Theorem gfsumz 16855

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

Hypothesis

Ref	Expression
gfsumz.z	⊢ 0 = (0g‘𝐺)

Assertion

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

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

Proof of Theorem gfsumz

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

Step	Hyp	Ref	Expression
1		mpteq1 4193	. . . 4 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ ∅ ↦ 0 ))
2	1	oveq2d 6065	. . 3 ⊢ (𝑤 = ∅ → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )))
3	2	eqeq1d 2241	. 2 ⊢ (𝑤 = ∅ → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = 0 ))
4		mpteq1 4193	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝑦 ↦ 0 ))
5	4	oveq2d 6065	. . 3 ⊢ (𝑤 = 𝑦 → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )))
6	5	eqeq1d 2241	. 2 ⊢ (𝑤 = 𝑦 → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ))
7		mpteq1 4193	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ))
8	7	oveq2d 6065	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )))
9	8	eqeq1d 2241	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = 0 ))
10		mpteq1 4193	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝐴 ↦ 0 ))
11	10	oveq2d 6065	. . 3 ⊢ (𝑤 = 𝐴 → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ 𝐴 ↦ 0 )))
12	11	eqeq1d 2241	. 2 ⊢ (𝑤 = 𝐴 → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ))
13		gfsum0 16850	. . . 4 ⊢ (𝐺 ∈ CMnd → (𝐺 Σgf ∅) = (0g‘𝐺))
14		mpt0 5485	. . . . 5 ⊢ (𝑘 ∈ ∅ ↦ 0 ) = ∅
15	14	oveq2i 6060	. . . 4 ⊢ (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = (𝐺 Σgf ∅)
16		gfsumz.z	. . . 4 ⊢ 0 = (0g‘𝐺)
17	13, 15, 16	3eqtr4g 2290	. . 3 ⊢ (𝐺 ∈ CMnd → (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = 0 )
18	17	adantr 276	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = 0 )
19		eqid 2232	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
20		eqid 2232	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
21		simplll 535	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐺 ∈ CMnd)
22		cmnmnd 14007	. . . . . . . . . 10 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
23	19, 16	mndidcl 13632	. . . . . . . . . 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 5831	. . . . . . 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 529	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
29		simprr 533	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
30	29	eldifbd 3222	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
31	19, 20, 21, 27, 28, 29, 30	gfsump1 16854	. . . . 5 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = ((𝐺 Σgf ((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 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = ((𝐺 Σgf ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦))(+g‘𝐺)((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧)))
33		ssun1 3381	. . . . . . . . 9 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
34	33	a1i 9	. . . . . . . 8 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
35	34	resmptd 5088	. . . . . . 7 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦) = (𝑘 ∈ 𝑦 ↦ 0 ))
36	35	oveq2d 6065	. . . . . 6 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σgf ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦)) = (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )))
37		simpr 110	. . . . . 6 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 )
38	36, 37	eqtrd 2265	. . . . 5 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σgf ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦)) = 0 )
39		eqid 2232	. . . . . . 7 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )
40		eqidd 2233	. . . . . . 7 ⊢ (𝑘 = 𝑧 → 0 = 0 )
41		vsnid 3720	. . . . . . . 8 ⊢ 𝑧 ∈ {𝑧}
42		elun2 3386	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43	41, 42	mp1i 10	. . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝑧 ∈ (𝑦 ∪ {𝑧}))
44	39, 40, 43, 24	fvmptd3 5770	. . . . . 6 ⊢ (𝐺 ∈ CMnd → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧) = 0 )
45	44	ad4antr 494	. . . . 5 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧) = 0 )
46	38, 45	oveq12d 6067	. . . 4 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ((𝐺 Σgf ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) ↾ 𝑦))(+g‘𝐺)((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )‘𝑧)) = ( 0 (+g‘𝐺) 0 ))
47	22	ad4antr 494	. . . . 5 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → 𝐺 ∈ Mnd)
48	19, 20, 16	mndlid 13637	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 0 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺) 0 ) = 0 )
49	47, 23, 48	syl2anc2 412	. . . 4 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → ( 0 (+g‘𝐺) 0 ) = 0 )
50	32, 46, 49	3eqtrd 2269	. . 3 ⊢ (((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ) → (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = 0 )
51	50	ex 115	. 2 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 → (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = 0 ))
52		simpr 110	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
53	3, 6, 9, 12, 18, 51, 52	findcard2sd 7148	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σgf (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203   ∖ cdif 3207   ∪ cun 3208   ⊆ wss 3210  ∅c0 3507  {csn 3688   ↦ cmpt 4170   ↾ cres 4750  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  Fincfn 6974  Basecbs 13201  +_gcplusg 13279  0_gc0g 13458  Mndcmnd 13618  CMndccmn 13990   Σ_gf cgfsu 16846

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-inn 9234  df-2 9292  df-n0 9493  df-z 9574  df-uz 9850  df-fz 10339  df-fzo 10473  df-seqfrec 10806  df-ihash 11134  df-ndx 13204  df-slot 13205  df-base 13207  df-plusg 13292  df-0g 13460  df-igsum 13461  df-mgm 13558  df-sgrp 13604  df-mnd 13619  df-minusg 13706  df-mulg 13826  df-cmn 13992  df-gfsum 16847

This theorem is referenced by: (None)