Theorem gsumress 13423

Description: The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither 𝐺 nor 𝐻 need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
gsumress.b	⊢ 𝐵 = (Base‘𝐺)
gsumress.o	⊢ + = (+g‘𝐺)
gsumress.h	⊢ 𝐻 = (𝐺 ↾s 𝑆)
gsumress.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gsumress.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
gsumress.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
gsumress.f	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
gsumress.z	⊢ (𝜑 → 0 ∈ 𝑆)
gsumress.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))

Assertion

Ref	Expression
gsumress	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥   𝑥,𝑆   𝑥,𝐻   𝑥, +   𝑥, 0

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem gsumress

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

Step	Hyp	Ref	Expression
1		gsumress.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
2		gsumress.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2229	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		gsumress.o	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
5		eqid 2229	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}
6	2, 3, 4, 5	mgmidsssn0 13412	. . . . . . . . . 10 ⊢ (𝐺 ∈ 𝑉 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g‘𝐺)})
7	1, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g‘𝐺)})
8		oveq1 6007	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
9	8	eqeq1d 2238	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
10	9	ovanraleqv 6024	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
11		gsumress.s	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
12		gsumress.z	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ 𝑆)
13	11, 12	sseldd 3225	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ 𝐵)
14		gsumress.c	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
15	14	ralrimiva 2603	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
16	10, 13, 15	elrabd 2961	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
17	7, 16	sseldd 3225	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {(0g‘𝐺)})
18		elsni 3684	. . . . . . . 8 ⊢ ( 0 ∈ {(0g‘𝐺)} → 0 = (0g‘𝐺))
19	17, 18	syl 14	. . . . . . 7 ⊢ (𝜑 → 0 = (0g‘𝐺))
20		gsumress.h	. . . . . . . . . . . . 13 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
21	20	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 = (𝐺 ↾s 𝑆))
22	2	a1i 9	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
23	21, 22, 1, 11	ressbas2d 13096	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (Base‘𝐻))
24	23, 12	basmexd 13088	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ V)
25		eqid 2229	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
26		eqid 2229	. . . . . . . . . . 11 ⊢ (0g‘𝐻) = (0g‘𝐻)
27		eqid 2229	. . . . . . . . . . 11 ⊢ (+g‘𝐻) = (+g‘𝐻)
28		eqid 2229	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}
29	25, 26, 27, 28	mgmidsssn0 13412	. . . . . . . . . 10 ⊢ (𝐻 ∈ V → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ {(0g‘𝐻)})
30	24, 29	syl 14	. . . . . . . . 9 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ {(0g‘𝐻)})
31	9	ovanraleqv 6024	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
32	11	sselda 3224	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
33	32, 14	syldan 282	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
34	33	ralrimiva 2603	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
35	31, 12, 34	elrabd 2961	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝑆 ∣ ∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
36	4	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → + = (+g‘𝐺))
37		basfn 13086	. . . . . . . . . . . . . . . . . 18 ⊢ Base Fn V
38		funfvex 5643	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun Base ∧ 𝐻 ∈ dom Base) → (Base‘𝐻) ∈ V)
39	38	funfni 5422	. . . . . . . . . . . . . . . . . 18 ⊢ ((Base Fn V ∧ 𝐻 ∈ V) → (Base‘𝐻) ∈ V)
40	37, 24, 39	sylancr 414	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐻) ∈ V)
41	23, 40	eqeltrd 2306	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ∈ V)
42	21, 36, 41, 1	ressplusgd 13157	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → + = (+g‘𝐻))
43	42	oveqd 6017	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐻)𝑥))
44	43	eqeq1d 2238	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g‘𝐻)𝑥) = 𝑥))
45	42	oveqd 6017	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐻)𝑦))
46	45	eqeq1d 2238	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g‘𝐻)𝑦) = 𝑥))
47	44, 46	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)))
48	23, 47	raleqbidv 2744	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)))
49	23, 48	rabeqbidv 2794	. . . . . . . . . 10 ⊢ (𝜑 → {𝑦 ∈ 𝑆 ∣ ∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
50	35, 49	eleqtrd 2308	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
51	30, 50	sseldd 3225	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {(0g‘𝐻)})
52		elsni 3684	. . . . . . . 8 ⊢ ( 0 ∈ {(0g‘𝐻)} → 0 = (0g‘𝐻))
53	51, 52	syl 14	. . . . . . 7 ⊢ (𝜑 → 0 = (0g‘𝐻))
54	19, 53	eqtr3d 2264	. . . . . 6 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
55	54	eqeq2d 2241	. . . . 5 ⊢ (𝜑 → (𝑧 = (0g‘𝐺) ↔ 𝑧 = (0g‘𝐻)))
56	55	anbi2d 464	. . . 4 ⊢ (𝜑 → ((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐺)) ↔ (𝐴 = ∅ ∧ 𝑧 = (0g‘𝐻))))
57	42	seqeq2d 10671	. . . . . . . . 9 ⊢ (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
58	57	fveq1d 5628	. . . . . . . 8 ⊢ (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
59	58	eqeq2d 2241	. . . . . . 7 ⊢ (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
60	59	anbi2d 464	. . . . . 6 ⊢ (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
61	60	rexbidv 2531	. . . . 5 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
62	61	exbidv 1871	. . . 4 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
63	56, 62	orbi12d 798	. . 3 ⊢ (𝜑 → (((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) ↔ ((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐻)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))))
64	63	iotabidv 5300	. 2 ⊢ (𝜑 → (℩𝑧((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))) = (℩𝑧((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐻)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))))
65		gsumress.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
66		gsumress.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
67	66, 11	fssd 5485	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
68	2, 3, 4, 1, 65, 67	igsumval 13418	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))))
69	23	feq3d 5461	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘𝐻)))
70	66, 69	mpbid 147	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐻))
71	25, 26, 27, 24, 65, 70	igsumval 13418	. 2 ⊢ (𝜑 → (𝐻 Σg 𝐹) = (℩𝑧((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐻)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))))
72	64, 68, 71	3eqtr4d 2272	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512  Vcvv 2799   ⊆ wss 3197  ∅c0 3491  {csn 3666  ℩cio 5275   Fn wfn 5312  ⟶wf 5313  ‘cfv 5317  (class class class)co 6000  ℤ_≥cuz 9718  ...cfz 10200  seqcseq 10664  Basecbs 13027   ↾_s cress 13028  +_gcplusg 13105  0_gc0g 13284   Σ_g cgsu 13285

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-neg 8316  df-inn 9107  df-2 9165  df-z 9443  df-uz 9719  df-seqfrec 10665  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-iress 13035  df-plusg 13118  df-0g 13286  df-igsum 13287

This theorem is referenced by:  gsumsubm  13522