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Theorem gsumfs2d 33058
Description: Express a finite sum over a two-dimensional range as a double sum. Version of gsum2d 19990 using finite support. (Contributed by Thierry Arnoux, 5-Oct-2025.)
Hypotheses
Ref Expression
gsumfs2d.p 𝑥𝜑
gsumfs2d.b 𝐵 = (Base‘𝑊)
gsumfs2d.1 0 = (0g𝑊)
gsumfs2d.r (𝜑 → Rel 𝐴)
gsumfs2d.2 (𝜑𝐹 finSupp 0 )
gsumfs2d.w (𝜑𝑊 ∈ CMnd)
gsumfs2d.3 (𝜑𝐹:𝐴𝐵)
gsumfs2d.a (𝜑𝐴𝑋)
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.
StepHypRef Expression
1 gsumfs2d.b . . . . 5 𝐵 = (Base‘𝑊)
2 gsumfs2d.1 . . . . 5 0 = (0g𝑊)
3 gsumfs2d.w . . . . . 6 (𝜑𝑊 ∈ CMnd)
43adantr 480 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → 𝑊 ∈ CMnd)
5 gsumfs2d.a . . . . . . 7 (𝜑𝐴𝑋)
65adantr 480 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → 𝐴𝑋)
76imaexd 7938 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → (𝐴 “ {𝑥}) ∈ V)
8 gsumfs2d.3 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
98ffnd 6737 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
109ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝐹 Fn 𝐴)
115ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝐴𝑋)
122fvexi 6920 . . . . . . 7 0 ∈ V
1312a1i 11 . . . . . 6 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 0 ∈ V)
14 simpr 484 . . . . . . . . 9 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥})))
1514eldifad 3963 . . . . . . . 8 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → 𝑦 ∈ (𝐴 “ {𝑥}))
16 vex 3484 . . . . . . . . . 10 𝑥 ∈ V
17 vex 3484 . . . . . . . . . 10 𝑦 ∈ V
1816, 17elimasn 6108 . . . . . . . . 9 (𝑦 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
1918biimpi 216 . . . . . . . 8 (𝑦 ∈ (𝐴 “ {𝑥}) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2015, 19syl 17 . . . . . . 7 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2114eldifbd 3964 . . . . . . . 8 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
2216, 17elimasn 6108 . . . . . . . . 9 (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
2322biimpri 228 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ) → 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
2421, 23nsyl 140 . . . . . . 7 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
2520, 24eldifd 3962 . . . . . 6 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
2610, 11, 13, 25fvdifsupp 8196 . . . . 5 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ ((𝐴 “ {𝑥}) ∖ ((𝐹 supp 0 ) “ {𝑥}))) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
27 gsumfs2d.2 . . . . . . . 8 (𝜑𝐹 finSupp 0 )
2827fsuppimpd 9409 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
2928adantr 480 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → (𝐹 supp 0 ) ∈ Fin)
30 imafi2 32723 . . . . . 6 ((𝐹 supp 0 ) ∈ Fin → ((𝐹 supp 0 ) “ {𝑥}) ∈ Fin)
3129, 30syl 17 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → ((𝐹 supp 0 ) “ {𝑥}) ∈ Fin)
328ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐹:𝐴𝐵)
3319adantl 481 . . . . . 6 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3432, 33ffvelcdmd 7105 . . . . 5 (((𝜑𝑥 ∈ dom (𝐹 supp 0 )) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐵)
35 suppssdm 8202 . . . . . . . 8 (𝐹 supp 0 ) ⊆ dom 𝐹
3635, 8fssdm 6755 . . . . . . 7 (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴)
3736adantr 480 . . . . . 6 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → (𝐹 supp 0 ) ⊆ 𝐴)
38 imass1 6119 . . . . . 6 ((𝐹 supp 0 ) ⊆ 𝐴 → ((𝐹 supp 0 ) “ {𝑥}) ⊆ (𝐴 “ {𝑥}))
3937, 38syl 17 . . . . 5 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → ((𝐹 supp 0 ) “ {𝑥}) ⊆ (𝐴 “ {𝑥}))
401, 2, 4, 7, 26, 31, 34, 39gsummptres2 33056 . . . 4 ((𝜑𝑥 ∈ dom (𝐹 supp 0 )) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))
4140mpteq2dva 5242 . . 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⟩)))))
4241oveq2d 7447 . 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⟩))))))
435dmexd 7925 . . 3 (𝜑 → dom 𝐴 ∈ V)
449ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐹 Fn 𝐴)
455ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐴𝑋)
4612a1i 11 . . . . . . 7 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 0 ∈ V)
4719adantl 481 . . . . . . . 8 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
48 simplr 769 . . . . . . . . . 10 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 )))
4948eldifbd 3964 . . . . . . . . 9 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ¬ 𝑥 ∈ dom (𝐹 supp 0 ))
5016, 17opeldm 5918 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ) → 𝑥 ∈ dom (𝐹 supp 0 ))
5149, 50nsyl 140 . . . . . . . 8 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
5247, 51eldifd 3962 . . . . . . 7 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
5344, 45, 46, 52fvdifsupp 8196 . . . . . 6 (((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
5453mpteq2dva 5242 . . . . 5 ((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) = (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 ))
5554oveq2d 7447 . . . 4 ((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )))
563cmnmndd 19822 . . . . 5 (𝜑𝑊 ∈ Mnd)
575adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → 𝐴𝑋)
5857imaexd 7938 . . . . 5 ((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝐴 “ {𝑥}) ∈ V)
592gsumz 18849 . . . . 5 ((𝑊 ∈ Mnd ∧ (𝐴 “ {𝑥}) ∈ V) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )) = 0 )
6056, 58, 59syl2an2r 685 . . . 4 ((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ 0 )) = 0 )
6155, 60eqtrd 2777 . . 3 ((𝜑𝑥 ∈ (dom 𝐴 ∖ dom (𝐹 supp 0 ))) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) = 0 )
62 dmfi 9375 . . . 4 ((𝐹 supp 0 ) ∈ Fin → dom (𝐹 supp 0 ) ∈ Fin)
6328, 62syl 17 . . 3 (𝜑 → dom (𝐹 supp 0 ) ∈ Fin)
643adantr 480 . . . 4 ((𝜑𝑥 ∈ dom 𝐴) → 𝑊 ∈ CMnd)
655adantr 480 . . . . 5 ((𝜑𝑥 ∈ dom 𝐴) → 𝐴𝑋)
6665imaexd 7938 . . . 4 ((𝜑𝑥 ∈ dom 𝐴) → (𝐴 “ {𝑥}) ∈ V)
678ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → 𝐹:𝐴𝐵)
6819adantl 481 . . . . . 6 (((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
6967, 68ffvelcdmd 7105 . . . . 5 (((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝐵)
7069fmpttd 7135 . . . 4 ((𝜑𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)):(𝐴 “ {𝑥})⟶𝐵)
7166mptexd 7244 . . . . 5 ((𝜑𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) ∈ V)
7270ffnd 6737 . . . . 5 ((𝜑𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) Fn (𝐴 “ {𝑥}))
7312a1i 11 . . . . 5 ((𝜑𝑥 ∈ dom 𝐴) → 0 ∈ V)
7428adantr 480 . . . . . 6 ((𝜑𝑥 ∈ dom 𝐴) → (𝐹 supp 0 ) ∈ Fin)
7574, 30syl 17 . . . . 5 ((𝜑𝑥 ∈ 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 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝜑)
78 simp-4r 784 . . . . . . . . 9 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑥 ∈ dom 𝐴)
79 simpr 484 . . . . . . . . . 10 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑦 = 𝑡)
80 simpllr 776 . . . . . . . . . 10 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑡 ∈ (𝐴 “ {𝑥}))
8179, 80eqeltrd 2841 . . . . . . . . 9 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → 𝑦 ∈ (𝐴 “ {𝑥}))
82 simplr 769 . . . . . . . . . 10 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥}))
8379, 82eqneltrd 2861 . . . . . . . . 9 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}))
849ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 𝐹 Fn 𝐴)
855ad3antrrr 730 . . . . . . . . . 10 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 𝐴𝑋)
8612a1i 11 . . . . . . . . . 10 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 0 ∈ V)
8768adantr 480 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ 𝐴)
8823con3i 154 . . . . . . . . . . . 12 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
8988adantl 481 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ¬ ⟨𝑥, 𝑦⟩ ∈ (𝐹 supp 0 ))
9087, 89eldifd 3962 . . . . . . . . . 10 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 ∖ (𝐹 supp 0 )))
9184, 85, 86, 90fvdifsupp 8196 . . . . . . . . 9 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑦 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥})) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
9277, 78, 81, 83, 91syl1111anc 841 . . . . . . . 8 (((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) ∧ 𝑦 = 𝑡) → (𝐹‘⟨𝑥, 𝑦⟩) = 0 )
93 simplr 769 . . . . . . . 8 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 𝑡 ∈ (𝐴 “ {𝑥}))
9412a1i 11 . . . . . . . 8 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) → 0 ∈ V)
9576, 92, 93, 94fvmptd2 7024 . . . . . . 7 ((((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) ∧ ¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥})) → ((𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))‘𝑡) = 0 )
9695ex 412 . . . . . 6 (((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) → (¬ 𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥}) → ((𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))‘𝑡) = 0 ))
9796orrd 864 . . . . 5 (((𝜑𝑥 ∈ dom 𝐴) ∧ 𝑡 ∈ (𝐴 “ {𝑥})) → (𝑡 ∈ ((𝐹 supp 0 ) “ {𝑥}) ∨ ((𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))‘𝑡) = 0 ))
9871, 72, 73, 75, 97finnzfsuppd 9413 . . . 4 ((𝜑𝑥 ∈ dom 𝐴) → (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩)) finSupp 0 )
991, 2, 64, 66, 70, 98gsumcl 19933 . . 3 ((𝜑𝑥 ∈ dom 𝐴) → (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))) ∈ 𝐵)
100 dmss 5913 . . . 4 ((𝐹 supp 0 ) ⊆ 𝐴 → dom (𝐹 supp 0 ) ⊆ dom 𝐴)
10136, 100syl 17 . . 3 (𝜑 → dom (𝐹 supp 0 ) ⊆ dom 𝐴)
1021, 2, 3, 43, 61, 63, 99, 101gsummptres2 33056 . 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⟩))))))
1038, 36feqresmpt 6978 . . . 4 (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑧)))
104103oveq2d 7447 . . 3 (𝜑 → (𝑊 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝑊 Σg (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑧))))
105 ssidd 4007 . . . 4 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 ))
1061, 2, 3, 5, 8, 105, 27gsumres 19931 . . 3 (𝜑 → (𝑊 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝑊 Σg 𝐹))
107 nfcv 2905 . . . 4 𝑦(𝐹𝑧)
108 gsumfs2d.p . . . 4 𝑥𝜑
109 fveq2 6906 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
110 gsumfs2d.r . . . . 5 (𝜑 → Rel 𝐴)
111 relss 5791 . . . . 5 ((𝐹 supp 0 ) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐹 supp 0 )))
11236, 110, 111sylc 65 . . . 4 (𝜑 → Rel (𝐹 supp 0 ))
1138adantr 480 . . . . 5 ((𝜑𝑧 ∈ (𝐹 supp 0 )) → 𝐹:𝐴𝐵)
11436sselda 3983 . . . . 5 ((𝜑𝑧 ∈ (𝐹 supp 0 )) → 𝑧𝐴)
115113, 114ffvelcdmd 7105 . . . 4 ((𝜑𝑧 ∈ (𝐹 supp 0 )) → (𝐹𝑧) ∈ 𝐵)
116107, 108, 1, 109, 112, 28, 3, 115gsummpt2d 33052 . . 3 (𝜑 → (𝑊 Σg (𝑧 ∈ (𝐹 supp 0 ) ↦ (𝐹𝑧))) = (𝑊 Σg (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))
117104, 106, 1163eqtr3d 2785 . 2 (𝜑 → (𝑊 Σg 𝐹) = (𝑊 Σg (𝑥 ∈ dom (𝐹 supp 0 ) ↦ (𝑊 Σg (𝑦 ∈ ((𝐹 supp 0 ) “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))
11842, 102, 1173eqtr4rd 2788 1 (𝜑 → (𝑊 Σg 𝐹) = (𝑊 Σg (𝑥 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑦 ∈ (𝐴 “ {𝑥}) ↦ (𝐹‘⟨𝑥, 𝑦⟩))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wnf 1783  wcel 2108  Vcvv 3480  cdif 3948  wss 3951  {csn 4626  cop 4632   class class class wbr 5143  cmpt 5225  dom cdm 5685  cres 5687  cima 5688  Rel wrel 5690   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401  Basecbs 17247  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747  CMndccmn 19798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800
This theorem is referenced by:  gsumwrd2dccat  33070
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