Theorem gsumfs2d 33146

Description: Express a finite sum over a two-dimensional range as a double sum. Version of gsum2d 19905 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 7860	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → (𝐴 “ {𝑥}) ∈ V)
8		gsumfs2d.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
9	8	ffnd 6664	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐴)
10	9	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝐹 Fn 𝐴)
11	5	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝐴 ∈ 𝑋)
12	2	fvexi 6849	. . . . . . 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 3914	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝑦 ∈ (𝐴 “ {𝑥}))
16		vex 3445	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17		vex 3445	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
18	16, 17	elimasn 6050	. . . . . . . . 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 3915	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
22	16, 17	elimasn 6050	. . . . . . . . 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 3913	. . . . . 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 8115	. . . . 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 9276	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → (𝐹 supp 0 ) ∈ Fin)
30		imafi2 9265	. . . . . 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 727	. . . . . 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 7032	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐵)
35		suppssdm 8121	. . . . . . . 8 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
36	35, 8	fssdm 6682	. . . . . . 7 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
37	36	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ dom (𝐹 supp 0 )) → (𝐹 supp 0 ) ⊆ 𝐴)
38		imass1 6061	. . . . . 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 33138	. . . 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 5192	. . 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 7376	. 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 7847	. . 3 ⊢ (𝜑 → dom 𝐴 ∈ V)
44	9	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐹 Fn 𝐴)
45	5	ad2antrr 727	. . . . . . 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 3915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ¬ 𝑥 ∈ dom (𝐹 supp 0 ))
50	16, 17	opeldm 5857	. . . . . . . . 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 3913	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
53	44, 45, 46, 52	fvdifsupp 8115	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
54	53	mpteq2dva 5192	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 ))
55	54	oveq2d 7376	. . . 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 19737	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ Mnd)
57	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → 𝐴 ∈ 𝑋)
58	57	imaexd 7860	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝐴 “ {𝑥}) ∈ V)
59	2	gsumz 18765	. . . . 5 ⊢ ((𝑊 ∈ Mnd ∧ (𝐴 “ {𝑥}) ∈ V) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )) = 0 )
60	56, 58, 59	syl2an2r 686	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )) = 0 )
61	55, 60	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = 0 )
62		dmfi 9239	. . . 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 7860	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝐴 “ {𝑥}) ∈ V)
67	8	ad2antrr 727	. . . . . 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 7032	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐵)
70	69	fmpttd 7062	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)):(𝐴 “ {𝑥})⟶𝐵)
71	66	mptexd 7172	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ V)
72	70	ffnd 6664	. . . . 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 2737	. . . . . . . 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 2837	. . . . . . . . 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 2857	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
84	9	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 𝐹 Fn 𝐴)
85	5	ad3antrrr 731	. . . . . . . . . 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 3913	. . . . . . . . . 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 8115	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
92	77, 78, 81, 83, 91	syl1111anc 841	. . . . . . . 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 6951	. . . . . . 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 864	. . . . 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 9280	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) finSupp 0 )
99	1, 2, 64, 66, 70, 98	gsumcl 19848	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ dom 𝐴) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) ∈ 𝐵)
100		dmss 5852	. . . 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 33138	. 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 6904	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑧)))
104	103	oveq2d 7376	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝑊 Σg (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑧))))
105		ssidd 3958	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
106	1, 2, 3, 5, 8, 105, 27	gsumres 19846	. . 3 ⊢ (𝜑 → (𝑊 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝑊 Σg 𝐹))
107		nfcv 2899	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑧)
108		gsumfs2d.p	. . . 4 ⊢ Ⅎ𝑥𝜑
109		fveq2 6835	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
110		gsumfs2d.r	. . . . 5 ⊢ (𝜑 → Rel 𝐴)
111		relss 5732	. . . . 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 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 supp 0 )) → 𝑧 ∈ 𝐴)
115	113, 114	ffvelcdmd 7032	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 supp 0 )) → (𝐹‘𝑧) ∈ 𝐵)
116	107, 108, 1, 109, 112, 28, 3, 115	gsummpt2d 33134	. . 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 2780	. 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 2783	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 1542  Ⅎwnf 1785   ∈ wcel 2114  Vcvv 3441   ∖ cdif 3899   ⊆ wss 3902  {csn 4581  ⟨cop 4587   class class class wbr 5099   ↦ cmpt 5180  dom cdm 5625   ↾ cres 5627   “ cima 5628  Rel wrel 5630   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7360   supp csupp 8104  Fincfn 8887   finSupp cfsupp 9268  Basecbs 17140  0_gc0g 17363   Σ_g cgsu 17364  Mndcmnd 18663  CMndccmn 19713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-fsupp 9269  df-oi 9419  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-fzo 13575  df-seq 13929  df-hash 14258  df-sets 17095  df-slot 17113  df-ndx 17125  df-base 17141  df-ress 17162  df-plusg 17194  df-0g 17365  df-gsum 17366  df-mre 17509  df-mrc 17510  df-acs 17512  df-mgm 18569  df-sgrp 18648  df-mnd 18664  df-submnd 18713  df-mulg 19002  df-cntz 19250  df-cmn 19715

This theorem is referenced by:  gsumwrd2dccat  33162