MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumval3a Structured version   Visualization version   GIF version

Theorem gsumval3a 18350
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 2651 . . 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 553 . . . . . 6 (𝜑 → (𝐹:𝐴𝐵𝐴𝑉))
10 fex 6530 . . . . . 6 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
119, 10syl 17 . . . . 5 (𝜑𝐹 ∈ V)
12 fvex 6239 . . . . . 6 (0g𝐺) ∈ V
132, 12eqeltri 2726 . . . . 5 0 ∈ V
14 suppimacnv 7351 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
1511, 13, 14sylancl 695 . . . 4 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
16 gsumval3.g . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
171, 2, 3, 4gsumvallem2 17419 . . . . . . . 8 (𝐺 ∈ Mnd → {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1816, 17syl 17 . . . . . . 7 (𝜑 → {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
1918eqcomd 2657 . . . . . 6 (𝜑 → { 0 } = {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
2019difeq2d 3761 . . . . 5 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
2120imaeq2d 5501 . . . 4 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
226, 15, 213eqtrd 2689 . . 3 (𝜑𝑊 = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
231, 2, 3, 4, 22, 16, 8, 7gsumval 17318 . 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)))))))
24 gsumval3a.n . . . 4 (𝜑𝑊 ≠ ∅)
2518sseq2d 3666 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 }))
265a1i 11 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 ))
279adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴𝐵𝐴𝑉))
2827, 10syl 17 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
2928, 13, 14sylancl 695 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
30 ffn 6083 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
317, 30syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
3231adantr 480 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
33 simpr 476 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 })
34 df-f 5930 . . . . . . . . . 10 (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
3532, 33, 34sylanbrc 699 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
36 disjdif 4073 . . . . . . . . 9 ({ 0 } ∩ (V ∖ { 0 })) = ∅
37 fimacnvdisj 6121 . . . . . . . . 9 ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (𝐹 “ (V ∖ { 0 })) = ∅)
3835, 36, 37sylancl 695 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 “ (V ∖ { 0 })) = ∅)
3926, 29, 383eqtrd 2689 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
4039ex 449 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
4125, 40sylbid 230 . . . . 5 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
4241necon3ad 2836 . . . 4 (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
4324, 42mpd 15 . . 3 (𝜑 → ¬ ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
4443iffalsed 4130 . 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))))))
45 gsumval3a.i . . 3 (𝜑 → ¬ 𝐴 ∈ ran ...)
4645iffalsed 4130 . 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)))))
4723, 44, 463eqtrd 2689 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 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  ifcif 4119  {csn 4210  ccnv 5142  ran crn 5144  cima 5146  ccom 5147  cio 5887   Fn wfn 5921  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690   supp csupp 7340  Fincfn 7997  1c1 9975  cuz 11725  ...cfz 12364  seqcseq 12841  #chash 13157  Basecbs 15904  +gcplusg 15988  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  Cntzccntz 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seq 12842  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342
This theorem is referenced by:  gsumval3lem2  18353
  Copyright terms: Public domain W3C validator