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Theorem gsumzmhm 19935
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 2726 . . . . . . . 8 (0g𝐻) = (0g𝐻)
43gsumz 18826 . . . . . . 7 ((𝐻 ∈ Mnd ∧ 𝐴𝑉) → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
51, 2, 4syl2anc 582 . . . . . 6 (𝜑 → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
65adantr 479 . . . . 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 18784 . . . . . . 7 (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾0 ) = (0g𝐻))
107, 9syl 17 . . . . . 6 (𝜑 → (𝐾0 ) = (0g𝐻))
1110adantr 479 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾0 ) = (0g𝐻))
126, 11eqtr4d 2769 . . . 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 18742 . . . . . . . . 9 (𝐺 ∈ Mnd → 0𝐵)
1613, 15syl 17 . . . . . . . 8 (𝜑0𝐵)
1716ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘𝐴) → 0𝐵)
18 gsumzmhm.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
198fvexi 6915 . . . . . . . . 9 0 ∈ V
2019a1i 11 . . . . . . . 8 (𝜑0 ∈ V)
2118, 2fexd 7244 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
22 suppimacnv 8188 . . . . . . . . . 10 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2321, 20, 22syl2anc 582 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
24 ssid 4002 . . . . . . . . 9 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
2523, 24eqsstrdi 4034 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2618, 2, 20, 25gsumcllem 19906 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
27 eqid 2726 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
2814, 27mhmf 18779 . . . . . . . . . 10 (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
297, 28syl 17 . . . . . . . . 9 (𝜑𝐾:𝐵⟶(Base‘𝐻))
3029feqmptd 6971 . . . . . . . 8 (𝜑𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
3130adantr 479 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
32 fveq2 6901 . . . . . . 7 (𝑥 = 0 → (𝐾𝑥) = (𝐾0 ))
3317, 26, 31, 32fmptco 7143 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (𝐾0 )))
3410mpteq2dv 5255 . . . . . . 7 (𝜑 → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3534adantr 479 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3633, 35eqtrd 2766 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (0g𝐻)))
3736oveq2d 7440 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))))
3826oveq2d 7440 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
398gsumz 18826 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4013, 2, 39syl2anc 582 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4140adantr 479 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4238, 41eqtrd 2766 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = 0 )
4342fveq2d 6905 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾0 ))
4412, 37, 433eqtr4d 2776 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
4544ex 411 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
4613adantr 479 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
47 eqid 2726 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
4814, 47mndcl 18735 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
49483expb 1117 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
5046, 49sylan 578 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
51 f1of1 6842 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5251ad2antll 727 . . . . . . . . . . 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 })))
53 cnvimass 6091 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
5418adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
5553, 54fssdm 6747 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
56 f1ss 6803 . . . . . . . . . . 11 ((𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })) ∧ (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
5752, 55, 56syl2anc 582 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
58 f1f 6798 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
5957, 58syl 17 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
60 fco 6752 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6118, 59, 60syl2an2r 683 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6261ffvelcdmda 7098 . . . . . . 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)
63 simprl 769 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
64 nnuz 12917 . . . . . . . 8 ℕ = (ℤ‘1)
6563, 64eleqtrdi 2836 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
667adantr 479 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
67 eqid 2726 . . . . . . . . . 10 (+g𝐻) = (+g𝐻)
6814, 47, 67mhmlin 18783 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥𝐵𝑦𝐵) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
69683expb 1117 . . . . . . . 8 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥𝐵𝑦𝐵)) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
7066, 69sylan 578 . . . . . . 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)))
71 coass 6276 . . . . . . . . 9 ((𝐾𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹𝑓))
7271fveq1i 6902 . . . . . . . 8 (((𝐾𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹𝑓))‘𝑥)
73 fvco3 7001 . . . . . . . . 9 (((𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹𝑓))‘𝑥) = (𝐾‘((𝐹𝑓)‘𝑥)))
7461, 73sylan 578 . . . . . . . 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)))
7572, 74eqtr2id 2779 . . . . . . 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))
7650, 62, 65, 70, 75seqhomo 14069 . . . . . 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 })))))
77 gsumzmhm.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
782adantr 479 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐴𝑉)
79 gsumzmhm.c . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8079adantr 479 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8125adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
82 f1ofo 6850 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
83 forn 6818 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8482, 83syl 17 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8584ad2antll 727 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8681, 85sseqtrrd 4021 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
87 eqid 2726 . . . . . . . 8 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
8814, 8, 47, 77, 46, 78, 54, 80, 63, 57, 86, 87gsumval3 19905 . . . . . . 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 })))))
8988fveq2d 6905 . . . . . 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 }))))))
90 eqid 2726 . . . . . . 7 (Cntz‘𝐻) = (Cntz‘𝐻)
911adantr 479 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
92 fco 6752 . . . . . . . 8 ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴𝐵) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9329, 54, 92syl2an2r 683 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9477, 90cntzmhm2 19336 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
957, 80, 94syl2an2r 683 . . . . . . . 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)))
96 rnco2 6264 . . . . . . . 8 ran (𝐾𝐹) = (𝐾 “ ran 𝐹)
9796fveq2i 6904 . . . . . . . 8 ((Cntz‘𝐻)‘ran (𝐾𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
9895, 96, 973sstr4g 4025 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran (𝐾𝐹) ⊆ ((Cntz‘𝐻)‘ran (𝐾𝐹)))
99 eldifi 4126 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 }))) → 𝑥𝐴)
100 fvco3 7001 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐾𝐹)‘𝑥) = (𝐾‘(𝐹𝑥)))
10154, 99, 100syl2an 594 . . . . . . . . . 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)))
10219a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 0 ∈ V)
10354, 81, 78, 102suppssr 8210 . . . . . . . . . . 11 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → (𝐹𝑥) = 0 )
104103fveq2d 6905 . . . . . . . . . 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 ))
10510ad2antrr 724 . . . . . . . . . 10 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → (𝐾0 ) = (0g𝐻))
106101, 104, 1053eqtrd 2770 . . . . . . . . 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))
10793, 106suppss 8208 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ (𝐹 “ (V ∖ { 0 })))
108107, 85sseqtrrd 4021 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ ran 𝑓)
109 eqid 2726 . . . . . . 7 (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻)) = (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻))
11027, 3, 67, 90, 91, 78, 93, 98, 63, 57, 108, 109gsumval3 19905 . . . . . 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 })))))
11176, 89, 1103eqtr4rd 2777 . . . . 5 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
112111expr 455 . . . 4 ((𝜑 ∧ (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
113112exlimdv 1929 . . 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))))
114113expimpd 452 . 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))))
115 gsumzmhm.w . . . . 5 (𝜑𝐹 finSupp 0 )
116115fsuppimpd 9413 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
11723, 116eqeltrrd 2827 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
118 fz1f1o 15714 . . 3 ((𝐹 “ (V ∖ { 0 })) ∈ Fin → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
119117, 118syl 17 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
12045, 114, 119mpjaod 858 1 (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1534  wex 1774  wcel 2099  Vcvv 3462  cdif 3944  wss 3947  c0 4325  {csn 4633   class class class wbr 5153  cmpt 5236  ccnv 5681  ran crn 5683  cima 5685  ccom 5686  wf 6550  1-1wf1 6551  ontowfo 6552  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424   supp csupp 8174  Fincfn 8974   finSupp cfsupp 9405  1c1 11159  cn 12264  cuz 12874  ...cfz 13538  seqcseq 14021  chash 14347  Basecbs 17213  +gcplusg 17266  0gc0g 17454   Σg cgsu 17455  Mndcmnd 18727   MndHom cmhm 18771  Cntzccntz 19309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-n0 12525  df-z 12611  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-0g 17456  df-gsum 17457  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-mhm 18773  df-cntz 19311
This theorem is referenced by:  gsummhm  19936  gsumzinv  19943
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