Theorem gsumzfi 14158

Description: Value of a finite group sum over the zero element. (Contributed by Jim Kingdon, 24-May-2026.)

Hypothesis

Ref	Expression
gsumz.z	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
gsumzfi	⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )

Distinct variable groups:   0 ,𝑘   𝐴,𝑘   𝑘,𝐺

Proof of Theorem gsumzfi

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpteq1 4215	. . . 4 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ ∅ ↦ 0 ))
2	1	oveq2d 6101	. . 3 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ ∅ ↦ 0 )))
3	2	eqeq1d 2247	. 2 ⊢ (𝑤 = ∅ → ((𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σg (𝑘 ∈ ∅ ↦ 0 )) = 0 ))
4		mpteq1 4215	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝑦 ↦ 0 ))
5	4	oveq2d 6101	. . 3 ⊢ (𝑤 = 𝑦 → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )))
6	5	eqeq1d 2247	. 2 ⊢ (𝑤 = 𝑦 → ((𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ))
7		mpteq1 4215	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ))
8	7	oveq2d 6101	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )))
9	8	eqeq1d 2247	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = 0 ))
10		mpteq1 4215	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝐴 ↦ 0 ))
11	10	oveq2d 6101	. . 3 ⊢ (𝑤 = 𝐴 → (𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
12	11	eqeq1d 2247	. 2 ⊢ (𝑤 = 𝐴 → ((𝐺 Σg (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ))
13		gsum0cmn 14154	. . . 4 ⊢ (𝐺 ∈ CMnd → (𝐺 Σg ∅) = (0g‘𝐺))
14		mpt0 5511	. . . . 5 ⊢ (𝑘 ∈ ∅ ↦ 0 ) = ∅
15	14	oveq2i 6096	. . . 4 ⊢ (𝐺 Σg (𝑘 ∈ ∅ ↦ 0 )) = (𝐺 Σg ∅)
16		gsumz.z	. . . 4 ⊢ 0 = (0g‘𝐺)
17	13, 15, 16	3eqtr4g 2296	. . 3 ⊢ (𝐺 ∈ CMnd → (𝐺 Σg (𝑘 ∈ ∅ ↦ 0 )) = 0 )
18	17	adantr 276	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σg (𝑘 ∈ ∅ ↦ 0 )) = 0 )
19		eqid 2238	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
20		eqid 2238	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
21		simplll 539	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐺 ∈ CMnd)
22		cmnmnd 14104	. . . . . . . . . 10 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
23	19, 16	mndidcl 13743	. . . . . . . . . 10 ⊢ (𝐺 ∈ Mnd → 0 ∈ (Base‘𝐺))
24	22, 23	syl 14	. . . . . . . . 9 ⊢ (𝐺 ∈ CMnd → 0 ∈ (Base‘𝐺))
25	24	adantr 276	. . . . . . . 8 ⊢ ((𝐺 ∈ CMnd ∧ 𝑘 ∈ (𝑦 ∪ {𝑧})) → 0 ∈ (Base‘𝐺))
26	25	fmpttd 5863	. . . . . . 7 ⊢ (𝐺 ∈ CMnd → (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ):(𝑦 ∪ {𝑧})⟶(Base‘𝐺))
27	21, 26	syl 14	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ):(𝑦 ∪ {𝑧})⟶(Base‘𝐺))
28		simplr 533	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
29		simprr 537	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
30	29	eldifbd 3232	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
31	19, 20, 21, 27, 28, 29, 30	gsump1 14157	. . . . 5 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = ((𝐺 Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦))(+g‘𝐺)((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧)))
32	31	adantr 276	. . . 4 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = ((𝐺 Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦))(+g‘𝐺)((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧)))
33		ssun1 3392	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
34	33	a1i 9	. . . . . . . 8 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
35	34	resmptd 5114	. . . . . . 7 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦) = (𝑘 ∈ 𝑦 ↦ 0 ))
36	35	oveq2d 6101	. . . . . 6 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦)) = (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )))
37		simpr 110	. . . . . 6 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 )
38	36, 37	eqtrd 2271	. . . . 5 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦)) = 0 )
39		eqid 2238	. . . . . . 7 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )
40		eqidd 2239	. . . . . . 7 ⊢ (𝑘 = 𝑧 → 0 = 0 )
41		vsnid 3741	. . . . . . . 8 ⊢ 𝑧 ∈ {𝑧}
42		elun2 3397	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43	41, 42	mp1i 10	. . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝑧 ∈ (𝑦 ∪ {𝑧}))
44	39, 40, 43, 24	fvmptd3 5799	. . . . . 6 ⊢ (𝐺 ∈ CMnd → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧) = 0 )
45	44	ad4antr 498	. . . . 5 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧) = 0 )
46	38, 45	oveq12d 6103	. . . 4 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ((𝐺 Σg ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦))(+g‘𝐺)((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧)) = ( 0 (+g‘𝐺) 0 ))
47	22	ad4antr 498	. . . . 5 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → 𝐺 ∈ Mnd)
48	19, 20, 16	mndlid 13748	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
49	47, 23, 48	syl2anc2 416	. . . 4 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ( 0 (+g‘𝐺) 0 ) = 0 )
50	32, 46, 49	3eqtrd 2275	. . 3 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = 0 )
51	50	ex 115	. 2 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg (𝑘 ∈ 𝑦 ↦ 0 )) = 0 → (𝐺 Σg (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = 0 ))
52		simpr 110	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
53	3, 6, 9, 12, 18, 51, 52	findcard2sd 7196	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ∖ cdif 3217   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  {csn 3709   ↦ cmpt 4192   ↾ cres 4776  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  Fincfn 7022  Basecbs 13352  +_gcplusg 13431  0_gc0g 13610  Mndcmnd 13729  CMndccmn 14087   Σ_g cgsu 14150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-ihash 11215  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-0g 13612  df-gzsum 13613  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-minusg 13809  df-mulg 13923  df-cmn 14089  df-gsumfi 14151

This theorem is used by: (None)