Theorem gsumress 18641

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 7367	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
2	1	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
3	2	ovanraleqv 7384	. . . . . . . 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 3923	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ 𝐵)
7		gsumress.c	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
8	7	ralrimiva 3130	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
9	3, 6, 8	elrabd 3637	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
10	9	snssd 4753	. . . . . 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 2737	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
14		gsumress.o	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
15		eqid 2737	. . . . . . . . 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 18631	. . . . . . . 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 3923	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐺)})
19		elsni 4585	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐺)} → 0 = (0g‘𝐺))
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐺))
21	20	sneqd 4580	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐺)})
22	17, 21	sseqtrrd 3960	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ { 0 })
23	10, 22	eqssd 3940	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
24	2	ovanraleqv 7384	. . . . . . . . 9 ⊢ (𝑦 = 0 → (∀𝑥 ∈ 𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
25	4	sselda 3922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝐵)
26	25, 7	syldan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
27	26	ralrimiva 3130	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
28	24, 5, 27	elrabd 3637	. . . . . . . 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 17199	. . . . . . . . . 10 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐻))
31	4, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 = (Base‘𝐻))
32		fvex 6847	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐻) ∈ V
33	31, 32	eqeltrdi 2845	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ V)
34	29, 14	ressplusg 17245	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ V → + = (+g‘𝐻))
35	33, 34	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → + = (+g‘𝐻))
36	35	oveqd 7377	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐻)𝑥))
37	36	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g‘𝐻)𝑥) = 𝑥))
38	35	oveqd 7377	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐻)𝑦))
39	38	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g‘𝐻)𝑦) = 𝑥))
40	37, 39	anbi12d 633	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)))
41	31, 40	raleqbidv 3312	. . . . . . . . 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 3408	. . . . . . . 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 2839	. . . . . . 7 ⊢ (𝜑 → 0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
44	43	snssd 4753	. . . . . 6 ⊢ (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
45	29	ovexi 7394	. . . . . . . . 9 ⊢ 𝐻 ∈ V
46	45	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐻 ∈ V)
47		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
48		eqid 2737	. . . . . . . . 9 ⊢ (0g‘𝐻) = (0g‘𝐻)
49		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝐻) = (+g‘𝐻)
50		eqid 2737	. . . . . . . . 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 18631	. . . . . . . 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 3923	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ {(0g‘𝐻)})
54		elsni 4585	. . . . . . . . 9 ⊢ ( 0 ∈ {(0g‘𝐻)} → 0 = (0g‘𝐻))
55	53, 54	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 = (0g‘𝐻))
56	55	sneqd 4580	. . . . . . 7 ⊢ (𝜑 → { 0 } = {(0g‘𝐻)})
57	52, 56	sseqtrrd 3960	. . . . . 6 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)} ⊆ { 0 })
58	44, 57	eqssd 3940	. . . . 5 ⊢ (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)})
59	23, 58	eqtr3d 2774	. . . 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 3955	. . 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 2774	. . 3 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
62	35	seqeq2d 13961	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g‘𝐻), 𝐹))
63	62	fveq1d 6836	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
64	63	eqeq2d 2748	. . . . . . . 8 ⊢ (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
65	64	anbi2d 631	. . . . . . 7 ⊢ (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
66	65	rexbidv 3162	. . . . . 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 1923	. . . . 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 6476	. . . 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 13961	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , (𝐹 ∘ 𝑓)) = seq1((+g‘𝐻), (𝐹 ∘ 𝑓)))
70	69	fveq1d 6836	. . . . . . . 8 ⊢ (𝜑 → (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))
71	70	eqeq2d 2748	. . . . . . 7 ⊢ (𝜑 → (𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) ↔ 𝑧 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 }))))))
72	71	anbi2d 631	. . . . . 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 1923	. . . . 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 6476	. . . 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 4489	. . 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 4496	. 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 4067	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ 𝐵 ∣ ∀𝑥 ∈ 𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
78	77	imaeq2d 6019	. . 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 6679	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
82	12, 13, 14, 15, 78, 11, 79, 81	gsumval 18636	. 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 4067	. . . 4 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g‘𝐻)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐻)𝑦) = 𝑥)}))
84	83	imaeq2d 6019	. . 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 6647	. . . 4 ⊢ (𝜑 → (𝐹:𝐴⟶𝑆 ↔ 𝐹:𝐴⟶(Base‘𝐻)))
86	80, 85	mpbid 232	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶(Base‘𝐻))
87	47, 48, 49, 50, 84, 46, 79, 86	gsumval 18636	. 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 2782	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ifcif 4467  {csn 4568  ^◡ccnv 5623  ran crn 5625   “ cima 5627   ∘ ccom 5628  ℩cio 6446  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7360  1c1 11030  ℤ_≥cuz 12779  ...cfz 13452  seqcseq 13954  ♯chash 14283  Basecbs 17170   ↾_s cress 17191  +_gcplusg 17211  0_gc0g 17393   Σ_g cgsu 17394

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-seq 13955  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-0g 17395  df-gsum 17396

This theorem is referenced by:  gsumsubm  18794  regsumfsum  21425  regsumsupp  21612  frlmgsum  21762  imasdsf1olem  24348  gsumsubg  33122  gsumzrsum  33141  ressply1evls1  33640  mplgsum  33712  mplmonprod  33713  esumpfinvallem  34234  sge0tsms  46826  aacllem  50288