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Theorem gsumval2 12980
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)
Assertion
Ref Expression
gsumval2 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))

Proof of Theorem gsumval2
Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval2.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2193 . . 3 (0g𝐺) = (0g𝐺)
3 gsumval2.p . . 3 + = (+g𝐺)
4 gsumval2.g . . 3 (𝜑𝐺𝑉)
5 gsumval2.n . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzel2 9597 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
75, 6syl 14 . . . 4 (𝜑𝑀 ∈ ℤ)
8 eluzelz 9601 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
95, 8syl 14 . . . 4 (𝜑𝑁 ∈ ℤ)
107, 9fzfigd 10502 . . 3 (𝜑 → (𝑀...𝑁) ∈ Fin)
11 gsumval2.f . . 3 (𝜑𝐹:(𝑀...𝑁)⟶𝐵)
121, 2, 3, 4, 10, 11igsumval 12973 . 2 (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
13 simprr 531 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))
14 simprl 529 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑀...𝑁) = (𝑚...𝑛))
15 eqcom 2195 . . . . . . . . . . . . . 14 ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑀...𝑁) = (𝑚...𝑛))
16 fzopth 10127 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑚) → ((𝑚...𝑛) = (𝑀...𝑁) ↔ (𝑚 = 𝑀𝑛 = 𝑁)))
1715, 16bitr3id 194 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ𝑚) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑚 = 𝑀𝑛 = 𝑁)))
1817adantr 276 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑚 = 𝑀𝑛 = 𝑁)))
1914, 18mpbid 147 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (𝑚 = 𝑀𝑛 = 𝑁))
2019simpld 112 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑚 = 𝑀)
2120seqeq1d 10524 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
2219simprd 114 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑛 = 𝑁)
2321, 22fveq12d 5561 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
2413, 23eqtrd 2226 . . . . . . 7 ((𝑛 ∈ (ℤ𝑚) ∧ ((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
2524rexlimiva 2606 . . . . . 6 (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
2625exlimiv 1609 . . . . 5 (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
277elexd 2773 . . . . . . . 8 (𝜑𝑀 ∈ V)
2827adantr 276 . . . . . . 7 ((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ V)
295adantr 276 . . . . . . . 8 ((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑁 ∈ (ℤ𝑀))
30 oveq2 5926 . . . . . . . . . . 11 (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁))
3130eqeq2d 2205 . . . . . . . . . 10 (𝑛 = 𝑁 → ((𝑀...𝑁) = (𝑀...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑁)))
32 fveq2 5554 . . . . . . . . . . 11 (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))
3332eqeq2d 2205 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
3431, 33anbi12d 473 . . . . . . . . 9 (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
3534adantl 277 . . . . . . . 8 (((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) ∧ 𝑛 = 𝑁) → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))))
36 eqidd 2194 . . . . . . . . 9 ((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → (𝑀...𝑁) = (𝑀...𝑁))
37 simpr 110 . . . . . . . . 9 ((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
3836, 37jca 306 . . . . . . . 8 ((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ((𝑀...𝑁) = (𝑀...𝑁) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
3929, 35, 38rspcedvd 2870 . . . . . . 7 ((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
40 fveq2 5554 . . . . . . . 8 (𝑚 = 𝑀 → (ℤ𝑚) = (ℤ𝑀))
41 oveq1 5925 . . . . . . . . . 10 (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛))
4241eqeq2d 2205 . . . . . . . . 9 (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
43 seqeq1 10521 . . . . . . . . . . 11 (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹))
4443fveq1d 5556 . . . . . . . . . 10 (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛))
4544eqeq2d 2205 . . . . . . . . 9 (𝑚 = 𝑀 → (𝑥 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛)))
4642, 45anbi12d 473 . . . . . . . 8 (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
4740, 46rexeqbidv 2707 . . . . . . 7 (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑥 = (seq𝑀( + , 𝐹)‘𝑛))))
4828, 39, 47spcedv 2849 . . . . . 6 ((𝜑𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))
4948ex 115 . . . . 5 (𝜑 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))))
5026, 49impbid2 143 . . . 4 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)))
51 eluzfz2 10098 . . . . . . . 8 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
525, 51syl 14 . . . . . . 7 (𝜑𝑁 ∈ (𝑀...𝑁))
53 n0i 3452 . . . . . . 7 (𝑁 ∈ (𝑀...𝑁) → ¬ (𝑀...𝑁) = ∅)
5452, 53syl 14 . . . . . 6 (𝜑 → ¬ (𝑀...𝑁) = ∅)
5554intnanrd 933 . . . . 5 (𝜑 → ¬ ((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g𝐺)))
56 biorf 745 . . . . 5 (¬ ((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g𝐺)) → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
5755, 56syl 14 . . . 4 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
5850, 57bitr3d 190 . . 3 (𝜑 → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
5958iotabidv 5237 . 2 (𝜑 → (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (℩𝑥(((𝑀...𝑁) = ∅ ∧ 𝑥 = (0g𝐺)) ∨ ∃𝑚𝑛 ∈ (ℤ𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛)))))
60 eqid 2193 . . 3 (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)
61 seqex 10520 . . . . 5 seq𝑀( + , 𝐹) ∈ V
62 fvexg 5573 . . . . 5 ((seq𝑀( + , 𝐹) ∈ V ∧ 𝑁 ∈ (ℤ𝑀)) → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
6361, 5, 62sylancr 414 . . . 4 (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) ∈ V)
64 eueq 2931 . . . . 5 ((seq𝑀( + , 𝐹)‘𝑁) ∈ V ↔ ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
6563, 64sylib 122 . . . 4 (𝜑 → ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁))
66 eqeq1 2200 . . . . 5 (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) → (𝑥 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)))
6766iota2 5244 . . . 4 (((seq𝑀( + , 𝐹)‘𝑁) ∈ V ∧ ∃!𝑥 𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) → ((seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁) ↔ (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)))
6863, 65, 67syl2anc 411 . . 3 (𝜑 → ((seq𝑀( + , 𝐹)‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁) ↔ (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁)))
6960, 68mpbii 148 . 2 (𝜑 → (℩𝑥𝑥 = (seq𝑀( + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐹)‘𝑁))
7012, 59, 693eqtr2d 2232 1 (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wex 1503  ∃!weu 2042  wcel 2164  wrex 2473  Vcvv 2760  c0 3446  cio 5213  wf 5250  cfv 5254  (class class class)co 5918  Fincfn 6794  cz 9317  cuz 9592  ...cfz 10074  seqcseq 10518  Basecbs 12618  +gcplusg 12695  0gc0g 12867   Σg cgsu 12868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-er 6587  df-en 6795  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-seqfrec 10519  df-ndx 12621  df-slot 12622  df-base 12624  df-0g 12869  df-igsum 12870
This theorem is referenced by:  gsumsplit1r  12981  gsumprval  12982  gsumwsubmcl  13068  gsumwmhm  13070  mulgnngsum  13197  gsumfzconst  13411
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