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Theorem gsumval2a 18004
Description: Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumval2.b 𝐵 = (Base‘𝐺)
gsumval2.p + = (+g𝐺)
gsumval2.g (𝜑𝐺𝑉)
gsumval2.n (𝜑𝑁 ∈ (ℤ𝑀))
gsumval2.f (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
gsumval2a.o 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
gsumval2a.f (𝜑 → ¬ ran 𝐹𝑂)
Assertion
Ref Expression
gsumval2a (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝑉   𝑥, + ,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem gsumval2a
Dummy variables 𝑧 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval2.b . . . 4 𝐵 = (Base‘𝐺)
2 eqid 2738 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 gsumval2a.o . . . 4 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 eqidd 2739 . . . 4 (𝜑 → (𝐹 “ (V ∖ 𝑂)) = (𝐹 “ (V ∖ 𝑂)))
6 gsumval2.g . . . 4 (𝜑𝐺𝑉)
7 ovexd 7199 . . . 4 (𝜑 → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . 4 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
91, 2, 3, 4, 5, 6, 7, 8gsumval 17996 . . 3 (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹𝑂, (0g𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂)))))))))
10 gsumval2a.f . . . . 5 (𝜑 → ¬ ran 𝐹𝑂)
1110iffalsed 4422 . . . 4 (𝜑 → if(ran 𝐹𝑂, (0g𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂)))))))) = if((𝑀...𝑁) ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂))))))))
12 fzf 12978 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
13 ffn 6498 . . . . . . 7 (...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn (ℤ × ℤ))
1412, 13ax-mp 5 . . . . . 6 ... Fn (ℤ × ℤ)
15 gsumval2.n . . . . . . 7 (𝜑𝑁 ∈ (ℤ𝑀))
16 eluzel2 12322 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1715, 16syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
18 eluzelz 12327 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
1915, 18syl 17 . . . . . 6 (𝜑𝑁 ∈ ℤ)
20 fnovrn 7333 . . . . . 6 ((... Fn (ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
2114, 17, 19, 20mp3an2i 1467 . . . . 5 (𝜑 → (𝑀...𝑁) ∈ ran ...)
2221iftrued 4419 . . . 4 (𝜑 → if((𝑀...𝑁) ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂))))))) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
2311, 22eqtrd 2773 . . 3 (𝜑 → if(ran 𝐹𝑂, (0g𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ 𝑂))))–1-1-onto→(𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ 𝑂)))))))) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
249, 23eqtrd 2773 . 2 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
25 fvex 6681 . . 3 (seq𝑀( + , 𝐹)‘𝑁) ∈ V
26 fzopth 13028 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
2715, 26syl 17 . . . . . . . . . 10 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
28 simpl 486 . . . . . . . . . . . . . 14 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑀 = 𝑚)
2928seqeq1d 13459 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹))
30 simpr 488 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑁 = 𝑛)
3129, 30fveq12d 6675 . . . . . . . . . . . 12 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛))
3231eqcomd 2744 . . . . . . . . . . 11 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33 eqeq1 2742 . . . . . . . . . . 11 (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)))
3432, 33syl5ibrcom 250 . . . . . . . . . 10 ((𝑀 = 𝑚𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3527, 34syl6bi 256 . . . . . . . . 9 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))))
3635impd 414 . . . . . . . 8 (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3736rexlimdvw 3199 . . . . . . 7 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3837exlimdv 1939 . . . . . 6 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3917adantr 484 . . . . . . . 8 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ)
40 oveq2 7172 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
4140eqcomd 2744 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛))
4241biantrurd 536 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
43 fveq2 6668 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
4443eqeq2d 2749 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4542, 44bitr3d 284 . . . . . . . . . 10 (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4645rspcev 3524 . . . . . . . . 9 ((𝑁 ∈ (ℤ𝑀) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
4715, 46sylan 583 . . . . . . . 8 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
48 fveq2 6668 . . . . . . . . . 10 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
49 oveq1 7171 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
5049eqeq2d 2749 . . . . . . . . . . 11 (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
51 seqeq1 13456 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
5251fveq1d 6670 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
5352eqeq2d 2749 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
5450, 53anbi12d 634 . . . . . . . . . 10 (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5548, 54rexeqbidv 3304 . . . . . . . . 9 (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5655spcegv 3499 . . . . . . . 8 (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
5739, 47, 56sylc 65 . . . . . . 7 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))
5857ex 416 . . . . . 6 (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
5938, 58impbid 215 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6059adantr 484 . . . 4 ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6160iota5 6316 . . 3 ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
6225, 61mpan2 691 . 2 (𝜑 → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
6324, 62eqtrd 2773 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2113  wral 3053  wrex 3054  {crab 3057  Vcvv 3397  cdif 3838  wss 3841  ifcif 4411  𝒫 cpw 4485   × cxp 5517  ccnv 5518  ran crn 5520  cima 5522  ccom 5523  cio 6289   Fn wfn 6328  wf 6329  1-1-ontowf1o 6332  cfv 6333  (class class class)co 7164  1c1 10609  cz 12055  cuz 12317  ...cfz 12974  seqcseq 13453  chash 13775  Basecbs 16579  +gcplusg 16661  0gc0g 16809   Σg cgsu 16810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-pre-lttri 10682  ax-pre-lttrn 10683
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-er 8313  df-en 8549  df-dom 8550  df-sdom 8551  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-neg 10944  df-z 12056  df-uz 12318  df-fz 12975  df-seq 13454  df-gsum 16812
This theorem is referenced by:  gsumval2  18005
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