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Theorem gsumzmhm 19054
 Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
gsumzmhm.b 𝐵 = (Base‘𝐺)
gsumzmhm.z 𝑍 = (Cntz‘𝐺)
gsumzmhm.g (𝜑𝐺 ∈ Mnd)
gsumzmhm.h (𝜑𝐻 ∈ Mnd)
gsumzmhm.a (𝜑𝐴𝑉)
gsumzmhm.k (𝜑𝐾 ∈ (𝐺 MndHom 𝐻))
gsumzmhm.f (𝜑𝐹:𝐴𝐵)
gsumzmhm.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzmhm.0 0 = (0g𝐺)
gsumzmhm.w (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
gsumzmhm (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))

Proof of Theorem gsumzmhm
Dummy variables 𝑘 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzmhm.h . . . . . . 7 (𝜑𝐻 ∈ Mnd)
2 gsumzmhm.a . . . . . . 7 (𝜑𝐴𝑉)
3 eqid 2798 . . . . . . . 8 (0g𝐻) = (0g𝐻)
43gsumz 17995 . . . . . . 7 ((𝐻 ∈ Mnd ∧ 𝐴𝑉) → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
51, 2, 4syl2anc 587 . . . . . 6 (𝜑 → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
65adantr 484 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
7 gsumzmhm.k . . . . . . 7 (𝜑𝐾 ∈ (𝐺 MndHom 𝐻))
8 gsumzmhm.0 . . . . . . . 8 0 = (0g𝐺)
98, 3mhm0 17959 . . . . . . 7 (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾0 ) = (0g𝐻))
107, 9syl 17 . . . . . 6 (𝜑 → (𝐾0 ) = (0g𝐻))
1110adantr 484 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾0 ) = (0g𝐻))
126, 11eqtr4d 2836 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (𝐾0 ))
13 gsumzmhm.g . . . . . . . . 9 (𝜑𝐺 ∈ Mnd)
14 gsumzmhm.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
1514, 8mndidcl 17921 . . . . . . . . 9 (𝐺 ∈ Mnd → 0𝐵)
1613, 15syl 17 . . . . . . . 8 (𝜑0𝐵)
1716ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘𝐴) → 0𝐵)
18 gsumzmhm.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
198fvexi 6660 . . . . . . . . 9 0 ∈ V
2019a1i 11 . . . . . . . 8 (𝜑0 ∈ V)
21 fex 6967 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2218, 2, 21syl2anc 587 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
23 suppimacnv 7827 . . . . . . . . . 10 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2422, 20, 23syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
25 ssid 3937 . . . . . . . . 9 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
2624, 25eqsstrdi 3969 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2718, 2, 20, 26gsumcllem 19025 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
28 eqid 2798 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
2914, 28mhmf 17956 . . . . . . . . . 10 (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
307, 29syl 17 . . . . . . . . 9 (𝜑𝐾:𝐵⟶(Base‘𝐻))
3130feqmptd 6709 . . . . . . . 8 (𝜑𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
3231adantr 484 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
33 fveq2 6646 . . . . . . 7 (𝑥 = 0 → (𝐾𝑥) = (𝐾0 ))
3417, 27, 32, 33fmptco 6869 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (𝐾0 )))
3510mpteq2dv 5127 . . . . . . 7 (𝜑 → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3635adantr 484 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3734, 36eqtrd 2833 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (0g𝐻)))
3837oveq2d 7152 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))))
3927oveq2d 7152 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
408gsumz 17995 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4113, 2, 40syl2anc 587 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4241adantr 484 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4339, 42eqtrd 2833 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = 0 )
4443fveq2d 6650 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾0 ))
4512, 38, 443eqtr4d 2843 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
4645ex 416 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
4713adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
48 eqid 2798 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
4914, 48mndcl 17914 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
50493expb 1117 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
5147, 50sylan 583 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
52 f1of1 6590 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5352ad2antll 728 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
54 cnvimass 5917 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
5518adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
5654, 55fssdm 6505 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
57 f1ss 6556 . . . . . . . . . . 11 ((𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })) ∧ (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
5853, 56, 57syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
59 f1f 6550 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
6058, 59syl 17 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
61 fco 6506 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6218, 60, 61syl2an2r 684 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6362ffvelrnda 6829 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐹𝑓)‘𝑥) ∈ 𝐵)
64 simprl 770 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
65 nnuz 12272 . . . . . . . 8 ℕ = (ℤ‘1)
6664, 65eleqtrdi 2900 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
677adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
68 eqid 2798 . . . . . . . . . 10 (+g𝐻) = (+g𝐻)
6914, 48, 68mhmlin 17958 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥𝐵𝑦𝐵) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
70693expb 1117 . . . . . . . 8 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥𝐵𝑦𝐵)) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
7167, 70sylan 583 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ (𝑥𝐵𝑦𝐵)) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
72 coass 6086 . . . . . . . . 9 ((𝐾𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹𝑓))
7372fveq1i 6647 . . . . . . . 8 (((𝐾𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹𝑓))‘𝑥)
74 fvco3 6738 . . . . . . . . 9 (((𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹𝑓))‘𝑥) = (𝐾‘((𝐹𝑓)‘𝑥)))
7562, 74sylan 583 . . . . . . . 8 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹𝑓))‘𝑥) = (𝐾‘((𝐹𝑓)‘𝑥)))
7673, 75syl5req 2846 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → (𝐾‘((𝐹𝑓)‘𝑥)) = (((𝐾𝐹) ∘ 𝑓)‘𝑥))
7751, 63, 66, 71, 76seqhomo 13416 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾‘(seq1((+g𝐺), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))) = (seq1((+g𝐻), ((𝐾𝐹) ∘ 𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
78 gsumzmhm.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
792adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐴𝑉)
80 gsumzmhm.c . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8180adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8226adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
83 f1ofo 6598 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
84 forn 6569 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8583, 84syl 17 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8685ad2antll 728 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8782, 86sseqtrrd 3956 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
88 eqid 2798 . . . . . . . 8 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
8914, 8, 48, 78, 47, 79, 55, 81, 64, 58, 87, 88gsumval3 19024 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐺 Σg 𝐹) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
9089fveq2d 6650 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾‘(seq1((+g𝐺), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))))
91 eqid 2798 . . . . . . 7 (Cntz‘𝐻) = (Cntz‘𝐻)
921adantr 484 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
93 fco 6506 . . . . . . . 8 ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴𝐵) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9430, 55, 93syl2an2r 684 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9578, 91cntzmhm2 18466 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
967, 81, 95syl2an2r 684 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
97 rnco2 6074 . . . . . . . 8 ran (𝐾𝐹) = (𝐾 “ ran 𝐹)
9897fveq2i 6649 . . . . . . . 8 ((Cntz‘𝐻)‘ran (𝐾𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
9996, 97, 983sstr4g 3960 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran (𝐾𝐹) ⊆ ((Cntz‘𝐻)‘ran (𝐾𝐹)))
100 eldifi 4054 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 }))) → 𝑥𝐴)
101 fvco3 6738 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐾𝐹)‘𝑥) = (𝐾‘(𝐹𝑥)))
10255, 100, 101syl2an 598 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹)‘𝑥) = (𝐾‘(𝐹𝑥)))
10319a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 0 ∈ V)
10455, 82, 79, 103suppssr 7847 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → (𝐹𝑥) = 0 )
105104fveq2d 6650 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → (𝐾‘(𝐹𝑥)) = (𝐾0 ))
10610ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → (𝐾0 ) = (0g𝐻))
107102, 105, 1063eqtrd 2837 . . . . . . . . 9 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹)‘𝑥) = (0g𝐻))
10894, 107suppss 7846 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ (𝐹 “ (V ∖ { 0 })))
109108, 86sseqtrrd 3956 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ ran 𝑓)
110 eqid 2798 . . . . . . 7 (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻)) = (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻))
11128, 3, 68, 91, 92, 79, 94, 99, 64, 58, 109, 110gsumval3 19024 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg (𝐾𝐹)) = (seq1((+g𝐻), ((𝐾𝐹) ∘ 𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
11277, 90, 1113eqtr4rd 2844 . . . . 5 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
113112expr 460 . . . 4 ((𝜑 ∧ (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
114113exlimdv 1934 . . 3 ((𝜑 ∧ (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
115114expimpd 457 . 2 (𝜑 → (((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 }))) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
116 gsumzmhm.w . . . . 5 (𝜑𝐹 finSupp 0 )
117116fsuppimpd 8827 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
11824, 117eqeltrrd 2891 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
119 fz1f1o 15062 . . 3 ((𝐹 “ (V ∖ { 0 })) ∈ Fin → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
120118, 119syl 17 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
12146, 115, 120mpjaod 857 1 (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5031   ↦ cmpt 5111  ◡ccnv 5519  ran crn 5521   “ cima 5523   ∘ ccom 5524  ⟶wf 6321  –1-1→wf1 6322  –onto→wfo 6323  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136   supp csupp 7816  Fincfn 8495   finSupp cfsupp 8820  1c1 10530  ℕcn 11628  ℤ≥cuz 12234  ...cfz 12888  seqcseq 13367  ♯chash 13689  Basecbs 16478  +gcplusg 16560  0gc0g 16708   Σg cgsu 16709  Mndcmnd 17906   MndHom cmhm 17949  Cntzccntz 18441 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-supp 7817  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-oadd 8092  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fsupp 8821  df-oi 8961  df-card 9355  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-fzo 13032  df-seq 13368  df-hash 13690  df-0g 16710  df-gsum 16711  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-cntz 18443 This theorem is referenced by:  gsummhm  19055  gsumzinv  19062
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