Theorem gsumpropd2lem 17096

Description: Lemma for gsumpropd2 17097. (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 480	. . . . . 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 2612	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g‘𝐺)𝑡) = 𝑡 ↔ (𝑠(+g‘𝐻)𝑡) = 𝑡))
5	3	oveqrspc2v 6572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑎(+g‘𝐻)𝑏))
6	5	oveqrspc2v 6572	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g‘𝐺)𝑠) = (𝑡(+g‘𝐻)𝑠))
7	6	ancom2s 840	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g‘𝐺)𝑠) = (𝑡(+g‘𝐻)𝑠))
8	7	eqeq1d 2612	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g‘𝐺)𝑠) = 𝑡 ↔ (𝑡(+g‘𝐻)𝑠) = 𝑡))
9	4, 8	anbi12d 743	. . . . . . 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 678	. . . . . 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 3131	. . . . 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 3169	. . . 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 3596	. . 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 2611	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
15	14, 1, 3	grpidpropd 17084	. . 3 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
16		simprl 790	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ≥‘𝑚))
17		gsumpropd2.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
18	17	ad2antrr 758	. . . . . . . . . . . 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 758	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
21		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
22		simplrr 797	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
23	21, 22	eleqtrrd 2691	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24		fvelrn 6260	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ dom 𝐹) → (𝐹‘𝑠) ∈ ran 𝐹)
25	20, 23, 24	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹‘𝑠) ∈ ran 𝐹)
26	18, 25	sseldd 3569	. . . . . . . . . . 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 747	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
29	3	adantlr 747	. . . . . . . . . . 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 12712	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
31	30	eqeq2d 2620	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
32	31	anassrs 678	. . . . . . . 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 671	. . . . . . 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 3031	. . . . . 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 1837	. . . . 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 5789	. . . 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 3690	. . . . . . . . . . . . . . 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 5385	. . . . . . . . . . . . . 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 2669	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = 𝐵)
42	41	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝐴) = (#‘𝐵))
43	42	fveq2d 6107	. . . . . . . . . . 11 ⊢ (𝜑 → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
45		simpr 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) → (#‘𝐵) ∈ (ℤ≥‘1))
46	17	ad3antrrr 762	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47		f1ofun 6052	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → Fun 𝑓)
48	47	ad3antlr 763	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → Fun 𝑓)
49		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ (1...(#‘𝐵)))
50		f1odm 6054	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → dom 𝑓 = (1...(#‘𝐴)))
51	50	ad3antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → dom 𝑓 = (1...(#‘𝐴)))
52	42	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(#‘𝐴)) = (1...(#‘𝐵)))
53	52	ad3antrrr 762	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (1...(#‘𝐴)) = (1...(#‘𝐵)))
54	51, 53	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → dom 𝑓 = (1...(#‘𝐵)))
55	49, 54	eleqtrrd 2691	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ dom 𝑓)
56		fvco 6184	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑓 ∧ 𝑎 ∈ dom 𝑓) → ((𝐹 ∘ 𝑓)‘𝑎) = (𝐹‘(𝑓‘𝑎)))
57	48, 55, 56	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) = (𝐹‘(𝑓‘𝑎)))
58	19	ad3antrrr 762	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → Fun 𝐹)
59		difpreima 6251	. . . . . . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 = ((◡𝐹 “ V) ∖ (◡𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})))
62		difss 3699	. . . . . . . . . . . . . . . . . . 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 3618	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ (◡𝐹 “ V))
64		dfdm4 5238	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝐹 = ran ◡𝐹
65		dfrn4 5513	. . . . . . . . . . . . . . . . . . 19 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
66	64, 65	eqtri 2632	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝐹 = (◡𝐹 “ V)
67	63, 66	syl6sseqr 3615	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
68	67	ad3antrrr 762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69		f1of 6050	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → 𝑓:(1...(#‘𝐴))⟶𝐴)
70	69	ad3antlr 763	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑓:(1...(#‘𝐴))⟶𝐴)
71	49, 53	eleqtrrd 2691	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → 𝑎 ∈ (1...(#‘𝐴)))
72	70, 71	ffvelrnd 6268	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝑓‘𝑎) ∈ 𝐴)
73	68, 72	sseldd 3569	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝑓‘𝑎) ∈ dom 𝐹)
74		fvelrn 6260	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ (𝑓‘𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓‘𝑎)) ∈ ran 𝐹)
75	58, 73, 74	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → (𝐹‘(𝑓‘𝑎)) ∈ ran 𝐹)
76	57, 75	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) ∈ ran 𝐹)
77	46, 76	sseldd 3569	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(#‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) ∈ (Base‘𝐺))
78		simpll 786	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ (#‘𝐵) ∈ (ℤ≥‘1)) → 𝜑)
79	27	caovclg 6724	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) ∈ (Base‘𝐺))
80	78, 79	sylan 487	. . . . . . . . . . . 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 487	. . . . . . . . . . . 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 12712	. . . . . . . . . . 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 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → ¬ (#‘𝐵) ∈ (ℤ≥‘1))
84		1z 11284	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
85		seqfn 12675	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℤ → seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1))
86		fndm 5904	. . . . . . . . . . . . . . . . 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 2680	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) ↔ (#‘𝐵) ∈ (ℤ≥‘1))
89	83, 88	sylnibr 318	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → ¬ (#‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)))
90		ndmfv 6128	. . . . . . . . . . . . . 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 12675	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℤ → seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1))
93		fndm 5904	. . . . . . . . . . . . . . . . 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 2680	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) ↔ (#‘𝐵) ∈ (ℤ≥‘1))
96	83, 95	sylnibr 318	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → ¬ (#‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)))
97		ndmfv 6128	. . . . . . . . . . . . . 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 2647	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (#‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
100	99	adantlr 747	. . . . . . . . . . 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 828	. . . . . . . . . 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 2644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(#‘𝐴)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(#‘𝐵)))
103	102	eqeq2d 2620	. . . . . . . 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 671	. . . . . . 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 6041	. . . . . . . . . 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 6042	. . . . . . . . . 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 267	. . . . . . . 8 ⊢ (𝜑 → (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 ↔ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵))
110	109	anbi1d 737	. . . . . . 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 267	. . . . . 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 1837	. . . . 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 5789	. . . 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 4056	. . 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 4063	. 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 2610	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
117		eqid 2610	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
118		eqid 2610	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
119		eqid 2610	. . 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 2611	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
124	116, 117, 118, 119, 120, 121, 122, 123	gsumvalx 17093	. 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 2610	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
126		eqid 2610	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
127		eqid 2610	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
128		eqid 2610	. . 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 17093	. 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 2654	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   ∘ ccom 5042  ℩cio 5766  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1c1 9816  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923   Σ_g cgsu 15924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-0g 15925  df-gsum 15926

This theorem is referenced by:  gsumpropd2  17097