Theorem nn0gsumfz 19954

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 6845	. . . 4 ⊢ 0 ∈ V
4	1, 3	jctir 526	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m ℕ0) ∧ 0 ∈ V))
5		nn0gsumfz.y	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
6		fsuppmapnn0ub 13952	. . 3 ⊢ ((𝐹 ∈ (𝐵 ↑m ℕ0) ∧ 0 ∈ V) → (𝐹 finSupp 0 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )))
7	4, 5, 6	sylc 65	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ))
8		eqidd 2742	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)))
9		simpr 486	. . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐺 ∈ CMnd)
13	1	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐹 ∈ (𝐵 ↑m ℕ0))
14		simpr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝑠 ∈ ℕ0)
15		eqid 2741	. . . . . . 7 ⊢ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))
16	10, 2, 12, 13, 14, 15	fsfnn0gsumfsffz 19953	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
17	16	imp 408	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
18	13	adantr 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑m ℕ0))
19		fz0ssnn0 13571	. . . . . . 7 ⊢ (0...𝑠) ⊆ ℕ0
20		elmapssres 8808	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐵 ↑m ℕ0) ∧ (0...𝑠) ⊆ ℕ0) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑m (0...𝑠)))
21	18, 19, 20	sylancl 593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑m (0...𝑠)))
22		eqeq1 2745	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝑓 = (𝐹 ↾ (0...𝑠)) ↔ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))))
23		oveq2 7368	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
24	23	eqeq2d 2752	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝑓) ↔ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
25	22, 24	3anbi13d 1447	. . . . . . 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 483	. . . . . 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 3555	. . . . 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 1473	. . . 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 414	. . 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 3154	. 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 15	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 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  Vcvv 3433   ⊆ wss 3885   class class class wbr 5075   ↾ cres 5623  ‘cfv 6489  (class class class)co 7360   ↑_m cmap 8767   finSupp cfsupp 9268  0cc0 11033   < clt 11174  ℕ₀cn0 12432  ...cfz 13456  Basecbs 17174  0_gc0g 17397   Σ_g cgsu 17398  CMndccmn 19750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-oi 9419  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-n0 12433  df-z 12520  df-uz 12784  df-fz 13457  df-fzo 13604  df-seq 13959  df-hash 14288  df-0g 17399  df-gsum 17400  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-cntz 19287  df-cmn 19752

This theorem is referenced by:  nn0gsumfz0  19955