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Theorem gsumval1 18608
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 2731 . . 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 4079 . . . 4 𝑂𝐵
10 fss 6733 . . . 4 ((𝐹:𝐴𝑂𝑂𝐵) → 𝐹:𝐴𝐵)
118, 9, 10sylancl 584 . . 3 (𝜑𝐹:𝐴𝐵)
121, 2, 3, 4, 5, 6, 7, 11gsumval 18602 . 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 6723 . . 3 (𝐹:𝐴𝑂 → ran 𝐹𝑂)
14 iftrue 4533 . . 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 2770 1 (𝜑 → (𝐺 Σg 𝐹) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wex 1779  wcel 2104  wral 3059  wrex 3068  {crab 3430  Vcvv 3472  cdif 3944  wss 3947  ifcif 4527  ccnv 5674  ran crn 5676  cima 5678  ccom 5679  cio 6492  wf 6538  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7411  1c1 11113  cuz 12826  ...cfz 13488  seqcseq 13970  chash 14294  Basecbs 17148  +gcplusg 17201  0gc0g 17389   Σg cgsu 17390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seq 13971  df-gsum 17392
This theorem is referenced by:  gsum0  18609  gsumval2  18611  gsumz  18753
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