Theorem nn0gsumfz 20112

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 6896	. . . 4 ⊢ 0 ∈ V
4	1, 3	jctir 530	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m ℕ0) ∧ 0 ∈ V))
5		nn0gsumfz.y	. . 3 ⊢ (𝜑 → 𝐹 finSupp 0 )
6		fsuppmapnn0ub 14061	. . 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 2763	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠)))
9		simpr 490	. . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐺 ∈ CMnd)
13	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝐹 ∈ (𝐵 ↑m ℕ0))
14		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → 𝑠 ∈ ℕ0)
15		eqid 2762	. . . . . . 7 ⊢ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))
16	10, 2, 12, 13, 14, 15	fsfnn0gsumfsffz 20111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 ) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
17	16	imp 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
18	13	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → 𝐹 ∈ (𝐵 ↑m ℕ0))
19		fz0ssnn0 13679	. . . . . . 7 ⊢ (0...𝑠) ⊆ ℕ0
20		elmapssres 8876	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐵 ↑m ℕ0) ∧ (0...𝑠) ⊆ ℕ0) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑m (0...𝑠)))
21	18, 19, 20	sylancl 598	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝐹‘𝑥) = 0 )) → (𝐹 ↾ (0...𝑠)) ∈ (𝐵 ↑m (0...𝑠)))
22		eqeq1 2766	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝑓 = (𝐹 ↾ (0...𝑠)) ↔ (𝐹 ↾ (0...𝑠)) = (𝐹 ↾ (0...𝑠))))
23		oveq2 7424	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝐹 ↾ (0...𝑠))))
24	23	eqeq2d 2773	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ (0...𝑠)) → ((𝐺 Σg 𝐹) = (𝐺 Σg 𝑓) ↔ (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ (0...𝑠)))))
25	22, 24	3anbi13d 1466	. . . . . . 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 487	. . . . . 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 3572	. . . . 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 1493	. . . 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 418	. . 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 3177	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902   class class class wbr 5107   ↾ cres 5661  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829   finSupp cfsupp 9334  0cc0 11127   < clt 11270  ℕ₀cn0 12531  ...cfz 13563  Basecbs 17305  0_gc0g 17528   Σ_g cgsu 17529  CMndccmn 19908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-oi 9485  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-fzo 13712  df-seq 14068  df-hash 14397  df-0g 17530  df-gsum 17531  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-cntz 19445  df-cmn 19910

This theorem is used by:  nn0gsumfz0  20113