Theorem gsumzfi 14135

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 4210	. . . 4 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ ∅ ↦ 0 ))
2	1	oveq2d 6091	. . 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 4210	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝑦 ↦ 0 ))
5	4	oveq2d 6091	. . 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 4210	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ))
8	7	oveq2d 6091	. . 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 4210	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝐴 ↦ 0 ))
11	10	oveq2d 6091	. . 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 14131	. . . 4 ⊢ (𝐺 ∈ CMnd → (𝐺 Σg ∅) = (0g‘𝐺))
14		mpt0 5506	. . . . 5 ⊢ (𝑘 ∈ ∅ ↦ 0 ) = ∅
15	14	oveq2i 6086	. . . 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 14081	. . . . . . . . . 10 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
23	19, 16	mndidcl 13720	. . . . . . . . . 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 5854	. . . . . . 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 14134	. . . . 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 5109	. . . . . . 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 6091	. . . . . 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 3737	. . . . . . . 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 5793	. . . . . 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 6093	. . . 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 13725	. . . . 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 7186	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209   ∖ cdif 3217   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  {csn 3705   ↦ cmpt 4187   ↾ cres 4771  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  Fincfn 7012  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587  Mndcmnd 13706  CMndccmn 14064   Σ_g cgsu 14127

This theorem was proved from 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-ihash 11193  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-minusg 13786  df-mulg 13900  df-cmn 14066  df-gsumfi 14128

This theorem is referenced by: (None)