Theorem nn0gsumfz 20057

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 = (0g‘𝐺)
nn0gsumfz.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
nn0gsumfz.f	⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0))
nn0gsumfz.y	⊢ (𝜑 → 𝐹 finSupp 0 )

Assertion

Ref	Expression
nn0gsumfz	⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐵 ↑m (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 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

Step	Hyp	Ref	Expression
1		nn0gsumfz.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m ℕ0))
2		nn0gsumfz.0	. . . . 5 ⊢ 0 = (0g‘𝐺)
3	2	fvexi 6899	. . . 4 ⊢ 0 ∈ V
4	1, 3	jctir 529	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m ℕ0) ∧ 0 ∈ V))
5		nn0gsumfz.y	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
6		fsuppmapnn0ub 14034	. . 3 ⊢ ((𝐹 ∈ (𝐵 ↑m ℕ0) ∧ 0 ∈ V) → (𝐹 finSupp 0 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )))
7	4, 5, 6	sylc 66	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ))
8		eqidd 2771	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)))
9		simpr 489	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ))
10		nn0gsumfz.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
11		nn0gsumfz.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd)
12	11	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐺 ∈ CMnd)
13	1	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐹 ∈ (𝐵 ↑m ℕ0))
14		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝑠 ∈ ℕ0)
15		eqid 2770	. . . . . . 7 ⊢ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))
16	10, 2, 12, 13, 14, 15	fsfnn0gsumfsffz 20056	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
17	16	imp 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
18	13	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑m ℕ0))
19		fz0ssnn0 13653	. . . . . . 7 ⊢ (0...𝑠) ⊆ ℕ0
20		elmapssres 8867	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐵 ↑m ℕ0) ∧ (0...𝑠) ⊆ ℕ0) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑m (0...𝑠)))
21	18, 19, 20	sylancl 597	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑m (0...𝑠)))
22		eqeq1 2774	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝑓 = (𝐹 ↾ (0...𝑠)) ↔ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))))
23		oveq2 7422	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
24	23	eqeq2d 2781	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝑓) ↔ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
25	22, 24	3anbi13d 1464	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) ↔ ((𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))))
26	25	adantl 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) ∧ 𝑓 = (𝐹 ↾ (0...𝑠))) → ((𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) ↔ ((𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))))
27	21, 26	rspcedv 3582	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (((𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))) → ∃𝑓 ∈ (𝐵 ↑m (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
28	8, 9, 17, 27	mp3and 1491	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∃𝑓 ∈ (𝐵 ↑m (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))
29	28	ex 417	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) → ∃𝑓 ∈ (𝐵 ↑m (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
30	29	reximdva 3185	. 2 ⊢ (𝜑 → (∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐵 ↑m (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓))))
31	7, 30	mpd 16	1 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐵 ↑m (0...𝑠))(𝑓 = (𝐹 ↾ (0...𝑠)) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ∃wrex 3096  Vcvv 3462   ⊆ wss 3913   class class class wbr 5114   ↾ cres 5667  ‘cfv 6540  (class class class)co 7414   ↑_m cmap 8827   finSupp cfsupp 9324  0cc0 11103   < clt 11246  ℕ₀cn0 12507  ...cfz 13538  Basecbs 17272  0_gc0g 17495   Σ_g cgsu 17496  CMndccmn 19853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736  ax-cnex 11159  ax-resscn 11160  ax-1cn 11161  ax-icn 11162  ax-addcl 11163  ax-addrcl 11164  ax-mulcl 11165  ax-mulrcl 11166  ax-mulcom 11167  ax-addass 11168  ax-mulass 11169  ax-distr 11170  ax-i2m1 11171  ax-1ne0 11172  ax-1rid 11173  ax-rnegex 11174  ax-rrecex 11175  ax-cnre 11176  ax-pre-lttri 11177  ax-pre-lttrn 11178  ax-pre-ltadd 11179  ax-pre-mulgt0 11180

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-nel 3072  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-se 5619  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8160  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8456  df-er 8697  df-map 8829  df-en 8947  df-dom 8948  df-sdom 8949  df-fin 8950  df-fsupp 9325  df-oi 9475  df-card 9928  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11446  df-neg 11447  df-nn 12237  df-n0 12508  df-z 12595  df-uz 12866  df-fz 13539  df-fzo 13686  df-seq 14041  df-hash 14370  df-0g 17497  df-gsum 17498  df-mgm 18701  df-sgrp 18780  df-mnd 18796  df-cntz 19390  df-cmn 19855

This theorem is referenced by:  nn0gsumfz0  20058