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Theorem gsumzmhm 18986
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 2818 . . . . . . . 8 (0g𝐻) = (0g𝐻)
43gsumz 17988 . . . . . . 7 ((𝐻 ∈ Mnd ∧ 𝐴𝑉) → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
51, 2, 4syl2anc 584 . . . . . 6 (𝜑 → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
65adantr 481 . . . . 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 17952 . . . . . . 7 (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾0 ) = (0g𝐻))
107, 9syl 17 . . . . . 6 (𝜑 → (𝐾0 ) = (0g𝐻))
1110adantr 481 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾0 ) = (0g𝐻))
126, 11eqtr4d 2856 . . . 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 17914 . . . . . . . . 9 (𝐺 ∈ Mnd → 0𝐵)
1613, 15syl 17 . . . . . . . 8 (𝜑0𝐵)
1716ad2antrr 722 . . . . . . 7 (((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘𝐴) → 0𝐵)
18 gsumzmhm.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
198fvexi 6677 . . . . . . . . 9 0 ∈ V
2019a1i 11 . . . . . . . 8 (𝜑0 ∈ V)
21 fex 6980 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2218, 2, 21syl2anc 584 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
23 suppimacnv 7830 . . . . . . . . . 10 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2422, 20, 23syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
25 ssid 3986 . . . . . . . . 9 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
2624, 25eqsstrdi 4018 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2718, 2, 20, 26gsumcllem 18957 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
28 eqid 2818 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
2914, 28mhmf 17949 . . . . . . . . . 10 (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
307, 29syl 17 . . . . . . . . 9 (𝜑𝐾:𝐵⟶(Base‘𝐻))
3130feqmptd 6726 . . . . . . . 8 (𝜑𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
3231adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
33 fveq2 6663 . . . . . . 7 (𝑥 = 0 → (𝐾𝑥) = (𝐾0 ))
3417, 27, 32, 33fmptco 6883 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (𝐾0 )))
3510mpteq2dv 5153 . . . . . . 7 (𝜑 → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3635adantr 481 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3734, 36eqtrd 2853 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (0g𝐻)))
3837oveq2d 7161 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))))
3927oveq2d 7161 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
408gsumz 17988 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4113, 2, 40syl2anc 584 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4241adantr 481 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4339, 42eqtrd 2853 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = 0 )
4443fveq2d 6667 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾0 ))
4512, 38, 443eqtr4d 2863 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
4645ex 413 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
4713adantr 481 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
48 eqid 2818 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
4914, 48mndcl 17907 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
50493expb 1112 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
5147, 50sylan 580 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
52 f1of1 6607 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5352ad2antll 725 . . . . . . . . . . 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 5942 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
5518adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
5654, 55fssdm 6523 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
57 f1ss 6573 . . . . . . . . . . 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 584 . . . . . . . . . 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 6568 . . . . . . . . . 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 6524 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6218, 60, 61syl2an2r 681 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6362ffvelrnda 6843 . . . . . . 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 767 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
65 nnuz 12269 . . . . . . . 8 ℕ = (ℤ‘1)
6664, 65eleqtrdi 2920 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
677adantr 481 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
68 eqid 2818 . . . . . . . . . 10 (+g𝐻) = (+g𝐻)
6914, 48, 68mhmlin 17951 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥𝐵𝑦𝐵) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
70693expb 1112 . . . . . . . 8 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥𝐵𝑦𝐵)) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
7167, 70sylan 580 . . . . . . 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 6111 . . . . . . . . 9 ((𝐾𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹𝑓))
7372fveq1i 6664 . . . . . . . 8 (((𝐾𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹𝑓))‘𝑥)
74 fvco3 6753 . . . . . . . . 9 (((𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹𝑓))‘𝑥) = (𝐾‘((𝐹𝑓)‘𝑥)))
7562, 74sylan 580 . . . . . . . 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 2866 . . . . . . 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 13405 . . . . . 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 481 . . . . . . . 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 481 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8226adantr 481 . . . . . . . . 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 6615 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
84 forn 6586 . . . . . . . . . . 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 725 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8782, 86sseqtrrd 4005 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
88 eqid 2818 . . . . . . . 8 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
8914, 8, 48, 78, 47, 79, 55, 81, 64, 58, 87, 88gsumval3 18956 . . . . . . 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 6667 . . . . . 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 2818 . . . . . . 7 (Cntz‘𝐻) = (Cntz‘𝐻)
921adantr 481 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
93 fco 6524 . . . . . . . 8 ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴𝐵) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9430, 55, 93syl2an2r 681 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9578, 91cntzmhm2 18408 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
967, 81, 95syl2an2r 681 . . . . . . . 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 6099 . . . . . . . 8 ran (𝐾𝐹) = (𝐾 “ ran 𝐹)
9897fveq2i 6666 . . . . . . . 8 ((Cntz‘𝐻)‘ran (𝐾𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
9996, 97, 983sstr4g 4009 . . . . . . 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 4100 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 }))) → 𝑥𝐴)
101 fvco3 6753 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐾𝐹)‘𝑥) = (𝐾‘(𝐹𝑥)))
10255, 100, 101syl2an 595 . . . . . . . . . 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 7850 . . . . . . . . . . 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 6667 . . . . . . . . . 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 722 . . . . . . . . . 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 2857 . . . . . . . . 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 7849 . . . . . . . 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 4005 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ ran 𝑓)
110 eqid 2818 . . . . . . 7 (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻)) = (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻))
11128, 3, 68, 91, 92, 79, 94, 99, 64, 58, 109, 110gsumval3 18956 . . . . . 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 2864 . . . . 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 457 . . . 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 1925 . . 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 454 . 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 8828 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
11824, 117eqeltrrd 2911 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
119 fz1f1o 15055 . . 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 854 1 (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  cdif 3930  wss 3933  c0 4288  {csn 4557   class class class wbr 5057  cmpt 5137  ccnv 5547  ran crn 5549  cima 5551  ccom 5552  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145   supp csupp 7819  Fincfn 8497   finSupp cfsupp 8821  1c1 10526  cn 11626  cuz 12231  ...cfz 12880  seqcseq 13357  chash 13678  Basecbs 16471  +gcplusg 16553  0gc0g 16701   Σg cgsu 16702  Mndcmnd 17899   MndHom cmhm 17942  Cntzccntz 18383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-0g 16703  df-gsum 16704  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-cntz 18385
This theorem is referenced by:  gsummhm  18987  gsumzinv  18994
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