Theorem gsumress 18537

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 7364	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
2	1	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
3	2	ovanraleqv 7381	. . . . . . . 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 3945	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
7		gsumress.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
8	7	ralrimiva 3143	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
9	3, 6, 8	elrabd 3647	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
10	9	snssd 4769	. . . . . 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 2736	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		gsumress.o	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
15		eqid 2736	. . . . . . . . 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 18527	. . . . . . . 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 3945	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐺)})
19		elsni 4603	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐺)} → 0 = (0g‘𝐺))
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐺))
21	20	sneqd 4598	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐺)})
22	17, 21	sseqtrrd 3985	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ { 0 })
23	10, 22	eqssd 3961	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
24	2	ovanraleqv 7381	. . . . . . . . 9 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
25	4	sselda 3944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
26	25, 7	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
27	26	ralrimiva 3143	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
28	24, 5, 27	elrabd 3647	. . . . . . . 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 17120	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻))
31	4, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘𝐻))
32		fvex 6855	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐻) ∈ V
33	31, 32	eqeltrdi 2846	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ V)
34	29, 14	ressplusg 17171	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ V → + = (+g‘𝐻))
35	33, 34	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → + = (+g‘𝐻))
36	35	oveqd 7374	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐻)𝑥))
37	36	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g‘𝐻)𝑥) = 𝑥))
38	35	oveqd 7374	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐻)𝑦))
39	38	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g‘𝐻)𝑦) = 𝑥))
40	37, 39	anbi12d 631	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)))
41	31, 40	raleqbidv 3319	. . . . . . . . 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 3424	. . . . . . . 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 2840	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
44	43	snssd 4769	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
45	29	ovexi 7391	. . . . . . . . 9 ⊢ 𝐻 ∈ V
46	45	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ V)
47		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
48		eqid 2736	. . . . . . . . 9 ⊢ (0g‘𝐻) = (0g‘𝐻)
49		eqid 2736	. . . . . . . . 9 ⊢ (+g‘𝐻) = (+g‘𝐻)
50		eqid 2736	. . . . . . . . 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 18527	. . . . . . . 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 3945	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐻)})
54		elsni 4603	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐻)} → 0 = (0g‘𝐻))
55	53, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐻))
56	55	sneqd 4598	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐻)})
57	52, 56	sseqtrrd 3985	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ { 0 })
58	44, 57	eqssd 3961	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
59	23, 58	eqtr3d 2778	. . . 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 3976	. . 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 2778	. . 3 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
62	35	seqeq2d 13913	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
63	62	fveq1d 6844	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
64	63	eqeq2d 2747	. . . . . . . 8 ⊢ (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
65	64	anbi2d 629	. . . . . . 7 ⊢ (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
66	65	rexbidv 3175	. . . . . 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 1924	. . . . 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 6480	. . . 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 13913	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , (𝐹 ∘ 𝑓)) = seq1((+g‘𝐻), (𝐹 ∘ 𝑓)))
70	69	fveq1d 6844	. . . . . . . 8 ⊢ (𝜑 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))
71	70	eqeq2d 2747	. . . . . . 7 ⊢ (𝜑 → (𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) ↔ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))))
72	71	anbi2d 629	. . . . . 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 1924	. . . . 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 6480	. . . 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 4507	. . 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 4514	. 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 4082	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
78	77	imaeq2d 6013	. . 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 6686	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
82	12, 13, 14, 15, 78, 11, 79, 81	gsumval 18532	. 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 4082	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}))
84	83	imaeq2d 6013	. . 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 6655	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘𝐻)))
86	80, 85	mpbid 231	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐻))
87	47, 48, 49, 50, 84, 46, 79, 86	gsumval 18532	. 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 2786	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ∖ cdif 3907   ⊆ wss 3910  ifcif 4486  {csn 4586  ^◡ccnv 5632  ran crn 5634   “ cima 5636   ∘ ccom 5637  ℩cio 6446  ⟶wf 6492  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357  1c1 11052  ℤ_≥cuz 12763  ...cfz 13424  seqcseq 13906  ♯chash 14230  Basecbs 17083   ↾_s cress 17112  +_gcplusg 17133  0_gc0g 17321   Σ_g cgsu 17322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-seq 13907  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324

This theorem is referenced by:  gsumsubm  18645  regsumfsum  20865  regsumsupp  21026  frlmgsum  21178  imasdsf1olem  23726  gsumsubg  31888  esumpfinvallem  32673  sge0tsms  44611  aacllem  47238