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Theorem gsumval2a 18711
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 2735 . . . 4 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . . 4 + = (+g𝐺)
4 gsumval2a.o . . . 4 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5 eqidd 2736 . . . 4 (𝜑 → (𝐹 “ (V ∖ 𝑂)) = (𝐹 “ (V ∖ 𝑂)))
6 gsumval2.g . . . 4 (𝜑𝐺𝑉)
7 ovexd 7466 . . . 4 (𝜑 → (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . 4 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
91, 2, 3, 4, 5, 6, 7, 8gsumval 18703 . . 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 4542 . . . 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 13548 . . . . . . 7 ...:(ℤ × ℤ)⟶𝒫 ℤ
13 ffn 6737 . . . . . . 7 (...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn (ℤ × ℤ))
1412, 13ax-mp 5 . . . . . 6 ... Fn (ℤ × ℤ)
15 gsumval2.n . . . . . . 7 (𝜑𝑁 ∈ (ℤ𝑀))
16 eluzel2 12881 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
1715, 16syl 17 . . . . . 6 (𝜑𝑀 ∈ ℤ)
18 eluzelz 12886 . . . . . . 7 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
1915, 18syl 17 . . . . . 6 (𝜑𝑁 ∈ ℤ)
20 fnovrn 7608 . . . . . 6 ((... Fn (ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...)
2114, 17, 19, 20mp3an2i 1465 . . . . 5 (𝜑 → (𝑀...𝑁) ∈ ran ...)
2221iftrued 4539 . . . 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 2775 . . 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 2775 . 2 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
25 fvex 6920 . . 3 (seq𝑀( + , 𝐹)‘𝑁) ∈ V
26 fzopth 13598 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
2715, 26syl 17 . . . . . . . . . 10 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚𝑁 = 𝑛)))
28 simpl 482 . . . . . . . . . . . . . 14 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑀 = 𝑚)
2928seqeq1d 14045 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹))
30 simpr 484 . . . . . . . . . . . . 13 ((𝑀 = 𝑚𝑁 = 𝑛) → 𝑁 = 𝑛)
3129, 30fveq12d 6914 . . . . . . . . . . . 12 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛))
3231eqcomd 2741 . . . . . . . . . . 11 ((𝑀 = 𝑚𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
33 eqeq1 2739 . . . . . . . . . . 11 (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)))
3432, 33syl5ibrcom 247 . . . . . . . . . 10 ((𝑀 = 𝑚𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3527, 34biimtrdi 253 . . . . . . . . 9 (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))))
3635impd 410 . . . . . . . 8 (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3736rexlimdvw 3158 . . . . . . 7 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3837exlimdv 1931 . . . . . 6 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
3917adantr 480 . . . . . . . 8 ((𝜑𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ)
40 oveq2 7439 . . . . . . . . . . . . 13 (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
4140eqcomd 2741 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛))
4241biantrurd 532 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
43 fveq2 6907 . . . . . . . . . . . 12 (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
4443eqeq2d 2746 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4542, 44bitr3d 281 . . . . . . . . . 10 (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
4645rspcev 3622 . . . . . . . . 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 6907 . . . . . . . . . 10 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
49 oveq1 7438 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
5049eqeq2d 2746 . . . . . . . . . . 11 (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
51 seqeq1 14042 . . . . . . . . . . . . 13 (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
5251fveq1d 6909 . . . . . . . . . . . 12 (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
5352eqeq2d 2746 . . . . . . . . . . 11 (𝑚 = 𝑀 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))
5450, 53anbi12d 632 . . . . . . . . . 10 (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5548, 54rexeqbidv 3345 . . . . . . . . 9 (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))))
5655spcegv 3597 . . . . . . . 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 412 . . . . . 6 (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))))
5938, 58impbid 212 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6059adantr 480 . . . 4 ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))
6160iota5 6546 . . 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 2775 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  wss 3963  ifcif 4531  𝒫 cpw 4605   × cxp 5687  ccnv 5688  ran crn 5690  cima 5692  ccom 5693  cio 6514   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  1c1 11154  cz 12611  cuz 12876  ...cfz 13544  seqcseq 14039  chash 14366  Basecbs 17245  +gcplusg 17298  0gc0g 17486   Σg cgsu 17487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-neg 11493  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-gsum 17489
This theorem is referenced by:  gsumval2  18712
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