f (-g𝐺)𝐹) = (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))))
354adantr 480 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → 𝐺 ∈ Grp)
3611, 5, 25grpsubid 19055 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3735, 32, 36syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3837mpteq2dva 5248 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))) = (𝑘𝐴 ↦ (0g𝐺)))
3934, 38eqtrd 2775 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹f (-g𝐺)𝐹) = (𝑘𝐴 ↦ (0g𝐺)))
4039oveq2d 7447 . . . . . . . . . 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 18996 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
434, 42syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (0g𝐺) ∈ 𝐵)
4443adantr 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (0g𝐺) ∈ 𝐵)
4544fmpttd 7135 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)):𝐴𝐵)
46 fconstmpt 5751 . . . . . . . . . . . 12 (𝐴 × {(0g𝐺)}) = (𝑘𝐴 ↦ (0g𝐺))
47 fvexd 6922 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐺) ∈ V)
4818, 47fczfsuppd 9424 . . . . . . . . . . . . 13 (𝜑 → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
4948adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
5046, 49eqbrtrrid 5184 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)) finSupp (0g𝐺))
5111, 5, 26, 41, 27, 45, 50, 8tsmsgsum 24163 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘𝐴 ↦ (0g𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}))
52 cmnmnd 19830 . . . . . . . . . . . . . 14 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5326, 52syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
545gsumz 18862 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5553, 27, 54syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5655sneqd 4643 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))} = {(0g𝐺)})
5756fveq2d 6911 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}) = ((cls‘𝐽)‘{(0g𝐺)}))
5840, 51, 573eqtrd 2779 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹f (-g𝐺)𝐹)) = ((cls‘𝐽)‘{(0g𝐺)}))
5931, 58eleqtrd 2841 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))
60 isabl 19817 . . . . . . . . . 10 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
614, 26, 60sylanbrc 583 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
6211subgss 19158 . . . . . . . . . 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 19867 . . . . . . . . 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 1341 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋)
6714, 66ersym 8756 . . . . . 6 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥)
6812releqg 19206 . . . . . . 7 Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
69 relelec 8791 . . . . . . 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 234 . . . . 5 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})))
72 eqid 2735 . . . . . . 7 ((cls‘𝐽)‘{(0g𝐺)}) = ((cls‘𝐽)‘{(0g𝐺)})
7311, 8, 5, 12, 72snclseqg 24140 . . . . . 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 2841 . . . 4 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
7675ex 412 . . 3 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
7776ssrdv 4001 . 2 (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
7811, 8, 15, 17, 18, 19, 22tsmscls 24162 . 2 (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
7977, 78eqssd 4013 1 (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  Rel wrel 5694  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695   Er wer 8741  [cec 8742   finSupp cfsupp 9399  Basecbs 17245  TopOpenctopn 17468  0gc0g 17486   Σg cgsu 17487  Mndcmnd 18760  Grpcgrp 18964  -gcsg 18966  SubGrpcsubg 19151   ~QG cqg 19153  CMndccmn 19813  Abelcabl 19814  TopSpctps 22954  clsccl 23042  TopGrpctgp 24095   tsums ctsu 24150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-gsum 17489  df-topgen 17490  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-eqg 19156  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cn 23251  df-cnp 23252  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-tmd 24096  df-tgp 24097  df-tsms 24151
This theorem is referenced by:  tgptsmscld  24175
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