Theorem gsumpropd2lem 17273

Description: Lemma for gsumpropd2 17274. (Contributed by Thierry Arnoux, 28-Jun-2017.)

Hypotheses

Ref	Expression
gsumpropd2.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
gsumpropd2.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
gsumpropd2.h	⊢ (𝜑 → 𝐻 ∈ 𝑋)
gsumpropd2.b	⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
gsumpropd2.c	⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
gsumpropd2.e	⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))
gsumpropd2.n	⊢ (𝜑 → Fun 𝐹)
gsumpropd2.r	⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
gsumprop2dlem.1	⊢ 𝐴 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)}))
gsumprop2dlem.2	⊢ 𝐵 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}))

Assertion

Ref	Expression
gsumpropd2lem	⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Distinct variable groups:   𝑡,𝑠,𝐹   𝐺,𝑠,𝑡   𝐻,𝑠,𝑡   𝜑,𝑠,𝑡

Allowed substitution hints:   𝐴(𝑡,𝑠)   𝐵(𝑡,𝑠)   𝑉(𝑡,𝑠)   𝑊(𝑡,𝑠)   𝑋(𝑡,𝑠)

Proof of Theorem gsumpropd2lem

Dummy variables 𝑎 𝑏 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumpropd2.b	. . . . 5 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻))
2	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝐺)) → (Base‘𝐺) = (Base‘𝐻))
3		gsumpropd2.e	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))
4	3	eqeq1d 2624	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g‘𝐺)𝑡) = 𝑡 ↔ (𝑠(+g‘𝐻)𝑡) = 𝑡))
5	3	oveqrspc2v 6673	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑎(+g‘𝐻)𝑏))
6	5	oveqrspc2v 6673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g‘𝐺)𝑠) = (𝑡(+g‘𝐻)𝑠))
7	6	ancom2s 844	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g‘𝐺)𝑠) = (𝑡(+g‘𝐻)𝑠))
8	7	eqeq1d 2624	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g‘𝐺)𝑠) = 𝑡 ↔ (𝑡(+g‘𝐻)𝑠) = 𝑡))
9	4, 8	anbi12d 747	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡) ↔ ((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)))
10	9	anassrs 680	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (Base‘𝐺)) ∧ 𝑡 ∈ (Base‘𝐺)) → (((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡) ↔ ((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)))
11	2, 10	raleqbidva 3154	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (Base‘𝐺)) → (∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡) ↔ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)))
12	1, 11	rabeqbidva 3196	. . . 4 ⊢ (𝜑 → {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)} = {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)})
13	12	sseq2d 3633	. . 3 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)} ↔ ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}))
14		eqidd 2623	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
15	14, 1, 3	grpidpropd 17261	. . 3 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
16		simprl 794	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ≥‘𝑚))
17		gsumpropd2.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
18	17	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → ran 𝐹 ⊆ (Base‘𝐺))
19		gsumpropd2.n	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun 𝐹)
20	19	ad2antrr 762	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
21		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
22		simplrr 801	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
23	21, 22	eleqtrrd 2704	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24		fvelrn 6352	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ dom 𝐹) → (𝐹‘𝑠) ∈ ran 𝐹)
25	20, 23, 24	syl2anc 693	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹‘𝑠) ∈ ran 𝐹)
26	18, 25	sseldd 3604	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹‘𝑠) ∈ (Base‘𝐺))
27		gsumpropd2.c	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
28	27	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
29	3	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡))
30	16, 26, 28, 29	seqfeq4 12850	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
31	30	eqeq2d 2632	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
32	31	anassrs 680	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ dom 𝐹 = (𝑚...𝑛)) → (𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
33	32	pm5.32da 673	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → ((dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)) ↔ (dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
34	33	rexbidva 3049	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
35	34	exbidv 1850	. . . . 5 ⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛)) ↔ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
36	35	iotabidv 5872	. . . 4 ⊢ (𝜑 → (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛))) = (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))))
37	12	difeq2d 3728	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)}) = (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}))
38	37	imaeq2d 5466	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})) = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)})))
39		gsumprop2dlem.1	. . . . . . . . . . . . . 14 ⊢ 𝐴 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)}))
40		gsumprop2dlem.2	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}))
41	38, 39, 40	3eqtr4g 2681	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = 𝐵)
42	41	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝐴) = (#‘𝐵))
43	42	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝜑 → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
44	43	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
45		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) → (#‘𝐵) ∈ (ℤ≥‘1))
46	17	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47		f1ofun 6139	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → Fun 𝑓)
48	47	ad3antlr 767	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → Fun 𝑓)
49		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ (1...(#‘𝐵)))
50		f1odm 6141	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → dom 𝑓 = (1...(#‘𝐴)))
51	50	ad3antlr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → dom 𝑓 = (1...(#‘𝐴)))
52	42	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(#‘𝐴)) = (1...(#‘𝐵)))
53	52	ad3antrrr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (1...(#‘𝐴)) = (1...(#‘𝐵)))
54	51, 53	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → dom 𝑓 = (1...(#‘𝐵)))
55	49, 54	eleqtrrd 2704	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ dom 𝑓)
56		fvco 6274	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑓 ∧ 𝑎 ∈ dom 𝑓) → ((𝐹 ∘ 𝑓)‘𝑎) = (𝐹‘(𝑓‘𝑎)))
57	48, 55, 56	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) = (𝐹‘(𝑓‘𝑎)))
58	19	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → Fun 𝐹)
59		difpreima 6343	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Fun 𝐹 → (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})) = ((◡𝐹 “ V) ∖ (◡𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})))
60	19, 59	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})) = ((◡𝐹 “ V) ∖ (◡𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})))
61	39, 60	syl5eq 2668	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 = ((◡𝐹 “ V) ∖ (◡𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})))
62		difss 3737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((◡𝐹 “ V) ∖ (◡𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})) ⊆ (◡𝐹 “ V)
63	61, 62	syl6eqss 3655	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ (◡𝐹 “ V))
64		dfdm4 5316	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝐹 = ran ◡𝐹
65		dfrn4 5595	. . . . . . . . . . . . . . . . . . 19 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
66	64, 65	eqtri 2644	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝐹 = (◡𝐹 “ V)
67	63, 66	syl6sseqr 3652	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
68	67	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69		f1of 6137	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(#‘𝐴))⟶𝐴)
70	69	ad3antlr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑓:(1...(#‘𝐴))⟶𝐴)
71	49, 53	eleqtrrd 2704	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ (1...(#‘𝐴)))
72	70, 71	ffvelrnd 6360	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝑓‘𝑎) ∈ 𝐴)
73	68, 72	sseldd 3604	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝑓‘𝑎) ∈ dom 𝐹)
74		fvelrn 6352	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ (𝑓‘𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓‘𝑎)) ∈ ran 𝐹)
75	58, 73, 74	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝐹‘(𝑓‘𝑎)) ∈ ran 𝐹)
76	57, 75	eqeltrd 2701	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) ∈ ran 𝐹)
77	46, 76	sseldd 3604	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) ∈ (Base‘𝐺))
78		simpll 790	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) → 𝜑)
79	27	caovclg 6826	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) ∈ (Base‘𝐺))
80	78, 79	sylan 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) ∈ (Base‘𝐺))
81	78, 5	sylan 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑎(+g‘𝐻)𝑏))
82	45, 77, 80, 81	seqfeq4 12850	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
83		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → ¬ (#‘𝐵) ∈ (ℤ≥‘1))
84		1z 11407	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
85		seqfn 12813	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℤ → seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1))
86		fndm 5990	. . . . . . . . . . . . . . . . 17 ⊢ (seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1) → dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) = (ℤ≥‘1))
87	84, 85, 86	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) = (ℤ≥‘1)
88	87	eleq2i 2693	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) ↔ (#‘𝐵) ∈ (ℤ≥‘1))
89	83, 88	sylnibr 319	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → ¬ (#‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)))
90		ndmfv 6218	. . . . . . . . . . . . . 14 ⊢ (¬ (#‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = ∅)
91	89, 90	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = ∅)
92		seqfn 12813	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℤ → seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1))
93		fndm 5990	. . . . . . . . . . . . . . . . 17 ⊢ (seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1) → dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) = (ℤ≥‘1))
94	84, 92, 93	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) = (ℤ≥‘1)
95	94	eleq2i 2693	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) ↔ (#‘𝐵) ∈ (ℤ≥‘1))
96	83, 95	sylnibr 319	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → ¬ (#‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)))
97		ndmfv 6218	. . . . . . . . . . . . . 14 ⊢ (¬ (#‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) → (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = ∅)
98	96, 97	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = ∅)
99	91, 98	eqtr4d 2659	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
100	99	adantlr 751	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
101	82, 100	pm2.61dan 832	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
102	44, 101	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
103	102	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)) ↔ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵))))
104	103	pm5.32da 673	. . . . . . 7 ⊢ (𝜑 → ((𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴))) ↔ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))))
105		f1oeq2 6128	. . . . . . . . . 10 ⊢ ((1...(#‘𝐴)) = (1...(#‘𝐵)) → (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐴))
106	52, 105	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐴))
107		f1oeq3 6129	. . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝑓:(1...(#‘𝐵))–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵))
108	41, 107	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:(1...(#‘𝐵))–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵))
109	106, 108	bitrd 268	. . . . . . . 8 ⊢ (𝜑 → (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵))
110	109	anbi1d 741	. . . . . . 7 ⊢ (𝜑 → ((𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵))) ↔ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))))
111	104, 110	bitrd 268	. . . . . 6 ⊢ (𝜑 → ((𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴))) ↔ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))))
112	111	exbidv 1850	. . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴))) ↔ ∃𝑓(𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))))
113	112	iotabidv 5872	. . . 4 ⊢ (𝜑 → (℩𝑥∃𝑓(𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)))) = (℩𝑥∃𝑓(𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))))
114	36, 113	ifeq12d 4106	. . 3 ⊢ (𝜑 → if(dom 𝐹 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴))))) = if(dom 𝐹 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵))))))
115	13, 15, 114	ifbieq12d 4113	. 2 ⊢ (𝜑 → if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)}, (0g‘𝐺), if(dom 𝐹 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)))))) = if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}, (0g‘𝐻), if(dom 𝐹 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))))))
116		eqid 2622	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
117		eqid 2622	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
118		eqid 2622	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
119		eqid 2622	. . 3 ⊢ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)} = {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)}
120	39	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})))
121		gsumpropd2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
122		gsumpropd2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
123		eqidd 2623	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
124	116, 117, 118, 119, 120, 121, 122, 123	gsumvalx 17270	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)}, (0g‘𝐺), if(dom 𝐹 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)))))))
125		eqid 2622	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
126		eqid 2622	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
127		eqid 2622	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
128		eqid 2622	. . 3 ⊢ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)} = {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}
129	40	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)})))
130		gsumpropd2.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
131	125, 126, 127, 128, 129, 130, 122, 123	gsumvalx 17270	. 2 ⊢ (𝜑 → (𝐻 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)}, (0g‘𝐻), if(dom 𝐹 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝐹 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))))))
132	115, 124, 131	3eqtr4d 2666	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   “ cima 5117   ∘ ccom 5118  ℩cio 5849  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  1c1 9937  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801  #chash 13117  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100   Σ_g cgsu 16101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-0g 16102  df-gsum 16103

This theorem is referenced by:  gsumpropd2  17274