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Theorem nn0gsumfz 18426
Description: Replacing a finitely supported function over the nonnegative integers by a function over a finite set of sequential integers in a finite group sum. (Contributed by AV, 9-Oct-2019.)
Hypotheses
Ref Expression
nn0gsumfz.b 𝐵 = (Base‘𝐺)
nn0gsumfz.0 0 = (0g𝐺)
nn0gsumfz.g (𝜑𝐺 ∈ CMnd)
nn0gsumfz.f (𝜑𝐹 ∈ (𝐵𝑚0))
nn0gsumfz.y (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
nn0gsumfz (𝜑 → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))
Distinct variable groups:   𝐵,𝑓   𝑓,𝐹,𝑠,𝑥   𝑓,𝐺   0 ,𝑓,𝑠,𝑥   𝜑,𝑓,𝑠
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥,𝑠)   𝐺(𝑥,𝑠)

Proof of Theorem nn0gsumfz
StepHypRef Expression
1 nn0gsumfz.f . . . 4 (𝜑𝐹 ∈ (𝐵𝑚0))
2 nn0gsumfz.0 . . . . 5 0 = (0g𝐺)
3 fvex 6239 . . . . 5 (0g𝐺) ∈ V
42, 3eqeltri 2726 . . . 4 0 ∈ V
51, 4jctir 560 . . 3 (𝜑 → (𝐹 ∈ (𝐵𝑚0) ∧ 0 ∈ V))
6 nn0gsumfz.y . . 3 (𝜑𝐹 finSupp 0 )
7 fsuppmapnn0ub 12835 . . 3 ((𝐹 ∈ (𝐵𝑚0) ∧ 0 ∈ V) → (𝐹 finSupp 0 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )))
85, 6, 7sylc 65 . 2 (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ))
9 eqidd 2652 . . . . 5 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)))
10 simpr 476 . . . . 5 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ))
11 nn0gsumfz.b . . . . . . 7 𝐵 = (Base‘𝐺)
12 nn0gsumfz.g . . . . . . . 8 (𝜑𝐺 ∈ CMnd)
1312adantr 480 . . . . . . 7 ((𝜑𝑠 ∈ ℕ0) → 𝐺 ∈ CMnd)
141adantr 480 . . . . . . 7 ((𝜑𝑠 ∈ ℕ0) → 𝐹 ∈ (𝐵𝑚0))
15 simpr 476 . . . . . . 7 ((𝜑𝑠 ∈ ℕ0) → 𝑠 ∈ ℕ0)
16 eqid 2651 . . . . . . 7 (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))
1711, 2, 13, 14, 15, 16fsfnn0gsumfsffz 18425 . . . . . 6 ((𝜑𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
1817imp 444 . . . . 5 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
1914adantr 480 . . . . . . 7 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → 𝐹 ∈ (𝐵𝑚0))
20 fz0ssnn0 12473 . . . . . . 7 (0...𝑠) ⊆ ℕ0
21 elmapssres 7924 . . . . . . 7 ((𝐹 ∈ (𝐵𝑚0) ∧ (0...𝑠) ⊆ ℕ0) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵𝑚 (0...𝑠)))
2219, 20, 21sylancl 695 . . . . . 6 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵𝑚 (0...𝑠)))
23 eqeq1 2655 . . . . . . . 8 (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝑓 = (𝐹 ↾ (0...𝑠)) ↔ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))))
24 oveq2 6698 . . . . . . . . 9 (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
2524eqeq2d 2661 . . . . . . . 8 (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝑓) ↔ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
2623, 253anbi13d 1441 . . . . . . 7 (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) ↔ ((𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))))
2726adantl 481 . . . . . 6 ((((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) ∧ 𝑓 = (𝐹 ↾ (0...𝑠))) → ((𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) ↔ ((𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))))
2822, 27rspcedv 3344 . . . . 5 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → (((𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))) → ∃𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
299, 10, 18, 28mp3and 1467 . . . 4 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → ∃𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))
3029ex 449 . . 3 ((𝜑𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) → ∃𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
3130reximdva 3046 . 2 (𝜑 → (∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
328, 31mpd 15 1 (𝜑 → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐵𝑚 (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  wss 3607   class class class wbr 4685  cres 5145  cfv 5926  (class class class)co 6690  𝑚 cmap 7899   finSupp cfsupp 8316  0cc0 9974   < clt 10112  0cn0 11330  ...cfz 12364  Basecbs 15904  0gc0g 16147   Σg cgsu 16148  CMndccmn 18239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-cntz 17796  df-cmn 18241
This theorem is referenced by:  nn0gsumfz0  18427
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