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Theorem gsumval3a 19019
 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 2824 . . 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 515 . . . . . 6 (𝜑 → (𝐹:𝐴𝐵𝐴𝑉))
10 fex 6977 . . . . . 6 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
119, 10syl 17 . . . . 5 (𝜑𝐹 ∈ V)
122fvexi 6672 . . . . 5 0 ∈ V
13 suppimacnv 7831 . . . . 5 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
1411, 12, 13sylancl 589 . . . 4 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
15 gsumval3.g . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
161, 2, 3, 4gsumvallem2 17994 . . . . . . . 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 2830 . . . . . 6 (𝜑 → { 0 } = {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
1918difeq2d 4084 . . . . 5 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
2019imaeq2d 5916 . . . 4 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
216, 14, 203eqtrd 2863 . . 3 (𝜑𝑊 = (𝐹 “ (V ∖ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
221, 2, 3, 4, 21, 15, 8, 7gsumval 17883 . 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 3984 . . . . . 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 484 . . . . . . . . . 10 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴𝐵𝐴𝑉))
2726, 10syl 17 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
2827, 12, 13sylancl 589 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
297ffnd 6503 . . . . . . . . . . 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 6347 . . . . . . . . . 10 (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
3330, 31, 32sylanbrc 586 . . . . . . . . 9 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
34 disjdif 4403 . . . . . . . . 9 ({ 0 } ∩ (V ∖ { 0 })) = ∅
35 fimacnvdisj 6545 . . . . . . . . 9 ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (𝐹 “ (V ∖ { 0 })) = ∅)
3633, 34, 35sylancl 589 . . . . . . . 8 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 “ (V ∖ { 0 })) = ∅)
3725, 28, 363eqtrd 2863 . . . . . . 7 ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
3837ex 416 . . . . . 6 (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
3924, 38sylbid 243 . . . . 5 (𝜑 → (ran 𝐹 ⊆ {𝑧𝐵 ∣ ∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
4039necon3ad 3027 . . . 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 4460 . 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 4460 . 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 2863 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 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  {crab 3137  Vcvv 3480   ∖ cdif 3916   ∩ cin 3918   ⊆ wss 3919  ∅c0 4275  ifcif 4449  {csn 4549  ◡ccnv 5541  ran crn 5543   “ cima 5545   ∘ ccom 5546  ℩cio 6300   Fn wfn 6338  ⟶wf 6339  –1-1-onto→wf1o 6342  ‘cfv 6343  (class class class)co 7145   supp csupp 7820  Fincfn 8499  1c1 10530  ℤ≥cuz 12236  ...cfz 12890  seqcseq 13369  ♯chash 13691  Basecbs 16479  +gcplusg 16561  0gc0g 16709   Σg cgsu 16710  Mndcmnd 17907  Cntzccntz 18441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-supp 7821  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-seq 13370  df-0g 16711  df-gsum 16712  df-mgm 17848  df-sgrp 17897  df-mnd 17908 This theorem is referenced by:  gsumval3lem2  19022
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