Theorem gsumresfi 14263

Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) (Revised by Jim Kingdon, 24-Sep-2026.)

Hypotheses

Ref	Expression
gsumresfi.b	⊢ 𝐵 = (Base‘𝐺)
gsumresfi.z	⊢ 0 = (0g‘𝐺)
gsumresfi.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsumresfi.a	⊢ (𝜑 → 𝐴 ∈ Fin)
gsumresfi.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumresfi.s	⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∖ 𝑊)(𝐹‘𝑥) = 0 )
gsumresfi.w	⊢ (𝜑 → 𝑊 ∈ Fin)
gsumresfi.ss	⊢ (𝜑 → 𝑊 ⊆ 𝐴)

Assertion

Ref	Expression
gsumresfi	⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))

Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐹   𝑥,𝑊

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐺(𝑥)

Proof of Theorem gsumresfi

Dummy variables 𝑤 𝑦 𝑧 𝑞 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumresfi.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2		frel 5541	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
3		resdm 5104	. . . . 5 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
4	1, 2, 3	3syl 17	. . . 4 ⊢ (𝜑 → (𝐹 ↾ dom 𝐹) = 𝐹)
5	1	fdmd 5543	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
6	5	reseq2d 5065	. . . 4 ⊢ (𝜑 → (𝐹 ↾ dom 𝐹) = (𝐹 ↾ 𝐴))
7	4, 6	eqtr3d 2273	. . 3 ⊢ (𝜑 → 𝐹 = (𝐹 ↾ 𝐴))
8	7	oveq2d 6104	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ↾ 𝐴)))
9		ineq2 3426	. . . . . 6 ⊢ (𝑤 = ∅ → (𝑊 ∩ 𝑤) = (𝑊 ∩ ∅))
10	9	reseq2d 5065	. . . . 5 ⊢ (𝑤 = ∅ → (𝐹 ↾ (𝑊 ∩ 𝑤)) = (𝐹 ↾ (𝑊 ∩ ∅)))
11	10	oveq2d 6104	. . . 4 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ (𝑊 ∩ ∅))))
12		reseq2 5060	. . . . 5 ⊢ (𝑤 = ∅ → (𝐹 ↾ 𝑤) = (𝐹 ↾ ∅))
13	12	oveq2d 6104	. . . 4 ⊢ (𝑤 = ∅ → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ ∅)))
14	11, 13	eqeq12d 2253	. . 3 ⊢ (𝑤 = ∅ → ((𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ 𝑤)) ↔ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ ∅))) = (𝐺 Σg (𝐹 ↾ ∅))))
15		ineq2 3426	. . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑊 ∩ 𝑤) = (𝑊 ∩ 𝑦))
16	15	reseq2d 5065	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ (𝑊 ∩ 𝑤)) = (𝐹 ↾ (𝑊 ∩ 𝑦)))
17	16	oveq2d 6104	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))))
18		reseq2 5060	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝑦))
19	18	oveq2d 6104	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ 𝑦)))
20	17, 19	eqeq12d 2253	. . 3 ⊢ (𝑤 = 𝑦 → ((𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ 𝑤)) ↔ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))))
21		ineq2 3426	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝑊 ∩ 𝑤) = (𝑊 ∩ (𝑦 ∪ {𝑧})))
22	21	reseq2d 5065	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ (𝑊 ∩ 𝑤)) = (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧}))))
23	22	oveq2d 6104	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))))
24		reseq2 5060	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑤) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
25	24	oveq2d 6104	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))))
26	23, 25	eqeq12d 2253	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ 𝑤)) ↔ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧})))))
27		ineq2 3426	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑊 ∩ 𝑤) = (𝑊 ∩ 𝐴))
28	27	reseq2d 5065	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ (𝑊 ∩ 𝑤)) = (𝐹 ↾ (𝑊 ∩ 𝐴)))
29	28	oveq2d 6104	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝐴))))
30		reseq2 5060	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝐹 ↾ 𝑤) = (𝐹 ↾ 𝐴))
31	30	oveq2d 6104	. . . 4 ⊢ (𝑤 = 𝐴 → (𝐺 Σg (𝐹 ↾ 𝑤)) = (𝐺 Σg (𝐹 ↾ 𝐴)))
32	29, 31	eqeq12d 2253	. . 3 ⊢ (𝑤 = 𝐴 → ((𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑤))) = (𝐺 Σg (𝐹 ↾ 𝑤)) ↔ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝐴))) = (𝐺 Σg (𝐹 ↾ 𝐴))))
33		in0 3557	. . . . . 6 ⊢ (𝑊 ∩ ∅) = ∅
34	33	reseq2i 5062	. . . . 5 ⊢ (𝐹 ↾ (𝑊 ∩ ∅)) = (𝐹 ↾ ∅)
35	34	oveq2i 6099	. . . 4 ⊢ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ ∅))) = (𝐺 Σg (𝐹 ↾ ∅))
36	35	a1i 9	. . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ ∅))) = (𝐺 Σg (𝐹 ↾ ∅)))
37		simpr 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦)))
38	1	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐹:𝐴⟶𝐵)
39		inss2 3452	. . . . . . . . . . . . 13 ⊢ (𝑊 ∩ 𝑦) ⊆ 𝑦
40		simprl 535	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ⊆ 𝐴)
41	39, 40	sstrid 3259	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑊 ∩ 𝑦) ⊆ 𝐴)
42	38, 41	feqresmpt 5760	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹 ↾ (𝑊 ∩ 𝑦)) = (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟)))
43	42	oveq2d 6104	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟))))
44	43	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟))))
45	37, 44	eqtr3d 2273	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟))))
46	45	oveq1d 6103	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) → ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺)(𝐹‘𝑧)) = ((𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟)))(+g‘𝐺)(𝐹‘𝑧)))
47	46	adantr 276	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺)(𝐹‘𝑧)) = ((𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟)))(+g‘𝐺)(𝐹‘𝑧)))
48		gsumresfi.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
49		eqid 2238	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
50		gsumresfi.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ CMnd)
51	50	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐺 ∈ CMnd)
52		simplr 533	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
53	38	adantr 276	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑟 ∈ 𝑦) → 𝐹:𝐴⟶𝐵)
54	40	sselda 3248	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑟 ∈ 𝑦) → 𝑟 ∈ 𝐴)
55	53, 54	ffvelcdmd 5847	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑟 ∈ 𝑦) → (𝐹‘𝑟) ∈ 𝐵)
56		simprr 537	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
57	56	eldifad 3231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
58	56	eldifbd 3232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
59	38, 57	ffvelcdmd 5847	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹‘𝑧) ∈ 𝐵)
60		fveq2 5698	. . . . . . . . . 10 ⊢ (𝑟 = 𝑧 → (𝐹‘𝑟) = (𝐹‘𝑧))
61	60	adantl 277	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑟 = 𝑧) → (𝐹‘𝑟) = (𝐹‘𝑧))
62	48, 49, 51, 52, 55, 57, 58, 59, 61	gsumunsnd 14260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝑟 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐹‘𝑟))) = ((𝐺 Σg (𝑟 ∈ 𝑦 ↦ (𝐹‘𝑟)))(+g‘𝐺)(𝐹‘𝑧)))
63	57	snssd 3860	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → {𝑧} ⊆ 𝐴)
64	40, 63	unssd 3405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
65	38, 64	feqresmpt 5760	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹 ↾ (𝑦 ∪ {𝑧})) = (𝑟 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐹‘𝑟)))
66	65	oveq2d 6104	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) = (𝐺 Σg (𝑟 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐹‘𝑟))))
67	38, 40	feqresmpt 5760	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹 ↾ 𝑦) = (𝑟 ∈ 𝑦 ↦ (𝐹‘𝑟)))
68	67	oveq2d 6104	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ 𝑦)) = (𝐺 Σg (𝑟 ∈ 𝑦 ↦ (𝐹‘𝑟))))
69	68	oveq1d 6103	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺)(𝐹‘𝑧)) = ((𝐺 Σg (𝑟 ∈ 𝑦 ↦ (𝐹‘𝑟)))(+g‘𝐺)(𝐹‘𝑧)))
70	62, 66, 69	3eqtr4d 2281	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺)(𝐹‘𝑧)))
71	70	ad2antrr 492	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺)(𝐹‘𝑧)))
72		indi 3478	. . . . . . . . . . . 12 ⊢ (𝑊 ∩ (𝑦 ∪ {𝑧})) = ((𝑊 ∩ 𝑦) ∪ (𝑊 ∩ {𝑧}))
73		snssi 3859	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑊 → {𝑧} ⊆ 𝑊)
74		dfss1 3435	. . . . . . . . . . . . . 14 ⊢ ({𝑧} ⊆ 𝑊 ↔ (𝑊 ∩ {𝑧}) = {𝑧})
75	73, 74	sylib 122	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑊 → (𝑊 ∩ {𝑧}) = {𝑧})
76	75	uneq2d 3383	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑊 → ((𝑊 ∩ 𝑦) ∪ (𝑊 ∩ {𝑧})) = ((𝑊 ∩ 𝑦) ∪ {𝑧}))
77	72, 76	eqtrid 2283	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑊 → (𝑊 ∩ (𝑦 ∪ {𝑧})) = ((𝑊 ∩ 𝑦) ∪ {𝑧}))
78	77	adantl 277	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝑊 ∩ (𝑦 ∪ {𝑧})) = ((𝑊 ∩ 𝑦) ∪ {𝑧}))
79	78	reseq2d 5065	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧}))) = (𝐹 ↾ ((𝑊 ∩ 𝑦) ∪ {𝑧})))
80	41, 63	unssd 3405	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝑊 ∩ 𝑦) ∪ {𝑧}) ⊆ 𝐴)
81	38, 80	feqresmpt 5760	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹 ↾ ((𝑊 ∩ 𝑦) ∪ {𝑧})) = (𝑟 ∈ ((𝑊 ∩ 𝑦) ∪ {𝑧}) ↦ (𝐹‘𝑟)))
82	81	ad2antrr 492	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐹 ↾ ((𝑊 ∩ 𝑦) ∪ {𝑧})) = (𝑟 ∈ ((𝑊 ∩ 𝑦) ∪ {𝑧}) ↦ (𝐹‘𝑟)))
83	79, 82	eqtrd 2271	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧}))) = (𝑟 ∈ ((𝑊 ∩ 𝑦) ∪ {𝑧}) ↦ (𝐹‘𝑟)))
84	83	oveq2d 6104	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑟 ∈ ((𝑊 ∩ 𝑦) ∪ {𝑧}) ↦ (𝐹‘𝑟))))
85		gsumresfi.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ Fin)
86	85	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑊 ∈ Fin)
87		gsumresfi.ss	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊 ⊆ 𝐴)
88	87	ad2antrr 492	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑊 ⊆ 𝐴)
89		gsumresfi.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ∈ Fin)
90	89	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐴 ∈ Fin)
91		fissfi 7267	. . . . . . . . . . . 12 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ∀𝑠 ∈ 𝐴 DECID 𝑠 ∈ 𝑦)
92	40, 90, 52, 91	syl3anc 1278	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑠 ∈ 𝐴 DECID 𝑠 ∈ 𝑦)
93		ssralv 3312	. . . . . . . . . . 11 ⊢ (𝑊 ⊆ 𝐴 → (∀𝑠 ∈ 𝐴 DECID 𝑠 ∈ 𝑦 → ∀𝑠 ∈ 𝑊 DECID 𝑠 ∈ 𝑦))
94	88, 92, 93	sylc 62	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑠 ∈ 𝑊 DECID 𝑠 ∈ 𝑦)
95		infidc 7252	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Fin ∧ ∀𝑠 ∈ 𝑊 DECID 𝑠 ∈ 𝑦) → (𝑊 ∩ 𝑦) ∈ Fin)
96	86, 94, 95	syl2anc 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝑊 ∩ 𝑦) ∈ Fin)
97	38	adantr 276	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑟 ∈ (𝑊 ∩ 𝑦)) → 𝐹:𝐴⟶𝐵)
98		elinel2 3416	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (𝑊 ∩ 𝑦) → 𝑟 ∈ 𝑦)
99	98, 54	sylan2 286	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑟 ∈ (𝑊 ∩ 𝑦)) → 𝑟 ∈ 𝐴)
100	97, 99	ffvelcdmd 5847	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑟 ∈ (𝑊 ∩ 𝑦)) → (𝐹‘𝑟) ∈ 𝐵)
101		elinel2 3416	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑊 ∩ 𝑦) → 𝑧 ∈ 𝑦)
102	58, 101	nsyl 637	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ (𝑊 ∩ 𝑦))
103	48, 49, 51, 96, 100, 57, 102, 59, 61	gsumunsnd 14260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝑟 ∈ ((𝑊 ∩ 𝑦) ∪ {𝑧}) ↦ (𝐹‘𝑟))) = ((𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟)))(+g‘𝐺)(𝐹‘𝑧)))
104	103	ad2antrr 492	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝑟 ∈ ((𝑊 ∩ 𝑦) ∪ {𝑧}) ↦ (𝐹‘𝑟))) = ((𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟)))(+g‘𝐺)(𝐹‘𝑧)))
105	84, 104	eqtrd 2271	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg (𝑟 ∈ (𝑊 ∩ 𝑦) ↦ (𝐹‘𝑟)))(+g‘𝐺)(𝐹‘𝑧)))
106	47, 71, 105	3eqtr4rd 2282	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))))
107	51	cmnmndd 14201	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝐺 ∈ Mnd)
108		gsumresfi.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
109	38, 40	fssresd 5569	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐹 ↾ 𝑦):𝑦⟶𝐵)
110	48, 108, 51, 52, 109	gsumclfi 14249	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝐵)
111	48, 49, 108	mndrid 13808	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝐺 Σg (𝐹 ↾ 𝑦)) ∈ 𝐵) → ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺) 0 ) = (𝐺 Σg (𝐹 ↾ 𝑦)))
112	107, 110, 111	syl2anc 415	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺) 0 ) = (𝐺 Σg (𝐹 ↾ 𝑦)))
113	112	ad2antrr 492	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺) 0 ) = (𝐺 Σg (𝐹 ↾ 𝑦)))
114	70	ad2antrr 492	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺)(𝐹‘𝑧)))
115		fveqeq2 5707	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = 0 ↔ (𝐹‘𝑧) = 0 ))
116		gsumresfi.s	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴 ∖ 𝑊)(𝐹‘𝑥) = 0 )
117	116	ad4antr 498	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → ∀𝑥 ∈ (𝐴 ∖ 𝑊)(𝐹‘𝑥) = 0 )
118	57	ad2antrr 492	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → 𝑧 ∈ 𝐴)
119		simpr 110	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → ¬ 𝑧 ∈ 𝑊)
120	118, 119	eldifd 3230	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → 𝑧 ∈ (𝐴 ∖ 𝑊))
121	115, 117, 120	rspcdva 2934	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → (𝐹‘𝑧) = 0 )
122	121	oveq2d 6104	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺)(𝐹‘𝑧)) = ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺) 0 ))
123	114, 122	eqtrd 2271	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))) = ((𝐺 Σg (𝐹 ↾ 𝑦))(+g‘𝐺) 0 ))
124		disjsn 3771	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑊)
125	124	biimpri 133	. . . . . . . . . . . 12 ⊢ (¬ 𝑧 ∈ 𝑊 → (𝑊 ∩ {𝑧}) = ∅)
126	125	uneq2d 3383	. . . . . . . . . . 11 ⊢ (¬ 𝑧 ∈ 𝑊 → ((𝑊 ∩ 𝑦) ∪ (𝑊 ∩ {𝑧})) = ((𝑊 ∩ 𝑦) ∪ ∅))
127		un0 3556	. . . . . . . . . . 11 ⊢ ((𝑊 ∩ 𝑦) ∪ ∅) = (𝑊 ∩ 𝑦)
128	126, 127	eqtrdi 2287	. . . . . . . . . 10 ⊢ (¬ 𝑧 ∈ 𝑊 → ((𝑊 ∩ 𝑦) ∪ (𝑊 ∩ {𝑧})) = (𝑊 ∩ 𝑦))
129	72, 128	eqtrid 2283	. . . . . . . . 9 ⊢ (¬ 𝑧 ∈ 𝑊 → (𝑊 ∩ (𝑦 ∪ {𝑧})) = (𝑊 ∩ 𝑦))
130	129	reseq2d 5065	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑊 → (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧}))) = (𝐹 ↾ (𝑊 ∩ 𝑦)))
131	130	oveq2d 6104	. . . . . . 7 ⊢ (¬ 𝑧 ∈ 𝑊 → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))))
132	131, 37	sylan9eqr 2293	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝐹 ↾ 𝑦)))
133	113, 123, 132	3eqtr4rd 2282	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) ∧ ¬ 𝑧 ∈ 𝑊) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))))
134		eleq1w 2299	. . . . . . . . 9 ⊢ (𝑞 = 𝑧 → (𝑞 ∈ 𝑊 ↔ 𝑧 ∈ 𝑊))
135	134	dcbid 850	. . . . . . . 8 ⊢ (𝑞 = 𝑧 → (DECID 𝑞 ∈ 𝑊 ↔ DECID 𝑧 ∈ 𝑊))
136		fissfi 7267	. . . . . . . . . 10 ⊢ ((𝑊 ⊆ 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝑊 ∈ Fin) → ∀𝑞 ∈ 𝐴 DECID 𝑞 ∈ 𝑊)
137	87, 89, 85, 136	syl3anc 1278	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑞 ∈ 𝐴 DECID 𝑞 ∈ 𝑊)
138	137	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∀𝑞 ∈ 𝐴 DECID 𝑞 ∈ 𝑊)
139	135, 138, 57	rspcdva 2934	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → DECID 𝑧 ∈ 𝑊)
140	139	adantr 276	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) → DECID 𝑧 ∈ 𝑊)
141		exmiddc 848	. . . . . 6 ⊢ (DECID 𝑧 ∈ 𝑊 → (𝑧 ∈ 𝑊 ∨ ¬ 𝑧 ∈ 𝑊))
142	140, 141	syl 14	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) → (𝑧 ∈ 𝑊 ∨ ¬ 𝑧 ∈ 𝑊))
143	106, 133, 142	mpjaodan 810	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦))) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧}))))
144	143	ex 115	. . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝑦))) = (𝐺 Σg (𝐹 ↾ 𝑦)) → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝐹 ↾ (𝑦 ∪ {𝑧})))))
145	14, 20, 26, 32, 36, 144, 89	findcard2sd 7200	. 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝐴))) = (𝐺 Σg (𝐹 ↾ 𝐴)))
146		df-ss 3233	. . . . 5 ⊢ (𝑊 ⊆ 𝐴 ↔ (𝑊 ∩ 𝐴) = 𝑊)
147	87, 146	sylib 122	. . . 4 ⊢ (𝜑 → (𝑊 ∩ 𝐴) = 𝑊)
148	147	reseq2d 5065	. . 3 ⊢ (𝜑 → (𝐹 ↾ (𝑊 ∩ 𝐴)) = (𝐹 ↾ 𝑊))
149	148	oveq2d 6104	. 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝑊 ∩ 𝐴))) = (𝐺 Σg (𝐹 ↾ 𝑊)))
150	8, 145, 149	3eqtr2rd 2278	1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝑊)) = (𝐺 Σg 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ∖ cdif 3217   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  {csn 3709   ↦ cmpt 4193  dom cdm 4776   ↾ cres 4778  Rel wrel 4781  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088  Fincfn 7026  Basecbs 13410  +_gcplusg 13490  0_gc0g 13669  Mndcmnd 13788  CMndccmn 14177   Σ_g cgsu 14240

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-ihash 11237  df-ndx 13413  df-slot 13414  df-base 13416  df-plusg 13503  df-0g 13671  df-gzsum 13672  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-minusg 13868  df-mulg 13982  df-cmn 14179  df-gsumfi 14241

This theorem is used by:  gsum2dfi  14273  psrlidmfi  15199  psrridmfi  15200