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Theorem gsumval3a 18690
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 2778 . . 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 . . . . . . 7 (𝜑𝐹:𝐴𝐵)
8 gsumval3.a . . . . . . 7 (𝜑𝐴𝑉)
97, 8jca 507 . . . . . 6 (𝜑 → (𝐹:𝐴𝐵𝐴𝑉))
10 fex 6761 . . . . . 6 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
119, 10syl 17 . . . . 5 (𝜑𝐹 ∈ V)
122fvexi 6460 . . . . 5 0 ∈ V
13 suppimacnv 7587 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
1411, 12, 13sylancl 580 . . . 4 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
15 gsumval3.g . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
161, 2, 3, 4gsumvallem2 17758 . . . . . . . 8 (𝐺 ∈ Mnd → {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1715, 16syl 17 . . . . . . 7 (𝜑 → {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1817eqcomd 2784 . . . . . 6 (𝜑 → { 0 } = {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
1918difeq2d 3951 . . . . 5 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
2019imaeq2d 5720 . . . 4 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
216, 14, 203eqtrd 2818 . . 3 (𝜑𝑊 = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
221, 2, 3, 4, 21, 15, 8, 7gsumval 17657 . 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)))))))
23 gsumval3a.n . . . 4 (𝜑𝑊 ≠ ∅)
2417sseq2d 3852 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 }))
255a1i 11 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 ))
269adantr 474 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴𝐵𝐴𝑉))
2726, 10syl 17 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
2827, 12, 13sylancl 580 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
297ffnd 6292 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
3029adantr 474 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
31 simpr 479 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 })
32 df-f 6139 . . . . . . . . . 10 (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
3330, 31, 32sylanbrc 578 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
34 disjdif 4264 . . . . . . . . 9 ({ 0 } ∩ (V ∖ { 0 })) = ∅
35 fimacnvdisj 6333 . . . . . . . . 9 ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (𝐹 “ (V ∖ { 0 })) = ∅)
3633, 34, 35sylancl 580 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 “ (V ∖ { 0 })) = ∅)
3725, 28, 363eqtrd 2818 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
3837ex 403 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
3924, 38sylbid 232 . . . . 5 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
4039necon3ad 2982 . . . 4 (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
4123, 40mpd 15 . . 3 (𝜑 → ¬ ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
4241iffalsed 4318 . 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 4318 . 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)))))
4522, 42, 443eqtrd 2818 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 386   = wceq 1601  wex 1823  wcel 2107  wne 2969  wral 3090  wrex 3091  {crab 3094  Vcvv 3398  cdif 3789  cin 3791  wss 3792  c0 4141  ifcif 4307  {csn 4398  ccnv 5354  ran crn 5356  cima 5358  ccom 5359  cio 6097   Fn wfn 6130  wf 6131  1-1-ontowf1o 6134  cfv 6135  (class class class)co 6922   supp csupp 7576  Fincfn 8241  1c1 10273  cuz 11992  ...cfz 12643  seqcseq 13119  chash 13435  Basecbs 16255  +gcplusg 16338  0gc0g 16486   Σg cgsu 16487  Mndcmnd 17680  Cntzccntz 18131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-seq 13120  df-0g 16488  df-gsum 16489  df-mgm 17628  df-sgrp 17670  df-mnd 17681
This theorem is referenced by:  gsumval3lem2  18693
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