Theorem gsumress 13038

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 2196	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		gsumress.o	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
5		eqid 2196	. . . . . . . . . . 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 13027	. . . . . . . . . 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 5929	. . . . . . . . . . . 12 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
9	8	eqeq1d 2205	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
10	9	ovanraleqv 5946	. . . . . . . . . 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 3184	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ 𝐵)
14		gsumress.c	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
15	14	ralrimiva 2570	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
16	10, 13, 15	elrabd 2922	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
17	7, 16	sseldd 3184	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {(0g‘𝐺)})
18		elsni 3640	. . . . . . . 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 12746	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 = (Base‘𝐻))
24	23, 12	basmexd 12738	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ V)
25		eqid 2196	. . . . . . . . . . 11 ⊢ (Base‘𝐻) = (Base‘𝐻)
26		eqid 2196	. . . . . . . . . . 11 ⊢ (0g‘𝐻) = (0g‘𝐻)
27		eqid 2196	. . . . . . . . . . 11 ⊢ (+g‘𝐻) = (+g‘𝐻)
28		eqid 2196	. . . . . . . . . . 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 13027	. . . . . . . . . 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 5946	. . . . . . . . . . 11 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
32	11	sselda 3183	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
33	32, 14	syldan 282	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
34	33	ralrimiva 2570	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
35	31, 12, 34	elrabd 2922	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝑆 ∣ ∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
36	4	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → + = (+g‘𝐺))
37		basfn 12736	. . . . . . . . . . . . . . . . . 18 ⊢ Base Fn V
38		funfvex 5575	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Fun Base ∧ 𝐻 ∈ dom Base) → (Base‘𝐻) ∈ V)
39	38	funfni 5358	. . . . . . . . . . . . . . . . . 18 ⊢ ((Base Fn V ∧ 𝐻 ∈ V) → (Base‘𝐻) ∈ V)
40	37, 24, 39	sylancr 414	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐻) ∈ V)
41	23, 40	eqeltrd 2273	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ∈ V)
42	21, 36, 41, 1	ressplusgd 12806	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → + = (+g‘𝐻))
43	42	oveqd 5939	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐻)𝑥))
44	43	eqeq1d 2205	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g‘𝐻)𝑥) = 𝑥))
45	42	oveqd 5939	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐻)𝑦))
46	45	eqeq1d 2205	. . . . . . . . . . . . 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 2709	. . . . . . . . . . 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 2758	. . . . . . . . . 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 2275	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
51	30, 50	sseldd 3184	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {(0g‘𝐻)})
52		elsni 3640	. . . . . . . 8 ⊢ ( 0 ∈ {(0g‘𝐻)} → 0 = (0g‘𝐻))
53	51, 52	syl 14	. . . . . . 7 ⊢ (𝜑 → 0 = (0g‘𝐻))
54	19, 53	eqtr3d 2231	. . . . . 6 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
55	54	eqeq2d 2208	. . . . 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 10546	. . . . . . . . 9 ⊢ (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
58	57	fveq1d 5560	. . . . . . . 8 ⊢ (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
59	58	eqeq2d 2208	. . . . . . 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 2498	. . . . 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 1839	. . . 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 794	. . 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 5241	. 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 5420	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
68	2, 3, 4, 1, 65, 67	igsumval 13033	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧((𝐴 = ∅ ∧ 𝑧 = (0g‘𝐺)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))))
69	23	feq3d 5396	. . . 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 13033	. 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 2239	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  Vcvv 2763   ⊆ wss 3157  ∅c0 3450  {csn 3622  ℩cio 5217   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  ℤ_≥cuz 9601  ...cfz 10083  seqcseq 10539  Basecbs 12678   ↾_s cress 12679  +_gcplusg 12755  0_gc0g 12927   Σ_g cgsu 12928

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-neg 8200  df-inn 8991  df-2 9049  df-z 9327  df-uz 9602  df-seqfrec 10540  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-0g 12929  df-igsum 12930

This theorem is referenced by:  gsumsubm  13126