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Theorem gsumval1 17100
 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 2611 . . 3 (𝜑 → (𝐹 “ (V ∖ 𝑂)) = (𝐹 “ (V ∖ 𝑂)))
6 gsumval1.g . . 3 (𝜑𝐺𝑉)
7 gsumval1.a . . 3 (𝜑𝐴𝑊)
8 gsumval1.f . . . 4 (𝜑𝐹:𝐴𝑂)
9 ssrab2 3650 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ 𝐵
104, 9eqsstri 3598 . . . 4 𝑂𝐵
11 fss 5969 . . . 4 ((𝐹:𝐴𝑂𝑂𝐵) → 𝐹:𝐴𝐵)
128, 10, 11sylancl 693 . . 3 (𝜑𝐹:𝐴𝐵)
131, 2, 3, 4, 5, 6, 7, 12gsumval 17094 . 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)))))))))
14 frn 5966 . . 3 (𝐹:𝐴𝑂 → ran 𝐹𝑂)
15 iftrue 4042 . . 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 )
168, 14, 153syl 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 )
1713, 16eqtrd 2644 1 (𝜑 → (𝐺 Σg 𝐹) = 0 )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ifcif 4036  ◡ccnv 5037  ran crn 5039   “ cima 5041   ∘ ccom 5042  ℩cio 5766  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1c1 9816  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-gsum 15926 This theorem is referenced by:  gsum0  17101  gsumval2  17103  gsumz  17197
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