Theorem gsum2dlem2 20191

Description: Lemma for gsum2d 20192. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by AV, 8-Jun-2019.)

Hypotheses

Ref	Expression
gsum2d.b	⊢ 𝐵 = (Base‘𝐺)
gsum2d.z	⊢ 0 = (0g‘𝐺)
gsum2d.g	⊢ (𝜑 → 𝐺 ∈ CMnd)
gsum2d.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsum2d.r	⊢ (𝜑 → Rel 𝐴)
gsum2d.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)
gsum2d.s	⊢ (𝜑 → dom 𝐴 ⊆ 𝐷)
gsum2d.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsum2d.w	⊢ (𝜑 → 𝐹 finSupp 0 )

Assertion

Ref	Expression
gsum2dlem2	⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))

Distinct variable groups:   𝑗,𝑘,𝐴   𝑗,𝐹,𝑘   𝑗,𝐺,𝑘   𝜑,𝑗,𝑘   𝐵,𝑗,𝑘   𝐷,𝑗,𝑘   0 ,𝑗,𝑘

Allowed substitution hints:   𝑉(𝑗, 𝑘)   𝑊(𝑗, 𝑘)

Proof of Theorem gsum2dlem2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsum2d.w	. . . 4 ⊢ (𝜑 → 𝐹 finSupp 0 )
2	1	fsuppimpd 9365	. . 3 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
3		dmfi 9328	. . 3 ⊢ ((𝐹 supp 0 ) ∈ Fin → dom (𝐹 supp 0 ) ∈ Fin)
4	2, 3	syl 18	. 2 ⊢ (𝜑 → dom (𝐹 supp 0 ) ∈ Fin)
5		reseq2 5964	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐴 ↾ 𝑥) = (𝐴 ↾ ∅))
6		res0 5973	. . . . . . . . 9 ⊢ (𝐴 ↾ ∅) = ∅
7	5, 6	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝐴 ↾ 𝑥) = ∅)
8	7	reseq2d 5969	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ ∅))
9		res0 5973	. . . . . . 7 ⊢ (𝐹 ↾ ∅) = ∅
10	8, 9	eqtrdi 2811	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐹 ↾ (𝐴 ↾ 𝑥)) = ∅)
11	10	oveq2d 7437	. . . . 5 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg ∅))
12		mpteq1 5193	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
13		mpt0 6682	. . . . . . 7 ⊢ (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅
14	12, 13	eqtrdi 2811	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅)
15	14	oveq2d 7437	. . . . 5 ⊢ (𝑥 = ∅ → (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg ∅))
16	11, 15	eqeq12d 2776	. . . 4 ⊢ (𝑥 = ∅ → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg ∅) = (𝐺 Σg ∅)))
17	16	imbi2d 343	. . 3 ⊢ (𝑥 = ∅ → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))))
18		reseq2 5964	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ↾ 𝑥) = (𝐴 ↾ 𝑦))
19	18	reseq2d 5969	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ (𝐴 ↾ 𝑦)))
20	19	oveq2d 7437	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))))
21		mpteq1 5193	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
22	21	oveq2d 7437	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
23	20, 22	eqeq12d 2776	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
24	23	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
25		reseq2 5964	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↾ 𝑥) = (𝐴 ↾ (𝑦 ∪ {𝑧})))
26	25	reseq2d 5969	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
27	26	oveq2d 7437	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))))
28		mpteq1 5193	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
29	28	oveq2d 7437	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
30	27, 29	eqeq12d 2776	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
31	30	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
32		reseq2 5964	. . . . . . 7 ⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐴 ↾ 𝑥) = (𝐴 ↾ dom (𝐹 supp 0 )))
33	32	reseq2d 5969	. . . . . 6 ⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 ))))
34	33	oveq2d 7437	. . . . 5 ⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))))
35		mpteq1 5193	. . . . . 6 ⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))
36	35	oveq2d 7437	. . . . 5 ⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))
37	34, 36	eqeq12d 2776	. . . 4 ⊢ (𝑥 = dom (𝐹 supp 0 ) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
38	37	imbi2d 343	. . 3 ⊢ (𝑥 = dom (𝐹 supp 0 ) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
39		eqidd 2761	. . 3 ⊢ (𝜑 → (𝐺 Σg ∅) = (𝐺 Σg ∅))
40		oveq1 7428	. . . . . 6 ⊢ ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
41		gsum2d.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
42		gsum2d.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
43		eqid 2760	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
44		gsum2d.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ CMnd)
45	44	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐺 ∈ CMnd)
46		gsum2d.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
47	46	resexd 6016	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
48	47	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V)
49		gsum2d.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
50		resss 5991	. . . . . . . . . . 11 ⊢ (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴
51		fssres 6749	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
52	49, 50, 51	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
53	52	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵)
54	49	ffund 6715	. . . . . . . . . . . 12 ⊢ (𝜑 → Fun 𝐹)
55	54	funresd 6584	. . . . . . . . . . 11 ⊢ (𝜑 → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
56	55	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))
57	2	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹 supp 0 ) ∈ Fin)
58	49, 46	fexd 7234	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
59	42	fvexi 6900	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
60		ressuppss 8204	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
61	58, 59, 60	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
62	61	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 ))
63	57, 62	ssfid 9264	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)
64	58	resexd 6016	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V)
65		isfsupp 9361	. . . . . . . . . . . 12 ⊢ (((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
66	64, 59, 65	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
67	66	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈ Fin)))
68	56, 63, 67	mpbir2and 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 )
69		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
70		disjsn 4668	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
71	69, 70	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
72	71	reseq2d 5969	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↾ (𝑦 ∩ {𝑧})) = (𝐴 ↾ ∅))
73		resindi 5985	. . . . . . . . . 10 ⊢ (𝐴 ↾ (𝑦 ∩ {𝑧})) = ((𝐴 ↾ 𝑦) ∩ (𝐴 ↾ {𝑧}))
74	72, 73, 6	3eqtr3g 2818	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↾ 𝑦) ∩ (𝐴 ↾ {𝑧})) = ∅)
75		resundi 5983	. . . . . . . . . 10 ⊢ (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴 ↾ 𝑦) ∪ (𝐴 ↾ {𝑧}))
76	75	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴 ↾ 𝑦) ∪ (𝐴 ↾ {𝑧})))
77	41, 42, 43, 45, 48, 53, 68, 74, 76	gsumsplit 20148	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})))))
78		ssun1 4176	. . . . . . . . . . 11 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
79		ssres2 5994	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (𝐴 ↾ 𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
80		resabs1 5996	. . . . . . . . . . 11 ⊢ ((𝐴 ↾ 𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)) = (𝐹 ↾ (𝐴 ↾ 𝑦)))
81	78, 79, 80	mp2b 10	. . . . . . . . . 10 ⊢ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)) = (𝐹 ↾ (𝐴 ↾ 𝑦))
82	81	oveq2i 7432	. . . . . . . . 9 ⊢ (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))
83		ssun2 4177	. . . . . . . . . . 11 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
84		ssres2 5994	. . . . . . . . . . 11 ⊢ ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → (𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})))
85		resabs1 5996	. . . . . . . . . . 11 ⊢ ((𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧})))
86	83, 84, 85	mp2b 10	. . . . . . . . . 10 ⊢ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧}))
87	86	oveq2i 7432	. . . . . . . . 9 ⊢ (𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))
88	82, 87	oveq12i 7433	. . . . . . . 8 ⊢ ((𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
89	77, 88	eqtrdi 2811	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
90		simprl 783	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
91		gsum2d.r	. . . . . . . . . . 11 ⊢ (𝜑 → Rel 𝐴)
92		gsum2d.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
93		gsum2d.s	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐴 ⊆ 𝐷)
94	41, 42, 44, 46, 91, 92, 93, 49, 1	gsum2dlem1 20190	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
95	94	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑗 ∈ 𝑦) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
96		vex 3454	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
97	96	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V)
98		sneq 4590	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑧 → {𝑗} = {𝑧})
99	98	imaeq2d 6051	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑧 → (𝐴 “ {𝑗}) = (𝐴 “ {𝑧}))
100		oveq1 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑧 → (𝑗𝐹𝑘) = (𝑧𝐹𝑘))
101	99, 100	mpteq12dv 5191	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑧 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))
102	101	oveq2d 7437	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑧 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))))
103	102	eleq1d 2845	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵 ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵))
104	103	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)))
105	104, 94	chvarvv 2021	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
106	105	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)
107	41, 43, 45, 90, 95, 97, 69, 106, 102	gsumunsn 20180	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))))
108	98	reseq2d 5969	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑧 → (𝐴 ↾ {𝑗}) = (𝐴 ↾ {𝑧}))
109	108	reseq2d 5969	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑧 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝐹 ↾ (𝐴 ↾ {𝑧})))
110	109	oveq2d 7437	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑧 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
111	102, 110	eqeq12d 2776	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
112	111	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗})))) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))))
113		imaexg 7926	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ {𝑗}) ∈ V)
114	46, 113	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 “ {𝑗}) ∈ V)
115		vex 3454	. . . . . . . . . . . . . . . 16 ⊢ 𝑗 ∈ V
116		vex 3454	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
117	115, 116	elimasn 6087	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) ↔ ⟨𝑗, 𝑘⟩ ∈ 𝐴)
118		df-ov 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑗𝐹𝑘) = (𝐹‘⟨𝑗, 𝑘⟩)
119	49	ffvelcdmda 7085	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝐹‘⟨𝑗, 𝑘⟩) ∈ 𝐵)
120	118, 119	eqeltrid 2864	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵)
121	117, 120	sylan2b 606	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵)
122	121	fmpttd 7116	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵)
123		funmpt 6579	. . . . . . . . . . . . . . 15 ⊢ Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))
124	123	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
125		rnfi 9333	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 supp 0 ) ∈ Fin → ran (𝐹 supp 0 ) ∈ Fin)
126	2, 125	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran (𝐹 supp 0 ) ∈ Fin)
127	117	biimpi 219	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝐴 “ {𝑗}) → ⟨𝑗, 𝑘⟩ ∈ 𝐴)
128	115, 116	opelrn 5924	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ) → 𝑘 ∈ ran (𝐹 supp 0 ))
129	128	con3i 155	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑘 ∈ ran (𝐹 supp 0 ) → ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 ))
130	127, 129	anim12i 625	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )))
131		eldif 4049	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )))
132		eldif 4049	. . . . . . . . . . . . . . . . . 18 ⊢ (⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (⟨𝑗, 𝑘⟩ ∈ 𝐴 ∧ ¬ ⟨𝑗, 𝑘⟩ ∈ (𝐹 supp 0 )))
133	130, 131, 132	3imtr4i 295	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
134		ssidd 3931	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
135	59	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 ∈ V)
136	49, 134, 46, 135	suppssr 8216	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘⟨𝑗, 𝑘⟩) = 0 )
137	118, 136	eqtrid 2807	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ⟨𝑗, 𝑘⟩ ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
138	133, 137	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 )
139	138, 114	suppss2 8221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 ))
140	126, 139	ssfid 9264	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)
141	114	mptexd 7231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V)
142		isfsupp 9361	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V ∧ 0 ∈ V) → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)))
143	141, 59, 142	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈ Fin)))
144	124, 140, 143	mpbir2and 726	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 )
145		2ndconst 8115	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ V → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗}))
146	115, 145	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗}))
147	41, 42, 44, 114, 122, 144, 146	gsumf1o 20136	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
148		1st2nd2 8041	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
149		xp1st 8034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st ‘𝑥) ∈ {𝑗})
150		elsni 4597	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥) ∈ {𝑗} → (1st ‘𝑥) = 𝑗)
151	149, 150	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st ‘𝑥) = 𝑗)
152	151	opeq1d 4845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩ = ⟨𝑗, (2nd ‘𝑥)⟩)
153	148, 152	eqtrd 2795	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = ⟨𝑗, (2nd ‘𝑥)⟩)
154	153	fveq2d 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹‘𝑥) = (𝐹‘⟨𝑗, (2nd ‘𝑥)⟩))
155		df-ov 7424	. . . . . . . . . . . . . . . 16 ⊢ (𝑗𝐹(2nd ‘𝑥)) = (𝐹‘⟨𝑗, (2nd ‘𝑥)⟩)
156	154, 155	eqtr4di 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹‘𝑥) = (𝑗𝐹(2nd ‘𝑥)))
157	156	mpteq2ia 5199	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd ‘𝑥)))
158	49	feqmptd 6954	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
159	158	reseq1d 5968	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗})))
160		resss 5991	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ↾ {𝑗}) ⊆ 𝐴
161		resmpt 6028	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ↾ {𝑗}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹‘𝑥)))
162	160, 161	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹‘𝑥))
163		ressn 6288	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ↾ {𝑗}) = ({𝑗} × (𝐴 “ {𝑗}))
164	163	mpteq1i 5195	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥))
165	162, 164	eqtri 2783	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥))
166	159, 165	eqtrdi 2811	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥)))
167		xp2nd 8035	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑗}))
168	167	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗}))) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑗}))
169		fo2nd 8023	. . . . . . . . . . . . . . . . . . 19 ⊢ 2nd :V–onto→V
170		fof 6797	. . . . . . . . . . . . . . . . . . 19 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
171	169, 170	mp1i 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 2nd :V⟶V)
172	171	feqmptd 6954	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 2nd = (𝑥 ∈ V ↦ (2nd ‘𝑥)))
173	172	reseq1d 5968	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))))
174		ssv 3932	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑗} × (𝐴 “ {𝑗})) ⊆ V
175		resmpt 6028	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑗} × (𝐴 “ {𝑗})) ⊆ V → ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd ‘𝑥)))
176	174, 175	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ V ↦ (2nd ‘𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd ‘𝑥))
177	173, 176	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd ‘𝑥)))
178		eqidd 2761	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))
179		oveq2 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (2nd ‘𝑥) → (𝑗𝐹𝑘) = (𝑗𝐹(2nd ‘𝑥)))
180	168, 177, 178, 179	fmptco 7131	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd ‘𝑥))))
181	157, 166, 180	3eqtr4a 2821	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))))
182	181	oveq2d 7437	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))))
183	147, 182	eqtr4d 2798	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))))
184	112, 183	chvarvv 2021	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
185	184	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))
186	185	oveq2d 7437	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
187	107, 186	eqtrd 2795	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))
188	89, 187	eqeq12d 2776	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))))
189	40, 188	imbitrrid 249	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
190	189	expcom 419	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜑 → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
191	190	a2d 30	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) → (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))))
192	17, 24, 31, 38, 39, 191	findcard2s 9185	. 2 ⊢ (dom (𝐹 supp 0 ) ∈ Fin → (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))
193	4, 192	mpcom 39	1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4584  ⟨cop 4834   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ cima 5653   ∘ ccom 5654  Rel wrel 5655  Fun wfun 6533  ⟶wf 6535  –onto→wfo 6537  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7421  1^st c1st 8000  2^nd c2nd 8001   supp csupp 8180  Fincfn 8977   finSupp cfsupp 9357  Basecbs 17393  +_gcplusg 17434  0_gc0g 17616   Σ_g cgsu 17617  CMndccmn 20000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-of 7694  df-om 7879  df-1st 8002  df-2nd 8003  df-supp 8181  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-fsupp 9358  df-oi 9508  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-n0 12611  df-z 12698  df-uz 12970  df-fz 13644  df-fzo 13793  df-seq 14149  df-hash 14479  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-0g 17618  df-gsum 17619  df-mre 17762  df-mrc 17763  df-acs 17765  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-submnd 18985  df-mulg 19284  df-cntz 19537  df-cmn 20002

This theorem is used by:  gsum2d  20192