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Theorem gsumval3a 18034
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 2514 . . 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 552 . . . . . 6 (𝜑 → (𝐹:𝐴𝐵𝐴𝑉))
10 fex 6271 . . . . . 6 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
119, 10syl 17 . . . . 5 (𝜑𝐹 ∈ V)
12 fvex 5997 . . . . . 6 (0g𝐺) ∈ V
132, 12eqeltri 2588 . . . . 5 0 ∈ V
14 suppimacnv 7068 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
1511, 13, 14sylancl 692 . . . 4 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
16 gsumval3.g . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
171, 2, 3, 4gsumvallem2 17087 . . . . . . . 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 2520 . . . . . 6 (𝜑 → { 0 } = {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
2019difeq2d 3594 . . . . 5 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
2120imaeq2d 5276 . . . 4 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
226, 15, 213eqtrd 2552 . . 3 (𝜑𝑊 = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
231, 2, 3, 4, 22, 16, 8, 7gsumval 16986 . 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 3500 . . . . . 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 479 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴𝐵𝐴𝑉))
2827, 10syl 17 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
2928, 13, 14sylancl 692 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
30 ffn 5843 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
317, 30syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝐴)
3231adantr 479 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
33 simpr 475 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 })
34 df-f 5693 . . . . . . . . . 10 (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
3532, 33, 34sylanbrc 694 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
36 disjdif 3895 . . . . . . . . 9 ({ 0 } ∩ (V ∖ { 0 })) = ∅
37 fimacnvdisj 5880 . . . . . . . . 9 ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (𝐹 “ (V ∖ { 0 })) = ∅)
3835, 36, 37sylancl 692 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 “ (V ∖ { 0 })) = ∅)
3926, 29, 383eqtrd 2552 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
4039ex 448 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
4125, 40sylbid 228 . . . . 5 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
4241necon3ad 2699 . . . 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 3950 . 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 3950 . 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 2552 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 382   = wceq 1474  wex 1694  wcel 1938  wne 2684  wral 2800  wrex 2801  {crab 2804  Vcvv 3077  cdif 3441  cin 3443  wss 3444  c0 3777  ifcif 3939  {csn 4028  ccnv 4931  ran crn 4933  cima 4935  ccom 4936  cio 5651   Fn wfn 5684  wf 5685  1-1-ontowf1o 5688  cfv 5689  (class class class)co 6426   supp csupp 7057  Fincfn 7717  1c1 9692  cuz 11427  ...cfz 12065  seqcseq 12531  #chash 12847  Basecbs 15579  +gcplusg 15652  0gc0g 15807   Σg cgsu 15808  Mndcmnd 17009  Cntzccntz 17463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-supp 7058  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-seq 12532  df-0g 15809  df-gsum 15810  df-mgm 16957  df-sgrp 16999  df-mnd 17010
This theorem is referenced by:  gsumval3lem2  18037
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