Theorem gsummptnn0fz 19952

Description: A final group sum over a function over the nonnegative integers (given as mapping) is equal to a final group sum over a finite interval of nonnegative integers. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 3-Jul-2022.)

Hypotheses

Ref	Expression
gsummptnn0fz.b	⊢ 𝐵 = (Base‘𝐺)
gsummptnn0fz.0	⊢ 0 = (0g‘𝐺)
gsummptnn0fz.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsummptnn0fz.f	⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)
gsummptnn0fz.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
gsummptnn0fz.u	⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ))

Assertion

Ref	Expression
gsummptnn0fz	⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ 𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶)))

Distinct variable groups:   𝐵,𝑘   𝑆,𝑘   0 ,𝑘

Allowed substitution hints:   𝜑(𝑘)   𝐶(𝑘)   𝐺(𝑘)

Proof of Theorem gsummptnn0fz

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsummptnn0fz.u	. . . 4 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ))
2		nfv 1921	. . . . 5 ⊢ Ⅎ𝑥(𝑆 < 𝑘 → 𝐶 = 0 )
3		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑘 𝑆 < 𝑥
4		nfcsb1v 3855	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶
5	4	nfeq1 2916	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 = 0
6	3, 5	nfim 1903	. . . . 5 ⊢ Ⅎ𝑘(𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )
7		breq2 5076	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑆 < 𝑘 ↔ 𝑆 < 𝑥))
8		csbeq1a 3845	. . . . . . 7 ⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶)
9	8	eqeq1d 2741	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐶 = 0 ↔ ⦋𝑥 / 𝑘⦌𝐶 = 0 ))
10	7, 9	imbi12d 345	. . . . 5 ⊢ (𝑘 = 𝑥 → ((𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )))
11	2, 6, 10	cbvralw 3281	. . . 4 ⊢ (∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))
12	1, 11	sylib 219	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))
13		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
14		gsummptnn0fz.f	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)
15	14	anim1ci 622	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵))
16		rspcsbela 4366	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵)
18	13, 17	jca 516	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵))
19	18	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵))
20		eqid 2739	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) = (𝑘 ∈ ℕ0 ↦ 𝐶)
21	20	fvmpts 6939	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶)
22	19, 21	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = ⦋𝑥 / 𝑘⦌𝐶)
23		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ⦋𝑥 / 𝑘⦌𝐶 = 0 )
24	22, 23	eqtrd 2774	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 )
25	24	ex 413	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (⦋𝑥 / 𝑘⦌𝐶 = 0 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ))
26	25	imim2d 57	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → ((𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 )))
27	26	ralimdva 3151	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 )))
28	12, 27	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ))
29		gsummptnn0fz.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
30		gsummptnn0fz.0	. . 3 ⊢ 0 = (0g‘𝐺)
31		gsummptnn0fz.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd)
32	20	fmpt 7051	. . . . 5 ⊢ (∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵 ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵)
33	14, 32	sylib 219	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵)
34	29	fvexi 6841	. . . . . 6 ⊢ 𝐵 ∈ V
35		nn0ex 12434	. . . . . 6 ⊢ ℕ0 ∈ V
36	34, 35	pm3.2i 471	. . . . 5 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V)
37		elmapg 8776	. . . . 5 ⊢ ((𝐵 ∈ V ∧ ℕ0 ∈ V) → ((𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0) ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵))
38	36, 37	mp1i 13	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0) ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵))
39	33, 38	mpbird 258	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0))
40		gsummptnn0fz.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
41		fz0ssnn0 13567	. . . . 5 ⊢ (0...𝑆) ⊆ ℕ0
42		resmpt 5989	. . . . 5 ⊢ ((0...𝑆) ⊆ ℕ0 → ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆)) = (𝑘 ∈ (0...𝑆) ↦ 𝐶))
43	41, 42	ax-mp 5	. . . 4 ⊢ ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆)) = (𝑘 ∈ (0...𝑆) ↦ 𝐶)
44	43	eqcomi 2748	. . 3 ⊢ (𝑘 ∈ (0...𝑆) ↦ 𝐶) = ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆))
45	29, 30, 31, 39, 40, 44	fsfnn0gsumfsffz 19949	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 ) → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ 𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶))))
46	28, 45	mpd 15	1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ ℕ0 ↦ 𝐶)) = (𝐺 Σg (𝑘 ∈ (0...𝑆) ↦ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431  ⦋csb 3831   ⊆ wss 3883   class class class wbr 5072   ↦ cmpt 5153   ↾ cres 5620  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  0cc0 11029   < clt 11170  ℕ₀cn0 12428  ...cfz 13452  Basecbs 17170  0_gc0g 17393   Σ_g cgsu 17394  CMndccmn 19746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-oi 9415  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-0g 17395  df-gsum 17396  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-cntz 19283  df-cmn 19748

This theorem is referenced by:  gsummptnn0fzfv  19953  telgsums  19959  gsummoncoe1  22294  pmatcollpwfi  22765  mp2pm2mplem4  22792