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Theorem gsumzmhm 18804
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 2775 . . . . . . . 8 (0g𝐻) = (0g𝐻)
43gsumz 17836 . . . . . . 7 ((𝐻 ∈ Mnd ∧ 𝐴𝑉) → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
51, 2, 4syl2anc 576 . . . . . 6 (𝜑 → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
65adantr 473 . . . . 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 17805 . . . . . . 7 (𝐾 ∈ (𝐺 MndHom 𝐻) → (𝐾0 ) = (0g𝐻))
107, 9syl 17 . . . . . 6 (𝜑 → (𝐾0 ) = (0g𝐻))
1110adantr 473 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾0 ) = (0g𝐻))
126, 11eqtr4d 2814 . . . 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 17770 . . . . . . . . 9 (𝐺 ∈ Mnd → 0𝐵)
1613, 15syl 17 . . . . . . . 8 (𝜑0𝐵)
1716ad2antrr 713 . . . . . . 7 (((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘𝐴) → 0𝐵)
18 gsumzmhm.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
198fvexi 6511 . . . . . . . . 9 0 ∈ V
2019a1i 11 . . . . . . . 8 (𝜑0 ∈ V)
21 fex 6813 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2218, 2, 21syl2anc 576 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
23 suppimacnv 7641 . . . . . . . . . 10 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2422, 20, 23syl2anc 576 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
25 ssid 3878 . . . . . . . . 9 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
2624, 25syl6eqss 3910 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2718, 2, 20, 26gsumcllem 18776 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
28 eqid 2775 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
2914, 28mhmf 17802 . . . . . . . . . 10 (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
307, 29syl 17 . . . . . . . . 9 (𝜑𝐾:𝐵⟶(Base‘𝐻))
3130feqmptd 6560 . . . . . . . 8 (𝜑𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
3231adantr 473 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
33 fveq2 6497 . . . . . . 7 (𝑥 = 0 → (𝐾𝑥) = (𝐾0 ))
3417, 27, 32, 33fmptco 6712 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (𝐾0 )))
3510mpteq2dv 5021 . . . . . . 7 (𝜑 → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3635adantr 473 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3734, 36eqtrd 2811 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (0g𝐻)))
3837oveq2d 6990 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))))
3927oveq2d 6990 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
408gsumz 17836 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4113, 2, 40syl2anc 576 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4241adantr 473 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4339, 42eqtrd 2811 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = 0 )
4443fveq2d 6501 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾0 ))
4512, 38, 443eqtr4d 2821 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
4645ex 405 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
4713adantr 473 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
48 eqid 2775 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
4914, 48mndcl 17763 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
50493expb 1100 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
5147, 50sylan 572 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
52 f1of1 6441 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5352ad2antll 716 . . . . . . . . . . 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 5787 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
5518adantr 473 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
5654, 55fssdm 6358 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
57 f1ss 6407 . . . . . . . . . . 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 576 . . . . . . . . . 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 6402 . . . . . . . . . 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 6359 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6218, 60, 61syl2an2r 672 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6362ffvelrnda 6674 . . . . . . 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 758 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
65 nnuz 12092 . . . . . . . 8 ℕ = (ℤ‘1)
6664, 65syl6eleq 2873 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
677adantr 473 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
68 eqid 2775 . . . . . . . . . 10 (+g𝐻) = (+g𝐻)
6914, 48, 68mhmlin 17804 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥𝐵𝑦𝐵) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
70693expb 1100 . . . . . . . 8 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥𝐵𝑦𝐵)) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
7167, 70sylan 572 . . . . . . 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 5955 . . . . . . . . 9 ((𝐾𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹𝑓))
7372fveq1i 6498 . . . . . . . 8 (((𝐾𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹𝑓))‘𝑥)
74 fvco3 6586 . . . . . . . . 9 (((𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹𝑓))‘𝑥) = (𝐾‘((𝐹𝑓)‘𝑥)))
7562, 74sylan 572 . . . . . . . 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 2824 . . . . . . 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 13229 . . . . . 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 473 . . . . . . . 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 473 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8226adantr 473 . . . . . . . . 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 6449 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
84 forn 6420 . . . . . . . . . . 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 716 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8782, 86sseqtr4d 3897 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
88 eqid 2775 . . . . . . . 8 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
8914, 8, 48, 78, 47, 79, 55, 81, 64, 58, 87, 88gsumval3 18775 . . . . . . 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 6501 . . . . . 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 2775 . . . . . . 7 (Cntz‘𝐻) = (Cntz‘𝐻)
921adantr 473 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
93 fco 6359 . . . . . . . 8 ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴𝐵) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9430, 55, 93syl2an2r 672 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9578, 91cntzmhm2 18235 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
967, 81, 95syl2an2r 672 . . . . . . . 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 5943 . . . . . . . 8 ran (𝐾𝐹) = (𝐾 “ ran 𝐹)
9897fveq2i 6500 . . . . . . . 8 ((Cntz‘𝐻)‘ran (𝐾𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
9996, 97, 983sstr4g 3901 . . . . . . 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 3992 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 }))) → 𝑥𝐴)
101 fvco3 6586 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐾𝐹)‘𝑥) = (𝐾‘(𝐹𝑥)))
10255, 100, 101syl2an 586 . . . . . . . . . 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 7661 . . . . . . . . . . 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 6501 . . . . . . . . . 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 713 . . . . . . . . . 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 2815 . . . . . . . . 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 7660 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ (𝐹 “ (V ∖ { 0 })))
109108, 86sseqtr4d 3897 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ ran 𝑓)
110 eqid 2775 . . . . . . 7 (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻)) = (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻))
11128, 3, 68, 91, 92, 79, 94, 99, 64, 58, 109, 110gsumval3 18775 . . . . . 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 2822 . . . . 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 449 . . . 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 1892 . . 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 446 . 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 8631 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
11824, 117eqeltrrd 2864 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
119 fz1f1o 14921 . . 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 846 1 (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wo 833   = wceq 1507  wex 1742  wcel 2048  Vcvv 3412  cdif 3825  wss 3828  c0 4177  {csn 4439   class class class wbr 4927  cmpt 5006  ccnv 5403  ran crn 5405  cima 5407  ccom 5408  wf 6182  1-1wf1 6183  ontowfo 6184  1-1-ontowf1o 6185  cfv 6186  (class class class)co 6974   supp csupp 7630  Fincfn 8302   finSupp cfsupp 8624  1c1 10332  cn 11435  cuz 12055  ...cfz 12705  seqcseq 13181  chash 13502  Basecbs 16333  +gcplusg 16415  0gc0g 16563   Σg cgsu 16564  Mndcmnd 17756   MndHom cmhm 17795  Cntzccntz 18210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-rep 5047  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277  ax-cnex 10387  ax-resscn 10388  ax-1cn 10389  ax-icn 10390  ax-addcl 10391  ax-addrcl 10392  ax-mulcl 10393  ax-mulrcl 10394  ax-mulcom 10395  ax-addass 10396  ax-mulass 10397  ax-distr 10398  ax-i2m1 10399  ax-1ne0 10400  ax-1rid 10401  ax-rnegex 10402  ax-rrecex 10403  ax-cnre 10404  ax-pre-lttri 10405  ax-pre-lttrn 10406  ax-pre-ltadd 10407  ax-pre-mulgt0 10408
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-nel 3071  df-ral 3090  df-rex 3091  df-reu 3092  df-rmo 3093  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-pss 3844  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-tp 4444  df-op 4446  df-uni 4711  df-int 4748  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-tr 5029  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-se 5364  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-isom 6195  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7498  df-2nd 7499  df-supp 7631  df-wrecs 7747  df-recs 7809  df-rdg 7847  df-1o 7901  df-oadd 7905  df-er 8085  df-map 8204  df-en 8303  df-dom 8304  df-sdom 8305  df-fin 8306  df-fsupp 8625  df-oi 8765  df-card 9158  df-pnf 10472  df-mnf 10473  df-xr 10474  df-ltxr 10475  df-le 10476  df-sub 10668  df-neg 10669  df-nn 11436  df-n0 11705  df-z 11791  df-uz 12056  df-fz 12706  df-fzo 12847  df-seq 13182  df-hash 13503  df-0g 16565  df-gsum 16566  df-mgm 17704  df-sgrp 17746  df-mnd 17757  df-mhm 17797  df-cntz 18212
This theorem is referenced by:  gsummhm  18805  gsumzinv  18812
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