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Theorem gsumval1 18367
Description: Value of the group sum operation when every element being summed is an identity of 𝐺. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumval1.b 𝐵 = (Base‘𝐺)
gsumval1.z 0 = (0g𝐺)
gsumval1.p + = (+g𝐺)
gsumval1.o 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
gsumval1.g (𝜑𝐺𝑉)
gsumval1.a (𝜑𝐴𝑊)
gsumval1.f (𝜑𝐹:𝐴𝑂)
Assertion
Ref Expression
gsumval1 (𝜑 → (𝐺 Σg 𝐹) = 0 )
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥, + ,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem gsumval1
Dummy variables 𝑓 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval1.b . . 3 𝐵 = (Base‘𝐺)
2 gsumval1.z . . 3 0 = (0g𝐺)
3 gsumval1.p . . 3 + = (+g𝐺)
4 gsumval1.o . . 3 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 eqidd 2739 . . 3 (𝜑 → (𝐹 “ (V ∖ 𝑂)) = (𝐹 “ (V ∖ 𝑂)))
6 gsumval1.g . . 3 (𝜑𝐺𝑉)
7 gsumval1.a . . 3 (𝜑𝐴𝑊)
8 gsumval1.f . . . 4 (𝜑𝐹:𝐴𝑂)
94ssrab3 4015 . . . 4 𝑂𝐵
10 fss 6617 . . . 4 ((𝐹:𝐴𝑂𝑂𝐵) → 𝐹:𝐴𝐵)
118, 9, 10sylancl 586 . . 3 (𝜑𝐹:𝐴𝐵)
121, 2, 3, 4, 5, 6, 7, 11gsumval 18361 . 2 (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂)))))))))
13 frn 6607 . . 3 (𝐹:𝐴𝑂 → ran 𝐹𝑂)
14 iftrue 4465 . . 3 (ran 𝐹𝑂 → if(ran 𝐹𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂)))))))) = 0 )
158, 13, 143syl 18 . 2 (𝜑 → if(ran 𝐹𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂)))))))) = 0 )
1612, 15eqtrd 2778 1 (𝜑 → (𝐺 Σg 𝐹) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  wral 3064  wrex 3065  {crab 3068  Vcvv 3432  cdif 3884  wss 3887  ifcif 4459  ccnv 5588  ran crn 5590  cima 5592  ccom 5593  cio 6389  wf 6429  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275  1c1 10872  cuz 12582  ...cfz 13239  seqcseq 13721  chash 14044  Basecbs 16912  +gcplusg 16962  0gc0g 17150   Σg cgsu 17151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seq 13722  df-gsum 17153
This theorem is referenced by:  gsum0  18368  gsumval2  18370  gsumz  18474
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