Theorem gsumress 17895

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

Step	Hyp	Ref	Expression
1		oveq1 7166	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
2	1	eqeq1d 2826	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
3	2	ovanraleqv 7183	. . . . . . . 8 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
4		gsumress.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
5		gsumress.z	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ 𝑆)
6	4, 5	sseldd 3971	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
7		gsumress.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
8	7	ralrimiva 3185	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
9	3, 6, 8	elrabd 3685	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
10	9	snssd 4745	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
11		gsumress.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
12		gsumress.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
13		eqid 2824	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		gsumress.o	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
15		eqid 2824	. . . . . . . . 9 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}
16	12, 13, 14, 15	mgmidsssn0 17885	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g‘𝐺)})
17	11, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g‘𝐺)})
18	17, 9	sseldd 3971	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐺)})
19		elsni 4587	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐺)} → 0 = (0g‘𝐺))
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐺))
21	20	sneqd 4582	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐺)})
22	17, 21	sseqtrrd 4011	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ { 0 })
23	10, 22	eqssd 3987	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
24	2	ovanraleqv 7183	. . . . . . . . 9 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
25	4	sselda 3970	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
26	25, 7	syldan 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
27	26	ralrimiva 3185	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
28	24, 5, 27	elrabd 3685	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝑆 ∣ ∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
29		gsumress.h	. . . . . . . . . . 11 ⊢ 𝐻 = (𝐺 ↾s 𝑆)
30	29, 12	ressbas2 16558	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻))
31	4, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘𝐻))
32		fvex 6686	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐻) ∈ V
33	31, 32	eqeltrdi 2924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ V)
34	29, 14	ressplusg 16615	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ V → + = (+g‘𝐻))
35	33, 34	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → + = (+g‘𝐻))
36	35	oveqd 7176	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐻)𝑥))
37	36	eqeq1d 2826	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g‘𝐻)𝑥) = 𝑥))
38	35	oveqd 7176	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐻)𝑦))
39	38	eqeq1d 2826	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g‘𝐻)𝑦) = 𝑥))
40	37, 39	anbi12d 632	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)))
41	31, 40	raleqbidv 3404	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)))
42	31, 41	rabeqbidv 3488	. . . . . . . 8 ⊢ (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)})
43	28, 42	eleqtrd 2918	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
44	43	snssd 4745	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
45	29	ovexi 7193	. . . . . . . . 9 ⊢ 𝐻 ∈ V
46	45	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ V)
47		eqid 2824	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
48		eqid 2824	. . . . . . . . 9 ⊢ (0g‘𝐻) = (0g‘𝐻)
49		eqid 2824	. . . . . . . . 9 ⊢ (+g‘𝐻) = (+g‘𝐻)
50		eqid 2824	. . . . . . . . 9 ⊢ {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)}
51	47, 48, 49, 50	mgmidsssn0 17885	. . . . . . . 8 ⊢ (𝐻 ∈ V → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ {(0g‘𝐻)})
52	46, 51	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ {(0g‘𝐻)})
53	52, 43	sseldd 3971	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐻)})
54		elsni 4587	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐻)} → 0 = (0g‘𝐻))
55	53, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐻))
56	55	sneqd 4582	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐻)})
57	52, 56	sseqtrrd 4011	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ { 0 })
58	44, 57	eqssd 3987	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
59	23, 58	eqtr3d 2861	. . . 4 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
60	59	sseq2d 4002	. . 3 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ↔ ran 𝐹 ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}))
61	20, 55	eqtr3d 2861	. . 3 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
62	35	seqeq2d 13379	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
63	62	fveq1d 6675	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
64	63	eqeq2d 2835	. . . . . . . 8 ⊢ (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
65	64	anbi2d 630	. . . . . . 7 ⊢ (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
66	65	rexbidv 3300	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
67	66	exbidv 1921	. . . . 5 ⊢ (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))))
68	67	iotabidv 6342	. . . 4 ⊢ (𝜑 → (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
69	35	seqeq2d 13379	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , (𝐹 ∘ 𝑓)) = seq1((+g‘𝐻), (𝐹 ∘ 𝑓)))
70	69	fveq1d 6675	. . . . . . . 8 ⊢ (𝜑 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))
71	70	eqeq2d 2835	. . . . . . 7 ⊢ (𝜑 → (𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) ↔ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))))
72	71	anbi2d 630	. . . . . 6 ⊢ (𝜑 → ((𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))) ↔ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))
73	72	exbidv 1921	. . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))) ↔ ∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))
74	73	iotabidv 6342	. . . 4 ⊢ (𝜑 → (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))) = (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))
75	68, 74	ifeq12d 4490	. . 3 ⊢ (𝜑 → if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))))) = if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))))))
76	60, 61, 75	ifbieq12d 4497	. 2 ⊢ (𝜑 → if(ran 𝐹 ⊆ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}, (0g‘𝐺), if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))) = if(ran 𝐹 ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}, (0g‘𝐻), if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))))
77	23	difeq2d 4102	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
78	77	imaeq2d 5932	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})))
79		gsumress.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
80		gsumress.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
81	80, 4	fssd 6531	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
82	12, 13, 14, 15, 78, 11, 79, 81	gsumval 17890	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}, (0g‘𝐺), if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))))
83	58	difeq2d 4102	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}))
84	83	imaeq2d 5932	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})))
85	31	feq3d 6504	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘𝐻)))
86	80, 85	mpbid 234	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐻))
87	47, 48, 49, 50, 84, 46, 79, 86	gsumval 17890	. 2 ⊢ (𝜑 → (𝐻 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}, (0g‘𝐻), if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))))
88	76, 82, 87	3eqtr4d 2869	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142  {crab 3145  Vcvv 3497   ∖ cdif 3936   ⊆ wss 3939  ifcif 4470  {csn 4570  ^◡ccnv 5557  ran crn 5559   “ cima 5561   ∘ ccom 5562  ℩cio 6315  ⟶wf 6354  –1-1-onto→wf1o 6357  ‘cfv 6358  (class class class)co 7159  1c1 10541  ℤ_≥cuz 12246  ...cfz 12895  seqcseq 13372  ♯chash 13693  Basecbs 16486   ↾_s cress 16487  +_gcplusg 16568  0_gc0g 16716   Σ_g cgsu 16717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8513  df-dom 8514  df-sdom 8515  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-seq 13373  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-0g 16718  df-gsum 16719

This theorem is referenced by:  gsumsubm  18002  regsumfsum  20616  regsumsupp  20769  frlmgsum  20919  imasdsf1olem  22986  gsumsubg  30688  esumpfinvallem  31337  sge0tsms  42669  aacllem  44909