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Theorem gsumval3a 19288
Description: Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by AV, 29-May-2019.)
Hypotheses
Ref Expression
gsumval3.b 𝐵 = (Base‘𝐺)
gsumval3.0 0 = (0g𝐺)
gsumval3.p + = (+g𝐺)
gsumval3.z 𝑍 = (Cntz‘𝐺)
gsumval3.g (𝜑𝐺 ∈ Mnd)
gsumval3.a (𝜑𝐴𝑉)
gsumval3.f (𝜑𝐹:𝐴𝐵)
gsumval3.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3a.t (𝜑𝑊 ∈ Fin)
gsumval3a.n (𝜑𝑊 ≠ ∅)
gsumval3a.w 𝑊 = (𝐹 supp 0 )
gsumval3a.i (𝜑 → ¬ 𝐴 ∈ ran ...)
Assertion
Ref Expression
gsumval3a (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
Distinct variable groups:   𝑥,𝑓, +   𝐴,𝑓,𝑥   𝜑,𝑓,𝑥   𝑥, 0   𝑓,𝐺,𝑥   𝑥,𝑉   𝐵,𝑓,𝑥   𝑓,𝐹,𝑥   𝑓,𝑊,𝑥
Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem gsumval3a
Dummy variables 𝑚 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval3.b . . 3 𝐵 = (Base‘𝐺)
2 gsumval3.0 . . 3 0 = (0g𝐺)
3 gsumval3.p . . 3 + = (+g𝐺)
4 eqid 2737 . . 3 {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}
5 gsumval3a.w . . . . 5 𝑊 = (𝐹 supp 0 )
65a1i 11 . . . 4 (𝜑𝑊 = (𝐹 supp 0 ))
7 gsumval3.f . . . . . 6 (𝜑𝐹:𝐴𝐵)
8 gsumval3.a . . . . . 6 (𝜑𝐴𝑉)
97, 8fexd 7043 . . . . 5 (𝜑𝐹 ∈ V)
102fvexi 6731 . . . . 5 0 ∈ V
11 suppimacnv 7916 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
129, 10, 11sylancl 589 . . . 4 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
13 gsumval3.g . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
141, 2, 3, 4gsumvallem2 18260 . . . . . . . 8 (𝐺 ∈ Mnd → {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1513, 14syl 17 . . . . . . 7 (𝜑 → {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1615eqcomd 2743 . . . . . 6 (𝜑 → { 0 } = {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
1716difeq2d 4037 . . . . 5 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
1817imaeq2d 5929 . . . 4 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
196, 12, 183eqtrd 2781 . . 3 (𝜑𝑊 = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
201, 2, 3, 4, 19, 13, 8, 7gsumval 18149 . 2 (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))))
21 gsumval3a.n . . . 4 (𝜑𝑊 ≠ ∅)
2215sseq2d 3933 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 }))
235a1i 11 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 ))
247, 8jca 515 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴𝐵𝐴𝑉))
2524adantr 484 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴𝐵𝐴𝑉))
26 fex 7042 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2725, 26syl 17 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
2827, 10, 11sylancl 589 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
297ffnd 6546 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
3029adantr 484 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
31 simpr 488 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 })
32 df-f 6384 . . . . . . . . . 10 (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
3330, 31, 32sylanbrc 586 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
34 disjdif 4386 . . . . . . . . 9 ({ 0 } ∩ (V ∖ { 0 })) = ∅
35 fimacnvdisj 6597 . . . . . . . . 9 ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (𝐹 “ (V ∖ { 0 })) = ∅)
3633, 34, 35sylancl 589 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 “ (V ∖ { 0 })) = ∅)
3723, 28, 363eqtrd 2781 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
3837ex 416 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
3922, 38sylbid 243 . . . . 5 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
4039necon3ad 2953 . . . 4 (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
4121, 40mpd 15 . . 3 (𝜑 → ¬ ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
4241iffalsed 4450 . 2 (𝜑 → if(ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))) = if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))))
43 gsumval3a.i . . 3 (𝜑 → ¬ 𝐴 ∈ ran ...)
4443iffalsed 4450 . 2 (𝜑 → if(𝐴 ∈ ran ..., (℩𝑥𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊))))) = (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
4520, 42, 443eqtrd 2781 1 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto𝑊𝑥 = (seq1( + , (𝐹𝑓))‘(♯‘𝑊)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1543  wex 1787  wcel 2110  wne 2940  wral 3061  wrex 3062  {crab 3065  Vcvv 3408  cdif 3863  cin 3865  wss 3866  c0 4237  ifcif 4439  {csn 4541  ccnv 5550  ran crn 5552  cima 5554  ccom 5555  cio 6336   Fn wfn 6375  wf 6376  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213   supp csupp 7903  Fincfn 8626  1c1 10730  cuz 12438  ...cfz 13095  seqcseq 13574  chash 13896  Basecbs 16760  +gcplusg 16802  0gc0g 16944   Σg cgsu 16945  Mndcmnd 18173  Cntzccntz 18709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-seq 13575  df-0g 16946  df-gsum 16947  df-mgm 18114  df-sgrp 18163  df-mnd 18174
This theorem is referenced by:  gsumval3lem2  19291
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