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Theorem nn0gsumfz 18734
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)
32fvexi 6448 . . . 4 0 ∈ V
41, 3jctir 518 . . 3 (𝜑 → (𝐹 ∈ (𝐵𝑚0) ∧ 0 ∈ V))
5 nn0gsumfz.y . . 3 (𝜑𝐹 finSupp 0 )
6 fsuppmapnn0ub 13090 . . 3 ((𝐹 ∈ (𝐵𝑚0) ∧ 0 ∈ V) → (𝐹 finSupp 0 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )))
74, 5, 6sylc 65 . 2 (𝜑 → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ))
8 eqidd 2827 . . . . 5 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)))
9 simpr 479 . . . . 5 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ))
10 nn0gsumfz.b . . . . . . 7 𝐵 = (Base‘𝐺)
11 nn0gsumfz.g . . . . . . . 8 (𝜑𝐺 ∈ CMnd)
1211adantr 474 . . . . . . 7 ((𝜑𝑠 ∈ ℕ0) → 𝐺 ∈ CMnd)
131adantr 474 . . . . . . 7 ((𝜑𝑠 ∈ ℕ0) → 𝐹 ∈ (𝐵𝑚0))
14 simpr 479 . . . . . . 7 ((𝜑𝑠 ∈ ℕ0) → 𝑠 ∈ ℕ0)
15 eqid 2826 . . . . . . 7 (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))
1610, 2, 12, 13, 14, 15fsfnn0gsumfsffz 18733 . . . . . 6 ((𝜑𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
1716imp 397 . . . . 5 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
1813adantr 474 . . . . . . 7 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → 𝐹 ∈ (𝐵𝑚0))
19 fz0ssnn0 12730 . . . . . . 7 (0...𝑠) ⊆ ℕ0
20 elmapssres 8148 . . . . . . 7 ((𝐹 ∈ (𝐵𝑚0) ∧ (0...𝑠) ⊆ ℕ0) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵𝑚 (0...𝑠)))
2118, 19, 20sylancl 582 . . . . . 6 (((𝜑𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵𝑚 (0...𝑠)))
22 eqeq1 2830 . . . . . . . 8 (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝑓 = (𝐹 ↾ (0...𝑠)) ↔ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))))
23 oveq2 6914 . . . . . . . . 9 (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
2423eqeq2d 2836 . . . . . . . 8 (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝑓) ↔ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
2522, 243anbi13d 1568 . . . . . . 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))))))
2625adantl 475 . . . . . 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))))))
2721, 26rspcedv 3531 . . . . 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))))
288, 9, 17, 27mp3and 1594 . . . 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)))
2928ex 403 . . 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))))
3029reximdva 3226 . 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))))
317, 30mpd 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 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3118  wrex 3119  Vcvv 3415  wss 3799   class class class wbr 4874  cres 5345  cfv 6124  (class class class)co 6906  𝑚 cmap 8123   finSupp cfsupp 8545  0cc0 10253   < clt 10392  0cn0 11619  ...cfz 12620  Basecbs 16223  0gc0g 16454   Σg cgsu 16455  CMndccmn 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rmo 3126  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-se 5303  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-isom 6133  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-supp 7561  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-fsupp 8546  df-oi 8685  df-card 9079  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-n0 11620  df-z 11706  df-uz 11970  df-fz 12621  df-fzo 12762  df-seq 13097  df-hash 13412  df-0g 16456  df-gsum 16457  df-mgm 17596  df-sgrp 17638  df-mnd 17649  df-cntz 18101  df-cmn 18549
This theorem is referenced by:  nn0gsumfz0  18735
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