Theorem gsummptnn0fz 19106

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 1915	. . . . 5 ⊢ Ⅎ𝑥(𝑆 < 𝑘 → 𝐶 = 0 )
3		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑘 𝑆 < 𝑥
4		nfcsb1v 3907	. . . . . . 7 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶
5	4	nfeq1 2993	. . . . . 6 ⊢ Ⅎ𝑘⦋𝑥 / 𝑘⦌𝐶 = 0
6	3, 5	nfim 1897	. . . . 5 ⊢ Ⅎ𝑘(𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )
7		breq2 5070	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝑆 < 𝑘 ↔ 𝑆 < 𝑥))
8		csbeq1a 3897	. . . . . . 7 ⊢ (𝑘 = 𝑥 → 𝐶 = ⦋𝑥 / 𝑘⦌𝐶)
9	8	eqeq1d 2823	. . . . . 6 ⊢ (𝑘 = 𝑥 → (𝐶 = 0 ↔ ⦋𝑥 / 𝑘⦌𝐶 = 0 ))
10	7, 9	imbi12d 347	. . . . 5 ⊢ (𝑘 = 𝑥 → ((𝑆 < 𝑘 → 𝐶 = 0 ) ↔ (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 )))
11	2, 6, 10	cbvralw 3441	. . . 4 ⊢ (∀𝑘 ∈ ℕ0 (𝑆 < 𝑘 → 𝐶 = 0 ) ↔ ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))
12	1, 11	sylib 220	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℕ0 (𝑆 < 𝑥 → ⦋𝑥 / 𝑘⦌𝐶 = 0 ))
13		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
14		gsummptnn0fz.f	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵)
15	14	anim1ci 617	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵))
16		rspcsbela 4387	. . . . . . . . . . 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 514	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵))
19	18	adantr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → (𝑥 ∈ ℕ0 ∧ ⦋𝑥 / 𝑘⦌𝐶 ∈ 𝐵))
20		eqid 2821	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 ↦ 𝐶) = (𝑘 ∈ ℕ0 ↦ 𝐶)
21	20	fvmpts 6771	. . . . . . . 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 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ⦋𝑥 / 𝑘⦌𝐶 = 0 )
24	22, 23	eqtrd 2856	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ⦋𝑥 / 𝑘⦌𝐶 = 0 ) → ((𝑘 ∈ ℕ0 ↦ 𝐶)‘𝑥) = 0 )
25	24	ex 415	. . . . 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 3177	. . 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 6874	. . . . 5 ⊢ (∀𝑘 ∈ ℕ0 𝐶 ∈ 𝐵 ↔ (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵)
33	14, 32	sylib 220	. . . 4 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶):ℕ0⟶𝐵)
34	29	fvexi 6684	. . . . . 6 ⊢ 𝐵 ∈ V
35		nn0ex 11904	. . . . . 6 ⊢ ℕ0 ∈ V
36	34, 35	pm3.2i 473	. . . . 5 ⊢ (𝐵 ∈ V ∧ ℕ0 ∈ V)
37		elmapg 8419	. . . . 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 259	. . 3 ⊢ (𝜑 → (𝑘 ∈ ℕ0 ↦ 𝐶) ∈ (𝐵 ↑m ℕ0))
40		gsummptnn0fz.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
41		fz0ssnn0 13003	. . . . 5 ⊢ (0...𝑆) ⊆ ℕ0
42		resmpt 5905	. . . . 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 2830	. . 3 ⊢ (𝑘 ∈ (0...𝑆) ↦ 𝐶) = ((𝑘 ∈ ℕ0 ↦ 𝐶) ↾ (0...𝑆))
45	29, 30, 31, 39, 40, 44	fsfnn0gsumfsffz 19103	. 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 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  Vcvv 3494  ⦋csb 3883   ⊆ wss 3936   class class class wbr 5066   ↦ cmpt 5146   ↾ cres 5557  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ↑_m cmap 8406  0cc0 10537   < clt 10675  ℕ₀cn0 11898  ...cfz 12893  Basecbs 16483  0_gc0g 16713   Σ_g cgsu 16714  CMndccmn 18906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-seq 13371  df-hash 13692  df-0g 16715  df-gsum 16716  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-cntz 18447  df-cmn 18908

This theorem is referenced by:  gsummptnn0fzfv  19107  telgsums  19113  gsummoncoe1  20472  pmatcollpwfi  21390  mp2pm2mplem4  21417