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Theorem gsumval2a 17722
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 2797 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 gsumval2a.o . . . 4 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 eqidd 2798 . . . 4 (𝜑 → (𝐹 “ (V ∖ 𝑂)) = (𝐹 “ (V ∖ 𝑂)))
6 gsumval2.g . . . 4 (𝜑𝐺𝑉)
7 ovexd 7057 . . . 4 (𝜑 → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . 4 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
91, 2, 3, 4, 5, 6, 7, 8gsumval 17714 . . 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 4398 . . . 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 12750 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
13 ffn 6389 . . . . . . 7 (...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn (ℤ × ℤ))
1412, 13ax-mp 5 . . . . . 6 ... Fn (ℤ × ℤ)
15 gsumval2.n . . . . . . 7 (𝜑𝑁 ∈ (ℤ𝑀))
16 eluzel2 12102 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1715, 16syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
18 eluzelz 12107 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
1915, 18syl 17 . . . . . 6 (𝜑𝑁 ∈ ℤ)
20 fnovrn 7186 . . . . . 6 ((... Fn (ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
2114, 17, 19, 20mp3an2i 1458 . . . . 5 (𝜑 → (𝑀...𝑁) ∈ ran ...)
2221iftrued 4395 . . . 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 2833 . . 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 2833 . 2 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
25 fvex 6558 . . 3 (seq𝑀( + , 𝐹)‘𝑁) ∈ V
26 fzopth 12798 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
2715, 26syl 17 . . . . . . . . . 10 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
28 simpl 483 . . . . . . . . . . . . . 14 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑀 = 𝑚)
2928seqeq1d 13229 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹))
30 simpr 485 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑁 = 𝑛)
3129, 30fveq12d 6552 . . . . . . . . . . . 12 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛))
3231eqcomd 2803 . . . . . . . . . . 11 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33 eqeq1 2801 . . . . . . . . . . 11 (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)))
3432, 33syl5ibrcom 248 . . . . . . . . . 10 ((𝑀 = 𝑚𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3527, 34syl6bi 254 . . . . . . . . 9 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))))
3635impd 411 . . . . . . . 8 (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3736rexlimdvw 3255 . . . . . . 7 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3837exlimdv 1915 . . . . . 6 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3917adantr 481 . . . . . . . 8 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ)
40 oveq2 7031 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
4140eqcomd 2803 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛))
4241biantrurd 533 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
43 fveq2 6545 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
4443eqeq2d 2807 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4542, 44bitr3d 282 . . . . . . . . . 10 (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4645rspcev 3561 . . . . . . . . 9 ((𝑁 ∈ (ℤ𝑀) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
4715, 46sylan 580 . . . . . . . 8 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
48 fveq2 6545 . . . . . . . . . 10 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
49 oveq1 7030 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
5049eqeq2d 2807 . . . . . . . . . . 11 (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
51 seqeq1 13226 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
5251fveq1d 6547 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
5352eqeq2d 2807 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
5450, 53anbi12d 630 . . . . . . . . . 10 (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5548, 54rexeqbidv 3364 . . . . . . . . 9 (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5655spcegv 3542 . . . . . . . 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 413 . . . . . 6 (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
5938, 58impbid 213 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6059adantr 481 . . . 4 ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6160iota5 6216 . . 3 ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
6225, 61mpan2 687 . 2 (𝜑 → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁))
6324, 62eqtrd 2833 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1525  wex 1765  wcel 2083  wral 3107  wrex 3108  {crab 3111  Vcvv 3440  cdif 3862  wss 3865  ifcif 4387  𝒫 cpw 4459   × cxp 5448  ccnv 5449  ran crn 5451  cima 5453  ccom 5454  cio 6194   Fn wfn 6227  wf 6228  1-1-ontowf1o 6231  cfv 6232  (class class class)co 7023  1c1 10391  cz 11835  cuz 12097  ...cfz 12746  seqcseq 13223  chash 13544  Basecbs 16316  +gcplusg 16398  0gc0g 16546   Σg cgsu 16547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-pre-lttri 10464  ax-pre-lttrn 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-po 5369  df-so 5370  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-neg 10726  df-z 11836  df-uz 12098  df-fz 12747  df-seq 13224  df-gsum 16549
This theorem is referenced by:  gsumval2  17723
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