Theorem nn0gsumfz 18380

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		fvex 6201	. . . . 5 ⊢ (0g‘𝐺) ∈ V
4	2, 3	eqeltri 2697	. . . 4 ⊢ 0 ∈ V
5	1, 4	jctir 561	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑𝑚 ℕ0) ∧ 0 ∈ V))
6		nn0gsumfz.y	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
7		fsuppmapnn0ub 12795	. . 3 ⊢ ((𝐹 ∈ (𝐵 ↑𝑚 ℕ0) ∧ 0 ∈ V) → (𝐹 finSupp 0 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )))
8	5, 6, 7	sylc 65	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ))
9		eqidd 2623	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)))
10		simpr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ))
11		nn0gsumfz.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
12		nn0gsumfz.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐺 ∈ CMnd)
14	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐹 ∈ (𝐵 ↑𝑚 ℕ0))
15		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝑠 ∈ ℕ0)
16		eqid 2622	. . . . . . 7 ⊢ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))
17	11, 2, 13, 14, 15, 16	fsfnn0gsumfsffz 18379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
18	17	imp 445	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
19	14	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑𝑚 ℕ0))
20		fz0ssnn0 12435	. . . . . . 7 ⊢ (0...𝑠) ⊆ ℕ0
21		elmapssres 7882	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐵 ↑𝑚 ℕ0) ∧ (0...𝑠) ⊆ ℕ0) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑𝑚 (0...𝑠)))
22	19, 20, 21	sylancl 694	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑𝑚 (0...𝑠)))
23		eqeq1 2626	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝑓 = (𝐹 ↾ (0...𝑠)) ↔ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))))
24		oveq2 6658	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
25	24	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝑓) ↔ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
26	23, 25	3anbi13d 1401	. . . . . . 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))))))
27	26	adantl 482	. . . . . 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))))))
28	22, 27	rspcedv 3313	. . . . 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))))
29	9, 10, 18, 28	mp3and 1427	. . . 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)))
30	29	ex 450	. . 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))))
31	30	reximdva 3017	. 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))))
32	8, 31	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857   finSupp cfsupp 8275  0cc0 9936   < clt 10074  ℕ₀cn0 11292  ...cfz 12326  Basecbs 15857  0_gc0g 16100   Σ_g cgsu 16101  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  nn0gsumfz0  18381