Theorem gsumress 18921

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 7458	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
2	1	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
3	2	ovanraleqv 7475	. . . . . . . 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 3914	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
7		gsumress.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
8	7	ralrimiva 3158	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
9	3, 6, 8	elrabd 3650	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
10	9	snssd 4749	. . . . . 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 2763	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		gsumress.o	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
15		eqid 2763	. . . . . . . . 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 18903	. . . . . . . 8 ⊢ (𝐺 ∈ 𝑉 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g‘𝐺)})
17	11, 16	syl 18	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g‘𝐺)})
18	17, 9	sseldd 3914	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐺)})
19		elsni 4601	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐺)} → 0 = (0g‘𝐺))
20	18, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐺))
21	20	sneqd 4596	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐺)})
22	17, 21	sseqtrrd 3950	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ { 0 })
23	10, 22	eqssd 3930	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
24	2	ovanraleqv 7475	. . . . . . . . 9 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
25	4	sselda 3913	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
26	25, 7	syldan 603	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
27	26	ralrimiva 3158	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
28	24, 5, 27	elrabd 3650	. . . . . . . 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 17460	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻))
31	4, 30	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘𝐻))
32		fvex 6929	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐻) ∈ V
33	31, 32	eqeltrdi 2872	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ V)
34	29, 14	ressplusg 17508	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ V → + = (+g‘𝐻))
35	33, 34	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → + = (+g‘𝐻))
36	35	oveqd 7468	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐻)𝑥))
37	36	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g‘𝐻)𝑥) = 𝑥))
38	35	oveqd 7468	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐻)𝑦))
39	38	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g‘𝐻)𝑦) = 𝑥))
40	37, 39	anbi12d 644	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)))
41	31, 40	raleqbidv 3338	. . . . . . . . 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 3433	. . . . . . . 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 2866	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
44	43	snssd 4749	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
45	29	ovexi 7485	. . . . . . . . 9 ⊢ 𝐻 ∈ V
46	45	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ V)
47		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
48		eqid 2763	. . . . . . . . 9 ⊢ (0g‘𝐻) = (0g‘𝐻)
49		eqid 2763	. . . . . . . . 9 ⊢ (+g‘𝐻) = (+g‘𝐻)
50		eqid 2763	. . . . . . . . 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 18903	. . . . . . . 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 18	. . . . . . 7 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ {(0g‘𝐻)})
53	52, 43	sseldd 3914	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐻)})
54		elsni 4601	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐻)} → 0 = (0g‘𝐻))
55	53, 54	syl 18	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐻))
56	55	sneqd 4596	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐻)})
57	52, 56	sseqtrrd 3950	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ { 0 })
58	44, 57	eqssd 3930	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
59	23, 58	eqtr3d 2800	. . . 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 3945	. . 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 2800	. . 3 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
62	35	seqeq2d 14191	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
63	62	fveq1d 6918	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
64	63	eqeq2d 2774	. . . . . . . 8 ⊢ (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
65	64	anbi2d 642	. . . . . . 7 ⊢ (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
66	65	rexbidv 3190	. . . . . 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 1956	. . . . 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 6549	. . . 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 14191	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , (𝐹 ∘ 𝑓)) = seq1((+g‘𝐻), (𝐹 ∘ 𝑓)))
70	69	fveq1d 6918	. . . . . . . 8 ⊢ (𝜑 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))
71	70	eqeq2d 2774	. . . . . . 7 ⊢ (𝜑 → (𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) ↔ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))))
72	71	anbi2d 642	. . . . . 6 ⊢ (𝜑 → ((𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))) ↔ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))
73	72	exbidv 1956	. . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))) ↔ ∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))
74	73	iotabidv 6549	. . . 4 ⊢ (𝜑 → (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))) = (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶1-1onto(◡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 }))))⟶1-1onto(◡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 }))))⟶1-1onto(◡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 }))))⟶1-1onto(◡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 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))))
77	23	difeq2d 4066	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
78	77	imaeq2d 6070	. . 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 6757	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
82	12, 13, 14, 15, 78, 11, 79, 81	gsumval 18916	. 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 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))))
83	58	difeq2d 4066	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}))
84	83	imaeq2d 6070	. . 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 6724	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘𝐻)))
86	80, 85	mpbid 235	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐻))
87	47, 48, 49, 50, 84, 46, 79, 86	gsumval 18916	. 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 }))))⟶1-1onto(◡𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))))))
88	76, 82, 87	3eqtr4d 2808	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ifcif 4485  {csn 4588  ^◡ccnv 5657  ran crn 5659   “ cima 5661   ∘ ccom 5662  ℩cio 6518  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  1c1 11240  ℤ_≥cuz 13005  ...cfz 13679  seqcseq 14184  ♯chash 14515  Basecbs 17431   ↾_s cress 17452  +_gcplusg 17472  0_gc0g 17660   Σ_g cgsu 17661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-seq 14185  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-0g 17662  df-gsum 17663

This theorem is used by:  gsumsubm  19081  regsumfsum  21795  regsumsupp  21983  frlmgsum  22133  imasdsf1olem  24747  gsumsubg  33706  gsumzrsum  33725  ressply1evls1  34200  mplgsum  34296  mplmonprod  34297  esumpfinvallem  34817  sge0tsms  47663  aacllem  51367