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Theorem gsumval2a 17548
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 2765 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 gsumval2a.o . . . 4 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 eqidd 2766 . . . 4 (𝜑 → (𝐹 “ (V ∖ 𝑂)) = (𝐹 “ (V ∖ 𝑂)))
6 gsumval2.g . . . 4 (𝜑𝐺𝑉)
7 ovexd 6878 . . . 4 (𝜑 → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . 4 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
91, 2, 3, 4, 5, 6, 7, 8gsumval 17540 . . 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 4256 . . . 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 gsumval2.n . . . . . . 7 (𝜑𝑁 ∈ (ℤ𝑀))
13 eluzel2 11894 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1412, 13syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
15 eluzelz 11899 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
1612, 15syl 17 . . . . . 6 (𝜑𝑁 ∈ ℤ)
17 fzf 12540 . . . . . . . 8 ...:(ℤ × ℤ)⟶𝒫 ℤ
18 ffn 6225 . . . . . . . 8 (...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn (ℤ × ℤ))
1917, 18ax-mp 5 . . . . . . 7 ... Fn (ℤ × ℤ)
20 fnovrn 7009 . . . . . . 7 ((... Fn (ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
2119, 20mp3an1 1572 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
2214, 16, 21syl2anc 579 . . . . 5 (𝜑 → (𝑀...𝑁) ∈ ran ...)
2322iftrued 4253 . . . 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))))
2411, 23eqtrd 2799 . . 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))))
259, 24eqtrd 2799 . 2 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
26 fvex 6390 . . 3 (seq𝑀( + , 𝐹)‘𝑁) ∈ V
27 fzopth 12588 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
2812, 27syl 17 . . . . . . . . . 10 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
29 simpl 474 . . . . . . . . . . . . . 14 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑀 = 𝑚)
3029seqeq1d 13017 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹))
31 simpr 477 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑁 = 𝑛)
3230, 31fveq12d 6384 . . . . . . . . . . . 12 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛))
3332eqcomd 2771 . . . . . . . . . . 11 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
34 eqeq1 2769 . . . . . . . . . . 11 (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)))
3533, 34syl5ibrcom 238 . . . . . . . . . 10 ((𝑀 = 𝑚𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3628, 35syl6bi 244 . . . . . . . . 9 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))))
3736impd 398 . . . . . . . 8 (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3837rexlimdvw 3181 . . . . . . 7 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3938exlimdv 2028 . . . . . 6 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4014adantr 472 . . . . . . . 8 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ)
41 oveq2 6852 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
4241eqcomd 2771 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛))
4342biantrurd 528 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
44 fveq2 6377 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
4544eqeq2d 2775 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4643, 45bitr3d 272 . . . . . . . . . 10 (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4746rspcev 3462 . . . . . . . . 9 ((𝑁 ∈ (ℤ𝑀) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
4812, 47sylan 575 . . . . . . . 8 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
49 fveq2 6377 . . . . . . . . . 10 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
50 oveq1 6851 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
5150eqeq2d 2775 . . . . . . . . . . 11 (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
52 seqeq1 13014 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
5352fveq1d 6379 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
5453eqeq2d 2775 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
5551, 54anbi12d 624 . . . . . . . . . 10 (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5649, 55rexeqbidv 3301 . . . . . . . . 9 (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5756spcegv 3447 . . . . . . . 8 (𝑀 ∈ ℤ → (∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
5840, 48, 57sylc 65 . . . . . . 7 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))
5958ex 401 . . . . . 6 (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
6039, 59impbid 203 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6160adantr 472 . . . 4 ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6261iota5 6053 . . 3 ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
6326, 62mpan2 682 . 2 (𝜑 → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
6425, 63eqtrd 2799 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3731  wss 3734  ifcif 4245  𝒫 cpw 4317   × cxp 5277  ccnv 5278  ran crn 5280  cima 5282  ccom 5283  cio 6031   Fn wfn 6065  wf 6066  1-1-ontowf1o 6069  cfv 6070  (class class class)co 6844  1c1 10192  cz 11626  cuz 11889  ...cfz 12536  seqcseq 13011  chash 13324  Basecbs 16133  +gcplusg 16217  0gc0g 16369   Σg cgsu 16370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-pre-lttri 10265  ax-pre-lttrn 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-po 5200  df-so 5201  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-neg 10525  df-z 11627  df-uz 11890  df-fz 12537  df-seq 13012  df-gsum 16372
This theorem is referenced by:  gsumval2  17549
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