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Theorem gsumval1 17198
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 2622 . . 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 3666 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ 𝐵
104, 9eqsstri 3614 . . . 4 𝑂𝐵
11 fss 6013 . . . 4 ((𝐹:𝐴𝑂𝑂𝐵) → 𝐹:𝐴𝐵)
128, 10, 11sylancl 693 . . 3 (𝜑𝐹:𝐴𝐵)
131, 2, 3, 4, 5, 6, 7, 12gsumval 17192 . 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 6010 . . 3 (𝐹:𝐴𝑂 → ran 𝐹𝑂)
15 iftrue 4064 . . 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 2655 1 (𝜑 → (𝐺 Σg 𝐹) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  ifcif 4058  ccnv 5073  ran crn 5075  cima 5077  ccom 5078  cio 5808  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  1c1 9881  cuz 11631  ...cfz 12268  seqcseq 12741  #chash 13057  Basecbs 15781  +gcplusg 15862  0gc0g 16021   Σg cgsu 16022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seq 12742  df-gsum 16024
This theorem is referenced by:  gsum0  17199  gsumval2  17201  gsumz  17295
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