Theorem gsumpropd2lem 18900

Description: Lemma for gsumpropd2 18901. (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 486	. . . . . 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 2765	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑠(+g‘𝐺)𝑡) = 𝑡 ↔ (𝑠(+g‘𝐻)𝑡) = 𝑡))
5	3	oveqrspc2v 7473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑎(+g‘𝐻)𝑏))
6	5	oveqrspc2v 7473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺))) → (𝑡(+g‘𝐺)𝑠) = (𝑡(+g‘𝐻)𝑠))
7	6	ancom2s 663	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑡(+g‘𝐺)𝑠) = (𝑡(+g‘𝐻)𝑠))
8	7	eqeq1d 2765	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → ((𝑡(+g‘𝐺)𝑠) = 𝑡 ↔ (𝑡(+g‘𝐻)𝑠) = 𝑡))
9	4, 8	anbi12d 644	. . . . . . 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 473	. . . . . 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 3329	. . . . 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 3432	. . . 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 3945	. . 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 2764	. . . 4 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺))
15	14, 1, 3	grpidpropd 18874	. . 3 ⊢ (𝜑 → (0g‘𝐺) = (0g‘𝐻))
16		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → 𝑛 ∈ (ℤ≥‘𝑚))
17		gsumpropd2.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺))
18	17	ad2antrr 739	. . . . . . . . . . . 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 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → Fun 𝐹)
21		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ (𝑚...𝑛))
22		simplrr 790	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → dom 𝐹 = (𝑚...𝑛))
23	21, 22	eleqtrrd 2867	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → 𝑠 ∈ dom 𝐹)
24		fvelrn 7103	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ 𝑠 ∈ dom 𝐹) → (𝐹‘𝑠) ∈ ran 𝐹)
25	20, 23, 24	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ 𝑠 ∈ (𝑚...𝑛)) → (𝐹‘𝑠) ∈ ran 𝐹)
26	18, 25	sseldd 3914	. . . . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺))
29	3	adantlr 728	. . . . . . . . . . 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 14224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛))
31	30	eqeq2d 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝑚) ∧ dom 𝐹 = (𝑚...𝑛))) → (𝑥 = (seq𝑚((+g‘𝐺), 𝐹)‘𝑛) ↔ 𝑥 = (seq𝑚((+g‘𝐻), 𝐹)‘𝑛)))
32	31	anassrs 473	. . . . . . . 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 590	. . . . . . 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 3188	. . . . . 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 1956	. . . . 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 6544	. . . 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 4066	. . . . . . . . . . . . . . 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 6065	. . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = 𝐵)
42	41	fveq2d 6915	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵))
43	42	fveq2d 6915	. . . . . . . . . . 11 ⊢ (𝜑 → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴)) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴)) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))
45		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) → (♯‘𝐵) ∈ (ℤ≥‘1))
46	17	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ran 𝐹 ⊆ (Base‘𝐺))
47		f1ofun 6851	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → Fun 𝑓)
48	47	ad3antlr 744	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝑓)
49		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐵)))
50		f1odm 6853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → dom 𝑓 = (1...(♯‘𝐴)))
51	50	ad3antlr 744	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐴)))
52	42	oveq2d 7462	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
53	52	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (1...(♯‘𝐴)) = (1...(♯‘𝐵)))
54	51, 53	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐵)))
55	49, 54	eleqtrrd 2867	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ dom 𝑓)
56		fvco 7009	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝑓 ∧ 𝑎 ∈ dom 𝑓) → ((𝐹 ∘ 𝑓)‘𝑎) = (𝐹‘(𝑓‘𝑎)))
57	48, 55, 56	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) = (𝐹‘(𝑓‘𝑎)))
58	19	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → Fun 𝐹)
59		difpreima 7091	. . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . 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	eqtrid 2810	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 = ((◡𝐹 “ V) ∖ (◡𝐹 “ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})))
62		difss 4075	. . . . . . . . . . . . . . . . . . 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	eqsstrdi 3957	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ (◡𝐹 “ V))
64		dfdm4 5885	. . . . . . . . . . . . . . . . . . 19 ⊢ dom 𝐹 = ran ◡𝐹
65		dfrn4 6210	. . . . . . . . . . . . . . . . . . 19 ⊢ ran ◡𝐹 = (◡𝐹 “ V)
66	64, 65	eqtri 2786	. . . . . . . . . . . . . . . . . 18 ⊢ dom 𝐹 = (◡𝐹 “ V)
67	63, 66	sseqtrrdi 3954	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
68	67	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝐴 ⊆ dom 𝐹)
69		f1of 6849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 → 𝑓:(1...(♯‘𝐴))⟶𝐴)
70	69	ad3antlr 744	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐴))⟶𝐴)
71	49, 53	eleqtrrd 2867	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → 𝑎 ∈ (1...(♯‘𝐴)))
72	70, 71	ffvelcdmd 7112	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑎) ∈ 𝐴)
73	68, 72	sseldd 3914	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑎) ∈ dom 𝐹)
74		fvelrn 7103	. . . . . . . . . . . . . . 15 ⊢ ((Fun 𝐹 ∧ (𝑓‘𝑎) ∈ dom 𝐹) → (𝐹‘(𝑓‘𝑎)) ∈ ran 𝐹)
75	58, 73, 74	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → (𝐹‘(𝑓‘𝑎)) ∈ ran 𝐹)
76	57, 75	eqeltrd 2864	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) ∈ ran 𝐹)
77	46, 76	sseldd 3914	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ 𝑎 ∈ (1...(♯‘𝐵))) → ((𝐹 ∘ 𝑓)‘𝑎) ∈ (Base‘𝐺))
78	27	caovclg 7640	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) ∈ (Base‘𝐺))
79	78	ad4ant14 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) ∈ (Base‘𝐺))
80	5	ad4ant14 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) ∧ (𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺))) → (𝑎(+g‘𝐺)𝑏) = (𝑎(+g‘𝐻)𝑏))
81	45, 77, 79, 80	seqfeq4 14224	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ (♯‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))
82		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ≥‘1)) → ¬ (♯‘𝐵) ∈ (ℤ≥‘1))
83		1z 12756	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
84		seqfn 14186	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℤ → seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1))
85		fndm 6667	. . . . . . . . . . . . . . . . 17 ⊢ (seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1) → dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) = (ℤ≥‘1))
86	83, 84, 85	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) = (ℤ≥‘1)
87	86	eleq2i 2856	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) ↔ (♯‘𝐵) ∈ (ℤ≥‘1))
88	82, 87	sylnibr 332	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ≥‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)))
89		ndmfv 6943	. . . . . . . . . . . . . 14 ⊢ (¬ (♯‘𝐵) ∈ dom seq1((+g‘𝐺), (𝐹 ∘ 𝑓)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = ∅)
90	88, 89	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = ∅)
91		seqfn 14186	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ ℤ → seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1))
92		fndm 6667	. . . . . . . . . . . . . . . . 17 ⊢ (seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) Fn (ℤ≥‘1) → dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) = (ℤ≥‘1))
93	83, 91, 92	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) = (ℤ≥‘1)
94	93	eleq2i 2856	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) ↔ (♯‘𝐵) ∈ (ℤ≥‘1))
95	82, 94	sylnibr 332	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ≥‘1)) → ¬ (♯‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)))
96		ndmfv 6943	. . . . . . . . . . . . . 14 ⊢ (¬ (♯‘𝐵) ∈ dom seq1((+g‘𝐻), (𝐹 ∘ 𝑓)) → (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = ∅)
97	95, 96	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = ∅)
98	90, 97	eqtr4d 2801	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (♯‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))
99	98	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) ∧ ¬ (♯‘𝐵) ∈ (ℤ≥‘1)) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))
100	81, 99	pm2.61dan 825	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐵)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))
101	44, 100	eqtrd 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴)) = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))
102	101	eqeq2d 2774	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴) → (𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴)) ↔ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵))))
103	102	pm5.32da 590	. . . . . . 7 ⊢ (𝜑 → ((𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))))
104	52	f1oeq2d 6845	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ↔ 𝑓:(1...(♯‘𝐵))⟶1-1onto𝐴))
105	41	f1oeq3d 6846	. . . . . . . . 9 ⊢ (𝜑 → (𝑓:(1...(♯‘𝐵))⟶1-1onto𝐴 ↔ 𝑓:(1...(♯‘𝐵))⟶1-1onto𝐵))
106	104, 105	bitrd 282	. . . . . . . 8 ⊢ (𝜑 → (𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ↔ 𝑓:(1...(♯‘𝐵))⟶1-1onto𝐵))
107	106	anbi1d 643	. . . . . . 7 ⊢ (𝜑 → ((𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵))) ↔ (𝑓:(1...(♯‘𝐵))⟶1-1onto𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))))
108	103, 107	bitrd 282	. . . . . 6 ⊢ (𝜑 → ((𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴))) ↔ (𝑓:(1...(♯‘𝐵))⟶1-1onto𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))))
109	108	exbidv 1956	. . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴))) ↔ ∃𝑓(𝑓:(1...(♯‘𝐵))⟶1-1onto𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))))
110	109	iotabidv 6544	. . . 4 ⊢ (𝜑 → (℩𝑥∃𝑓(𝑓:(1...(♯‘𝐴))⟶1-1onto𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴)))) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝐵))⟶1-1onto𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))))
111	36, 110	ifeq12d 4507	. . 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))⟶1-1onto𝐴 ∧ 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))⟶1-1onto𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵))))))
112	13, 15, 111	ifbieq12d 4514	. 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))⟶1-1onto𝐴 ∧ 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))⟶1-1onto𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))))))
113		eqid 2763	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
114		eqid 2763	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
115		eqid 2763	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
116		eqid 2763	. . 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)}
117	39	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})))
118		gsumpropd2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
119		gsumpropd2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
120		eqidd 2764	. . 3 ⊢ (𝜑 → dom 𝐹 = dom 𝐹)
121	113, 114, 115, 116, 117, 118, 119, 120	gsumvalx 18897	. 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))⟶1-1onto𝐴 ∧ 𝑥 = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘𝐴)))))))
122		eqid 2763	. . 3 ⊢ (Base‘𝐻) = (Base‘𝐻)
123		eqid 2763	. . 3 ⊢ (0g‘𝐻) = (0g‘𝐻)
124		eqid 2763	. . 3 ⊢ (+g‘𝐻) = (+g‘𝐻)
125		eqid 2763	. . 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)}
126	40	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)})))
127		gsumpropd2.h	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
128	122, 123, 124, 125, 126, 127, 119, 120	gsumvalx 18897	. 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))⟶1-1onto𝐵 ∧ 𝑥 = (seq1((+g‘𝐻), (𝐹 ∘ 𝑓))‘(♯‘𝐵)))))))
129	112, 121, 128	3eqtr4d 2808	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  ifcif 4485  ^◡ccnv 5655  dom cdm 5656  ran crn 5657   “ cima 5659   ∘ ccom 5660  ℩cio 6513  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  1c1 11231  ℤcz 12723  ℤ_≥cuz 12995  ...cfz 13669  seqcseq 14174  ♯chash 14504  Basecbs 17419  +_gcplusg 17460  0_gc0g 17642   Σ_g cgsu 17643

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 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

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-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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-er 8748  df-en 9005  df-dom 9006  df-sdom 9007  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-n0 12637  df-z 12724  df-uz 12996  df-fz 13670  df-seq 14175  df-0g 17644  df-gsum 17645

This theorem is used by:  gsumpropd2  18901