Theorem gfsumz 14112

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 4199	. . . 4 ⊢ (𝑤 = ∅ → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ ∅ ↦ 0 ))
2	1	oveq2d 6074	. . 3 ⊢ (𝑤 = ∅ → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )))
3	2	eqeq1d 2243	. 2 ⊢ (𝑤 = ∅ → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = 0 ))
4		mpteq1 4199	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝑦 ↦ 0 ))
5	4	oveq2d 6074	. . 3 ⊢ (𝑤 = 𝑦 → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )))
6	5	eqeq1d 2243	. 2 ⊢ (𝑤 = 𝑦 → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ 𝑦 ↦ 0 )) = 0 ))
7		mpteq1 4199	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ))
8	7	oveq2d 6074	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )))
9	8	eqeq1d 2243	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )) = 0 ))
10		mpteq1 4199	. . . 4 ⊢ (𝑤 = 𝐴 → (𝑘 ∈ 𝑤 ↦ 0 ) = (𝑘 ∈ 𝐴 ↦ 0 ))
11	10	oveq2d 6074	. . 3 ⊢ (𝑤 = 𝐴 → (𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = (𝐺 Σgf (𝑘 ∈ 𝐴 ↦ 0 )))
12	11	eqeq1d 2243	. 2 ⊢ (𝑤 = 𝐴 → ((𝐺 Σgf (𝑘 ∈ 𝑤 ↦ 0 )) = 0 ↔ (𝐺 Σgf (𝑘 ∈ 𝐴 ↦ 0 )) = 0 ))
13		gfsum0 14107	. . . 4 ⊢ (𝐺 ∈ CMnd → (𝐺 Σgf ∅) = (0g‘𝐺))
14		mpt0 5491	. . . . 5 ⊢ (𝑘 ∈ ∅ ↦ 0 ) = ∅
15	14	oveq2i 6069	. . . 4 ⊢ (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = (𝐺 Σgf ∅)
16		gfsumz.z	. . . 4 ⊢ 0 = (0g‘𝐺)
17	13, 15, 16	3eqtr4g 2292	. . 3 ⊢ (𝐺 ∈ CMnd → (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = 0 )
18	17	adantr 276	. 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σgf (𝑘 ∈ ∅ ↦ 0 )) = 0 )
19		eqid 2234	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
20		eqid 2234	. . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺)
21		simplll 535	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐺 ∈ CMnd)
22		cmnmnd 14057	. . . . . . . . . 10 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
23	19, 16	mndidcl 13694	. . . . . . . . . 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 5837	. . . . . . 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 3226	. . . . . 6 ⊢ ((((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
31	19, 20, 21, 27, 28, 29, 30	gfsump1 14111	. . . . 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 3386	. . . . . . . . 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 5094	. . . . . . 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 6074	. . . . . 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 2267	. . . . 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 2234	. . . . . . 7 ⊢ (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 ) = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ 0 )
40		eqidd 2235	. . . . . . 7 ⊢ (𝑘 = 𝑧 → 0 = 0 )
41		vsnid 3726	. . . . . . . 8 ⊢ 𝑧 ∈ {𝑧}
42		elun2 3391	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43	41, 42	mp1i 10	. . . . . . 7 ⊢ (𝐺 ∈ CMnd → 𝑧 ∈ (𝑦 ∪ {𝑧}))
44	39, 40, 43, 24	fvmptd3 5776	. . . . . 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 6076	. . . 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 13699	. . . . 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 2271	. . 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 7162	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝐴 ∈ Fin) → (𝐺 Σgf (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205   ∖ cdif 3211   ∪ cun 3212   ⊆ wss 3214  ∅c0 3512  {csn 3694   ↦ cmpt 4176   ↾ cres 4756  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  Fincfn 6988  Basecbs 13299  +_gcplusg 13377  0_gc0g 13556  Mndcmnd 13680  CMndccmn 14040   Σ_gf cgfsu 14103

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-inn 9258  df-2 9316  df-n0 9517  df-z 9598  df-uz 9875  df-fz 10365  df-fzo 10502  df-seqfrec 10837  df-ihash 11167  df-ndx 13302  df-slot 13303  df-base 13305  df-plusg 13390  df-0g 13558  df-igsum 13559  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-minusg 13762  df-mulg 13876  df-cmn 14042  df-gfsum 14104

This theorem is referenced by: (None)