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Theorem tgptsmscls 22146
Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 22106, 0nsg 17832. (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 472 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp)
3 tgpgrp 22075 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5 eqid 2752 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
650subg 17812 . . . . . . . . . 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 22105 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ {(0g𝐺)} ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺))
102, 7, 9syl2anc 696 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺))
11 tgptsmscls.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
12 eqid 2752 . . . . . . . . 9 (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
1311, 12eqger 17837 . . . . . . . 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 22077 . . . . . . . . . . 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 22131 . . . . . . . . 9 (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
2120sselda 3736 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥𝐵)
22 tgptsmscls.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐺 tsums 𝐹))
2320, 22sseldd 3737 . . . . . . . . 9 (𝜑𝑋𝐵)
2423adantr 472 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋𝐵)
25 eqid 2752 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
2615adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd)
2718adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴𝑉)
2819adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴𝐵)
2922adantr 472 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹))
30 simpr 479 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹))
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 22145 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ (𝐺 tsums (𝐹𝑓 (-g𝐺)𝐹)))
3228ffvelrnda 6514 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (𝐹𝑘) ∈ 𝐵)
3328feqmptd 6403 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
3427, 32, 32, 33, 33offval2 7071 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹𝑓 (-g𝐺)𝐹) = (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))))
354adantr 472 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → 𝐺 ∈ Grp)
3611, 5, 25grpsubid 17692 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3735, 32, 36syl2anc 696 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3837mpteq2dva 4888 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))) = (𝑘𝐴 ↦ (0g𝐺)))
3934, 38eqtrd 2786 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹𝑓 (-g𝐺)𝐹) = (𝑘𝐴 ↦ (0g𝐺)))
4039oveq2d 6821 . . . . . . . . . 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 17643 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
434, 42syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (0g𝐺) ∈ 𝐵)
4443adantr 472 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (0g𝐺) ∈ 𝐵)
45 eqid 2752 . . . . . . . . . . . 12 (𝑘𝐴 ↦ (0g𝐺)) = (𝑘𝐴 ↦ (0g𝐺))
4644, 45fmptd 6540 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)):𝐴𝐵)
47 fconstmpt 5312 . . . . . . . . . . . 12 (𝐴 × {(0g𝐺)}) = (𝑘𝐴 ↦ (0g𝐺))
48 fvexd 6356 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐺) ∈ V)
4918, 48fczfsuppd 8450 . . . . . . . . . . . . 13 (𝜑 → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
5049adantr 472 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
5147, 50syl5eqbrr 4832 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)) finSupp (0g𝐺))
5211, 5, 26, 41, 27, 46, 51, 8tsmsgsum 22135 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘𝐴 ↦ (0g𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}))
53 cmnmnd 18400 . . . . . . . . . . . . . 14 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5426, 53syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
555gsumz 17567 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5654, 27, 55syl2anc 696 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5756sneqd 4325 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))} = {(0g𝐺)})
5857fveq2d 6348 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}) = ((cls‘𝐽)‘{(0g𝐺)}))
5940, 52, 583eqtrd 2790 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹𝑓 (-g𝐺)𝐹)) = ((cls‘𝐽)‘{(0g𝐺)}))
6031, 59eleqtrd 2833 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))
61 isabl 18389 . . . . . . . . . 10 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
624, 26, 61sylanbrc 701 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
6311subgss 17788 . . . . . . . . . 10 (((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺) → ((cls‘𝐽)‘{(0g𝐺)}) ⊆ 𝐵)
6410, 63syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g𝐺)}) ⊆ 𝐵)
6511, 25, 12eqgabl 18432 . . . . . . . . 9 ((𝐺 ∈ Abel ∧ ((cls‘𝐽)‘{(0g𝐺)}) ⊆ 𝐵) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋 ↔ (𝑥𝐵𝑋𝐵 ∧ (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))))
6662, 64, 65syl2anc 696 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋 ↔ (𝑥𝐵𝑋𝐵 ∧ (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))))
6721, 24, 60, 66mpbir3and 1425 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋)
6814, 67ersym 7915 . . . . . 6 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥)
6912releqg 17834 . . . . . . 7 Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
70 relelec 7946 . . . . . . 7 (Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) → (𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) ↔ 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥))
7169, 70ax-mp 5 . . . . . 6 (𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) ↔ 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥)
7268, 71sylibr 224 . . . . 5 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})))
73 eqid 2752 . . . . . . 7 ((cls‘𝐽)‘{(0g𝐺)}) = ((cls‘𝐽)‘{(0g𝐺)})
7411, 8, 5, 12, 73snclseqg 22112 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑋𝐵) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = ((cls‘𝐽)‘{𝑋}))
752, 24, 74syl2anc 696 . . . . 5 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = ((cls‘𝐽)‘{𝑋}))
7672, 75eleqtrd 2833 . . . 4 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
7776ex 449 . . 3 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
7877ssrdv 3742 . 2 (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
7911, 8, 15, 17, 18, 19, 22tsmscls 22134 . 2 (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
8078, 79eqssd 3753 1 (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  Vcvv 3332  wss 3707  {csn 4313   class class class wbr 4796  cmpt 4873   × cxp 5256  Rel wrel 5263  wf 6037  cfv 6041  (class class class)co 6805  𝑓 cof 7052   Er wer 7900  [cec 7901   finSupp cfsupp 8432  Basecbs 16051  TopOpenctopn 16276  0gc0g 16294   Σg cgsu 16295  Mndcmnd 17487  Grpcgrp 17615  -gcsg 17617  SubGrpcsubg 17781   ~QG cqg 17783  CMndccmn 18385  Abelcabl 18386  TopSpctps 20930  clsccl 21016  TopGrpctgp 22068   tsums ctsu 22122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-supp 7456  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-ec 7905  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8433  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-n0 11477  df-z 11562  df-uz 11872  df-fz 12512  df-fzo 12652  df-seq 12988  df-hash 13304  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-gsum 16297  df-topgen 16298  df-plusf 17434  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-mhm 17528  df-submnd 17529  df-grp 17618  df-minusg 17619  df-sbg 17620  df-subg 17784  df-eqg 17786  df-ghm 17851  df-cntz 17942  df-cmn 18387  df-abl 18388  df-fbas 19937  df-fg 19938  df-top 20893  df-topon 20910  df-topsp 20931  df-bases 20944  df-cld 21017  df-ntr 21018  df-cls 21019  df-nei 21096  df-cn 21225  df-cnp 21226  df-tx 21559  df-hmeo 21752  df-fil 21843  df-fm 21935  df-flim 21936  df-flf 21937  df-tmd 22069  df-tgp 22070  df-tsms 22123
This theorem is referenced by:  tgptsmscld  22147
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