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Theorem tgptsmscls 22173
Description: A sum in a topological group is uniquely determined up to a coset of cls({0}), which is a normal subgroup by clsnsg 22133, 0nsg 17847. (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 466 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp)
3 tgpgrp 22102 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
42, 3syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Grp)
5 eqid 2771 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
650subg 17827 . . . . . . . . . 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 22132 . . . . . . . . 9 ((𝐺 ∈ TopGrp ∧ {(0g𝐺)} ∈ (SubGrp‘𝐺)) → ((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺))
102, 7, 9syl2anc 573 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(0g𝐺)}) ∈ (SubGrp‘𝐺))
11 tgptsmscls.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
12 eqid 2771 . . . . . . . . 9 (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
1311, 12eqger 17852 . . . . . . . 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 22104 . . . . . . . . . . 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 22158 . . . . . . . . 9 (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵)
2120sselda 3752 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥𝐵)
22 tgptsmscls.x . . . . . . . . . 10 (𝜑𝑋 ∈ (𝐺 tsums 𝐹))
2320, 22sseldd 3753 . . . . . . . . 9 (𝜑𝑋𝐵)
2423adantr 466 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋𝐵)
25 eqid 2771 . . . . . . . . . 10 (-g𝐺) = (-g𝐺)
2615adantr 466 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd)
2718adantr 466 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴𝑉)
2819adantr 466 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴𝐵)
2922adantr 466 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋 ∈ (𝐺 tsums 𝐹))
30 simpr 471 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹))
3111, 25, 26, 2, 27, 28, 28, 29, 30tsmssub 22172 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ (𝐺 tsums (𝐹𝑓 (-g𝐺)𝐹)))
3228ffvelrnda 6502 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (𝐹𝑘) ∈ 𝐵)
3328feqmptd 6391 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
3427, 32, 32, 33, 33offval2 7061 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹𝑓 (-g𝐺)𝐹) = (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))))
354adantr 466 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → 𝐺 ∈ Grp)
3611, 5, 25grpsubid 17707 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3735, 32, 36syl2anc 573 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → ((𝐹𝑘)(-g𝐺)(𝐹𝑘)) = (0g𝐺))
3837mpteq2dva 4878 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ ((𝐹𝑘)(-g𝐺)(𝐹𝑘))) = (𝑘𝐴 ↦ (0g𝐺)))
3934, 38eqtrd 2805 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐹𝑓 (-g𝐺)𝐹) = (𝑘𝐴 ↦ (0g𝐺)))
4039oveq2d 6809 . . . . . . . . . 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 17658 . . . . . . . . . . . . . 14 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
434, 42syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (0g𝐺) ∈ 𝐵)
4443adantr 466 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) ∧ 𝑘𝐴) → (0g𝐺) ∈ 𝐵)
45 eqid 2771 . . . . . . . . . . . 12 (𝑘𝐴 ↦ (0g𝐺)) = (𝑘𝐴 ↦ (0g𝐺))
4644, 45fmptd 6527 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)):𝐴𝐵)
47 fconstmpt 5303 . . . . . . . . . . . 12 (𝐴 × {(0g𝐺)}) = (𝑘𝐴 ↦ (0g𝐺))
48 fvexd 6344 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐺) ∈ V)
4918, 48fczfsuppd 8449 . . . . . . . . . . . . 13 (𝜑 → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
5049adantr 466 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐴 × {(0g𝐺)}) finSupp (0g𝐺))
5147, 50syl5eqbrr 4822 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑘𝐴 ↦ (0g𝐺)) finSupp (0g𝐺))
5211, 5, 26, 41, 27, 46, 51, 8tsmsgsum 22162 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝑘𝐴 ↦ (0g𝐺))) = ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}))
53 cmnmnd 18415 . . . . . . . . . . . . . 14 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5426, 53syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Mnd)
555gsumz 17582 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5654, 27, 55syl2anc 573 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 Σg (𝑘𝐴 ↦ (0g𝐺))) = (0g𝐺))
5756sneqd 4328 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → {(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))} = {(0g𝐺)})
5857fveq2d 6336 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{(𝐺 Σg (𝑘𝐴 ↦ (0g𝐺)))}) = ((cls‘𝐽)‘{(0g𝐺)}))
5940, 52, 583eqtrd 2809 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums (𝐹𝑓 (-g𝐺)𝐹)) = ((cls‘𝐽)‘{(0g𝐺)}))
6031, 59eleqtrd 2852 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))
61 isabl 18404 . . . . . . . . . 10 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
624, 26, 61sylanbrc 572 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ Abel)
6311subgss 17803 . . . . . . . . . 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 18447 . . . . . . . . 9 ((𝐺 ∈ Abel ∧ ((cls‘𝐽)‘{(0g𝐺)}) ⊆ 𝐵) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋 ↔ (𝑥𝐵𝑋𝐵 ∧ (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))))
6662, 64, 65syl2anc 573 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → (𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋 ↔ (𝑥𝐵𝑋𝐵 ∧ (𝑋(-g𝐺)𝑥) ∈ ((cls‘𝐽)‘{(0g𝐺)}))))
6721, 24, 60, 66mpbir3and 1427 . . . . . . 7 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑋)
6814, 67ersym 7908 . . . . . 6 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑋(𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))𝑥)
6912releqg 17849 . . . . . . 7 Rel (𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)}))
70 relelec 7939 . . . . . . 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 2771 . . . . . . 7 ((cls‘𝐽)‘{(0g𝐺)}) = ((cls‘𝐽)‘{(0g𝐺)})
7411, 8, 5, 12, 73snclseqg 22139 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑋𝐵) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = ((cls‘𝐽)‘{𝑋}))
752, 24, 74syl2anc 573 . . . . 5 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → [𝑋](𝐺 ~QG ((cls‘𝐽)‘{(0g𝐺)})) = ((cls‘𝐽)‘{𝑋}))
7672, 75eleqtrd 2852 . . . 4 ((𝜑𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋}))
7776ex 397 . . 3 (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → 𝑥 ∈ ((cls‘𝐽)‘{𝑋})))
7877ssrdv 3758 . 2 (𝜑 → (𝐺 tsums 𝐹) ⊆ ((cls‘𝐽)‘{𝑋}))
7911, 8, 15, 17, 18, 19, 22tsmscls 22161 . 2 (𝜑 → ((cls‘𝐽)‘{𝑋}) ⊆ (𝐺 tsums 𝐹))
8078, 79eqssd 3769 1 (𝜑 → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723  {csn 4316   class class class wbr 4786  cmpt 4863   × cxp 5247  Rel wrel 5254  wf 6027  cfv 6031  (class class class)co 6793  𝑓 cof 7042   Er wer 7893  [cec 7894   finSupp cfsupp 8431  Basecbs 16064  TopOpenctopn 16290  0gc0g 16308   Σg cgsu 16309  Mndcmnd 17502  Grpcgrp 17630  -gcsg 17632  SubGrpcsubg 17796   ~QG cqg 17798  CMndccmn 18400  Abelcabl 18401  TopSpctps 20957  clsccl 21043  TopGrpctgp 22095   tsums ctsu 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-of 7044  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-ec 7898  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-gsum 16311  df-topgen 16312  df-plusf 17449  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17799  df-eqg 17801  df-ghm 17866  df-cntz 17957  df-cmn 18402  df-abl 18403  df-fbas 19958  df-fg 19959  df-top 20919  df-topon 20936  df-topsp 20958  df-bases 20971  df-cld 21044  df-ntr 21045  df-cls 21046  df-nei 21123  df-cn 21252  df-cnp 21253  df-tx 21586  df-hmeo 21779  df-fil 21870  df-fm 21962  df-flim 21963  df-flf 21964  df-tmd 22096  df-tgp 22097  df-tsms 22150
This theorem is referenced by:  tgptsmscld  22174
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