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Theorem gsumzmhm 18258
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 2621 . . . . . . . 8 (0g𝐻) = (0g𝐻)
43gsumz 17295 . . . . . . 7 ((𝐻 ∈ Mnd ∧ 𝐴𝑉) → (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))) = (0g𝐻))
51, 2, 4syl2anc 692 . . . . . 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 17264 . . . . . . 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 2658 . . . 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 17229 . . . . . . . . 9 (𝐺 ∈ Mnd → 0𝐵)
1613, 15syl 17 . . . . . . . 8 (𝜑0𝐵)
1716ad2antrr 761 . . . . . . 7 (((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) ∧ 𝑘𝐴) → 0𝐵)
18 gsumzmhm.f . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
19 fvex 6158 . . . . . . . . . 10 (0g𝐺) ∈ V
208, 19eqeltri 2694 . . . . . . . . 9 0 ∈ V
2120a1i 11 . . . . . . . 8 (𝜑0 ∈ V)
22 fex 6444 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2318, 2, 22syl2anc 692 . . . . . . . . . 10 (𝜑𝐹 ∈ V)
24 suppimacnv 7251 . . . . . . . . . 10 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2523, 21, 24syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
26 ssid 3603 . . . . . . . . 9 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
2725, 26syl6eqss 3634 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2818, 2, 21, 27gsumcllem 18230 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
29 eqid 2621 . . . . . . . . . . 11 (Base‘𝐻) = (Base‘𝐻)
3014, 29mhmf 17261 . . . . . . . . . 10 (𝐾 ∈ (𝐺 MndHom 𝐻) → 𝐾:𝐵⟶(Base‘𝐻))
317, 30syl 17 . . . . . . . . 9 (𝜑𝐾:𝐵⟶(Base‘𝐻))
3231feqmptd 6206 . . . . . . . 8 (𝜑𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
3332adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐾 = (𝑥𝐵 ↦ (𝐾𝑥)))
34 fveq2 6148 . . . . . . 7 (𝑥 = 0 → (𝐾𝑥) = (𝐾0 ))
3517, 28, 33, 34fmptco 6351 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (𝐾0 )))
3610mpteq2dv 4705 . . . . . . 7 (𝜑 → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3736adantr 481 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑘𝐴 ↦ (𝐾0 )) = (𝑘𝐴 ↦ (0g𝐻)))
3835, 37eqtrd 2655 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾𝐹) = (𝑘𝐴 ↦ (0g𝐻)))
3938oveq2d 6620 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐻 Σg (𝑘𝐴 ↦ (0g𝐻))))
4028oveq2d 6620 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
418gsumz 17295 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4213, 2, 41syl2anc 692 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4342adantr 481 . . . . . 6 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
4440, 43eqtrd 2655 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = 0 )
4544fveq2d 6152 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐾‘(𝐺 Σg 𝐹)) = (𝐾0 ))
4612, 39, 453eqtr4d 2665 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
4746ex 450 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
4813adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
49 eqid 2621 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
5014, 49mndcl 17222 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
51503expb 1263 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
5248, 51sylan 488 . . . . . . 7 (((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
5318adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
54 f1of1 6093 . . . . . . . . . . . 12 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5554ad2antll 764 . . . . . . . . . . 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 })))
56 cnvimass 5444 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
57 fdm 6008 . . . . . . . . . . . . 13 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5853, 57syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → dom 𝐹 = 𝐴)
5956, 58syl5sseq 3632 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
60 f1ss 6063 . . . . . . . . . . 11 ((𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })) ∧ (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
6155, 59, 60syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
62 f1f 6058 . . . . . . . . . 10 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
6361, 62syl 17 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
64 fco 6015 . . . . . . . . 9 ((𝐹:𝐴𝐵𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6553, 63, 64syl2anc 692 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹𝑓):(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵)
6665ffvelrnda 6315 . . . . . . 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)
67 simprl 793 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
68 nnuz 11667 . . . . . . . 8 ℕ = (ℤ‘1)
6967, 68syl6eleq 2708 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (#‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
707adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐾 ∈ (𝐺 MndHom 𝐻))
71 eqid 2621 . . . . . . . . . 10 (+g𝐻) = (+g𝐻)
7214, 49, 71mhmlin 17263 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥𝐵𝑦𝐵) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
73723expb 1263 . . . . . . . 8 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ (𝑥𝐵𝑦𝐵)) → (𝐾‘(𝑥(+g𝐺)𝑦)) = ((𝐾𝑥)(+g𝐻)(𝐾𝑦)))
7470, 73sylan 488 . . . . . . 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)))
75 coass 5613 . . . . . . . . 9 ((𝐾𝐹) ∘ 𝑓) = (𝐾 ∘ (𝐹𝑓))
7675fveq1i 6149 . . . . . . . 8 (((𝐾𝐹) ∘ 𝑓)‘𝑥) = ((𝐾 ∘ (𝐹𝑓))‘𝑥)
77 fvco3 6232 . . . . . . . . 9 (((𝐹𝑓):(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐵𝑥 ∈ (1...(#‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐾 ∘ (𝐹𝑓))‘𝑥) = (𝐾‘((𝐹𝑓)‘𝑥)))
7865, 77sylan 488 . . . . . . . 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)))
7976, 78syl5req 2668 . . . . . . 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))
8052, 66, 69, 74, 79seqhomo 12788 . . . . . 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 })))))
81 gsumzmhm.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
822adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐴𝑉)
83 gsumzmhm.c . . . . . . . . 9 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8483adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
8527adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
86 f1ofo 6101 . . . . . . . . . . 11 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
87 forn 6075 . . . . . . . . . . 11 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8886, 87syl 17 . . . . . . . . . 10 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
8988ad2antll 764 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
9085, 89sseqtr4d 3621 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
91 eqid 2621 . . . . . . . 8 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
9214, 8, 49, 81, 48, 82, 53, 84, 67, 61, 90, 91gsumval3 18229 . . . . . . 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 })))))
9392fveq2d 6152 . . . . . 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 }))))))
94 eqid 2621 . . . . . . 7 (Cntz‘𝐻) = (Cntz‘𝐻)
951adantr 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐻 ∈ Mnd)
9631adantr 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐾:𝐵⟶(Base‘𝐻))
97 fco 6015 . . . . . . . 8 ((𝐾:𝐵⟶(Base‘𝐻) ∧ 𝐹:𝐴𝐵) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9896, 53, 97syl2anc 692 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐾𝐹):𝐴⟶(Base‘𝐻))
9981, 94cntzmhm2 17693 . . . . . . . . 9 ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐾 “ ran 𝐹) ⊆ ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹)))
10070, 84, 99syl2anc 692 . . . . . . . 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)))
101 rnco2 5601 . . . . . . . 8 ran (𝐾𝐹) = (𝐾 “ ran 𝐹)
102101fveq2i 6151 . . . . . . . 8 ((Cntz‘𝐻)‘ran (𝐾𝐹)) = ((Cntz‘𝐻)‘(𝐾 “ ran 𝐹))
103100, 101, 1023sstr4g 3625 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran (𝐾𝐹) ⊆ ((Cntz‘𝐻)‘ran (𝐾𝐹)))
104 eldifi 3710 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 }))) → 𝑥𝐴)
105 fvco3 6232 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐾𝐹)‘𝑥) = (𝐾‘(𝐹𝑥)))
10653, 104, 105syl2an 494 . . . . . . . . . 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)))
10720a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 0 ∈ V)
10853, 85, 82, 107suppssr 7271 . . . . . . . . . . 11 (((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → (𝐹𝑥) = 0 )
109108fveq2d 6152 . . . . . . . . . 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 ))
11010ad2antrr 761 . . . . . . . . . 10 (((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (𝐴 ∖ (𝐹 “ (V ∖ { 0 })))) → (𝐾0 ) = (0g𝐻))
111106, 109, 1103eqtrd 2659 . . . . . . . . 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))
11298, 111suppss 7270 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ (𝐹 “ (V ∖ { 0 })))
113112, 89sseqtr4d 3621 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐾𝐹) supp (0g𝐻)) ⊆ ran 𝑓)
114 eqid 2621 . . . . . . 7 (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻)) = (((𝐾𝐹) ∘ 𝑓) supp (0g𝐻))
11529, 3, 71, 94, 95, 82, 98, 103, 67, 61, 113, 114gsumval3 18229 . . . . . 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 })))))
11680, 93, 1153eqtr4rd 2666 . . . . 5 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
117116expr 642 . . . 4 ((𝜑 ∧ (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹))))
118117exlimdv 1858 . . 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))))
119118expimpd 628 . 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))))
120 gsumzmhm.w . . . . 5 (𝜑𝐹 finSupp 0 )
121120fsuppimpd 8226 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
12225, 121eqeltrrd 2699 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
123 fz1f1o 14374 . . 3 ((𝐹 “ (V ∖ { 0 })) ∈ Fin → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
124122, 123syl 17 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
12547, 119, 124mpjaod 396 1 (𝜑 → (𝐻 Σg (𝐾𝐹)) = (𝐾‘(𝐺 Σg 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cdif 3552  wss 3555  c0 3891  {csn 4148   class class class wbr 4613  cmpt 4673  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604   supp csupp 7240  Fincfn 7899   finSupp cfsupp 8219  1c1 9881  cn 10964  cuz 11631  ...cfz 12268  seqcseq 12741  #chash 13057  Basecbs 15781  +gcplusg 15862  0gc0g 16021   Σg cgsu 16022  Mndcmnd 17215   MndHom cmhm 17254  Cntzccntz 17669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-0g 16023  df-gsum 16024  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-cntz 17671
This theorem is referenced by:  gsummhm  18259  gsumzinv  18266
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