Theorem gsumfs2d 33040

Description: Express a finite sum over a two-dimensional range as a double sum. Version of gsum2d 20004 using finite support. (Contributed by Thierry Arnoux, 5-Oct-2025.)

Hypotheses

Ref	Expression
gsumfs2d.p	⊢ Ⅎ𝑥𝜑
gsumfs2d.b	⊢ 𝐵 = (Base‘𝑊)
gsumfs2d.1	⊢ 0 = (0g‘𝑊)
gsumfs2d.r	⊢ (𝜑 → Rel 𝐴)
gsumfs2d.2	⊢ (𝜑 → 𝐹 finSupp 0 )
gsumfs2d.w	⊢ (𝜑 → 𝑊 ∈ CMnd)
gsumfs2d.3	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumfs2d.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)

Assertion

Ref	Expression
gsumfs2d	⊢ (𝜑 → (𝑊 Σg 𝐹) = (𝑊 Σg (𝑥 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))

Distinct variable groups:   𝑥, 0 ,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑊,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑋(𝑥,𝑦)

Proof of Theorem gsumfs2d

Dummy variables 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumfs2d.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
2		gsumfs2d.1	. . . . 5 ⊢ 0 = (0g‘𝑊)
3		gsumfs2d.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CMnd)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → 𝑊 ∈ CMnd)
5		gsumfs2d.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → 𝐴 ∈ 𝑋)
7	6	imaexd 7938	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → (𝐴 “ {𝑥}) ∈ V)
8		gsumfs2d.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
9	8	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
10	9	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝐹 Fn 𝐴)
11	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝐴 ∈ 𝑋)
12	2	fvexi 6920	. . . . . . 7 ⊢ 0 ∈ V
13	12	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 0 ∈ V)
14		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥})))
15	14	eldifad 3974	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝑦 ∈ (𝐴 “ {𝑥}))
16		vex 3481	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		vex 3481	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
18	16, 17	elimasn 6109	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
19	18	biimpi 216	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
20	15, 19	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
21	14	eldifbd 3975	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
22	16, 17	elimasn 6109	. . . . . . . . 9 ⊢ (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
23	22	biimpri 228	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ) → 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
24	21, 23	nsyl 140	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
25	20, 24	eldifd 3973	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
26	10, 11, 13, 25	fvdifsupp 8194	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
27		gsumfs2d.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹 finSupp 0 )
28	27	fsuppimpd 9406	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → (𝐹 supp 0 ) ∈ Fin)
30		imafi2 32728	. . . . . 6 ⊢ ((𝐹 supp 0 ) ∈ Fin → ((𝐹 supp 0 ) “ {𝑥}) ∈ Fin)
31	29, 30	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → ((𝐹 supp 0 ) “ {𝑥}) ∈ Fin)
32	8	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐹:𝐴⟶𝐵)
33	19	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
34	32, 33	ffvelcdmd 7104	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐵)
35		suppssdm 8200	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
36	35, 8	fssdm 6755	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
37	36	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → (𝐹 supp 0 ) ⊆ 𝐴)
38		imass1 6121	. . . . . 6 ⊢ ((𝐹 supp 0 ) ⊆ 𝐴 → ((𝐹 supp 0 ) “ {𝑥}) ⊆ (𝐴 “ {𝑥}))
39	37, 38	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → ((𝐹 supp 0 ) “ {𝑥}) ⊆ (𝐴 “ {𝑥}))
40	1, 2, 4, 7, 26, 31, 34, 39	gsummptres2 33038	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
41	40	mpteq2dva 5247	. . 3 ⊢ (𝜑 → (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)))) = (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)))))
42	41	oveq2d 7446	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))) = (𝑊 Σg (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))
43	5	dmexd 7925	. . 3 ⊢ (𝜑 → dom 𝐴 ∈ V)
44	9	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐹 Fn 𝐴)
45	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐴 ∈ 𝑋)
46	12	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 0 ∈ V)
47	19	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
48		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 )))
49	48	eldifbd 3975	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ¬ 𝑥 ∈ dom (𝐹 supp 0 ))
50	16, 17	opeldm 5920	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ) → 𝑥 ∈ dom (𝐹 supp 0 ))
51	49, 50	nsyl 140	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
52	47, 51	eldifd 3973	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
53	44, 45, 46, 52	fvdifsupp 8194	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
54	53	mpteq2dva 5247	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 ))
55	54	oveq2d 7446	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )))
56	3	cmnmndd 19836	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Mnd)
57	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → 𝐴 ∈ 𝑋)
58	57	imaexd 7938	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝐴 “ {𝑥}) ∈ V)
59	2	gsumz 18861	. . . . 5 ⊢ ((𝑊 ∈ Mnd ∧ (𝐴 “ {𝑥}) ∈ V) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )) = 0 )
60	56, 58, 59	syl2an2r 685	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )) = 0 )
61	55, 60	eqtrd 2774	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = 0 )
62		dmfi 9372	. . . 4 ⊢ ((𝐹 supp 0 ) ∈ Fin → dom (𝐹 supp 0 ) ∈ Fin)
63	28, 62	syl 17	. . 3 ⊢ (𝜑 → dom (𝐹 supp 0 ) ∈ Fin)
64	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
65	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → 𝐴 ∈ 𝑋)
66	65	imaexd 7938	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝐴 “ {𝑥}) ∈ V)
67	8	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐹:𝐴⟶𝐵)
68	19	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
69	67, 68	ffvelcdmd 7104	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐵)
70	69	fmpttd 7134	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)):(𝐴 “ {𝑥})⟶𝐵)
71	66	mptexd 7243	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ V)
72	70	ffnd 6737	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) Fn (𝐴 “ {𝑥}))
73	12	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → 0 ∈ V)
74	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝐹 supp 0 ) ∈ Fin)
75	74, 30	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → ((𝐹 supp 0 ) “ {𝑥}) ∈ Fin)
76		eqid 2734	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))
77		simp-4l 783	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝜑)
78		simp-4r 784	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑥 ∈ dom 𝐴)
79		simpr 484	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑦 = 𝑡)
80		simpllr 776	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑡 ∈ (𝐴 “ {𝑥}))
81	79, 80	eqeltrd 2838	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑦 ∈ (𝐴 “ {𝑥}))
82		simplr 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥}))
83	79, 82	eqneltrd 2858	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
84	9	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 𝐹 Fn 𝐴)
85	5	ad3antrrr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 𝐴 ∈ 𝑋)
86	12	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 0 ∈ V)
87	68	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
88	23	con3i 154	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
89	88	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
90	87, 89	eldifd 3973	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
91	84, 85, 86, 90	fvdifsupp 8194	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
92	77, 78, 81, 83, 91	syl1111anc 840	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
93		simplr 769	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 𝑡 ∈ (𝐴 “ {𝑥}))
94	12	a1i 11	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 0 ∈ V)
95	76, 92, 93, 94	fvmptd2 7023	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ((𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))‘𝑡) = 0 )
96	95	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) → (¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥}) → ((𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))‘𝑡) = 0 ))
97	96	orrd 863	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) → (𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥}) ∨ ((𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))‘𝑡) = 0 ))
98	71, 72, 73, 75, 97	finnzfsuppd 9410	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) finSupp 0 )
99	1, 2, 64, 66, 70, 98	gsumcl 19947	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) ∈ 𝐵)
100		dmss 5915	. . . 4 ⊢ ((𝐹 supp 0 ) ⊆ 𝐴 → dom (𝐹 supp 0 ) ⊆ dom 𝐴)
101	36, 100	syl 17	. . 3 ⊢ (𝜑 → dom (𝐹 supp 0 ) ⊆ dom 𝐴)
102	1, 2, 3, 43, 61, 63, 99, 101	gsummptres2 33038	. 2 ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))) = (𝑊 Σg (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))
103	8, 36	feqresmpt 6977	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑧)))
104	103	oveq2d 7446	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝑊 Σg (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑧))))
105		ssidd 4018	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
106	1, 2, 3, 5, 8, 105, 27	gsumres 19945	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝑊 Σg 𝐹))
107		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑧)
108		gsumfs2d.p	. . . 4 ⊢ Ⅎ𝑥𝜑
109		fveq2 6906	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
110		gsumfs2d.r	. . . . 5 ⊢ (𝜑 → Rel 𝐴)
111		relss 5793	. . . . 5 ⊢ ((𝐹 supp 0 ) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐹 supp 0 )))
112	36, 110, 111	sylc 65	. . . 4 ⊢ (𝜑 → Rel (𝐹 supp 0 ))
113	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 supp 0 )) → 𝐹:𝐴⟶𝐵)
114	36	sselda 3994	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 supp 0 )) → 𝑧 ∈ 𝐴)
115	113, 114	ffvelcdmd 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 supp 0 )) → (𝐹‘𝑧) ∈ 𝐵)
116	107, 108, 1, 109, 112, 28, 3, 115	gsummpt2d 33034	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑧))) = (𝑊 Σg (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))
117	104, 106, 116	3eqtr3d 2782	. 2 ⊢ (𝜑 → (𝑊 Σg 𝐹) = (𝑊 Σg (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))
118	42, 102, 117	3eqtr4rd 2785	1 ⊢ (𝜑 → (𝑊 Σg 𝐹) = (𝑊 Σg (𝑥 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1536  Ⅎwnf 1779   ∈ wcel 2105  Vcvv 3477   ∖ cdif 3959   ⊆ wss 3962  {csn 4630  ⟨cop 4636   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5688   ↾ cres 5690   “ cima 5691  Rel wrel 5693   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   supp csupp 8183  Fincfn 8983   finSupp cfsupp 9398  Basecbs 17244  0_gc0g 17485   Σ_g cgsu 17486  Mndcmnd 18759  CMndccmn 19812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814

This theorem is referenced by:  gsumwrd2dccat  33052