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Theorem gsumzsplit 18150
 Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)
Hypotheses
Ref Expression
gsumzsplit.b 𝐵 = (Base‘𝐺)
gsumzsplit.0 0 = (0g𝐺)
gsumzsplit.p + = (+g𝐺)
gsumzsplit.z 𝑍 = (Cntz‘𝐺)
gsumzsplit.g (𝜑𝐺 ∈ Mnd)
gsumzsplit.a (𝜑𝐴𝑉)
gsumzsplit.f (𝜑𝐹:𝐴𝐵)
gsumzsplit.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzsplit.w (𝜑𝐹 finSupp 0 )
gsumzsplit.i (𝜑 → (𝐶𝐷) = ∅)
gsumzsplit.u (𝜑𝐴 = (𝐶𝐷))
Assertion
Ref Expression
gsumzsplit (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹𝐶)) + (𝐺 Σg (𝐹𝐷))))

Proof of Theorem gsumzsplit
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 gsumzsplit.b . . 3 𝐵 = (Base‘𝐺)
2 gsumzsplit.0 . . 3 0 = (0g𝐺)
3 gsumzsplit.p . . 3 + = (+g𝐺)
4 gsumzsplit.z . . 3 𝑍 = (Cntz‘𝐺)
5 gsumzsplit.g . . 3 (𝜑𝐺 ∈ Mnd)
6 gsumzsplit.a . . 3 (𝜑𝐴𝑉)
7 gsumzsplit.f . . . 4 (𝜑𝐹:𝐴𝐵)
8 fvex 6113 . . . . . 6 (0g𝐺) ∈ V
92, 8eqeltri 2684 . . . . 5 0 ∈ V
109a1i 11 . . . 4 (𝜑0 ∈ V)
11 gsumzsplit.w . . . 4 (𝜑𝐹 finSupp 0 )
127, 6, 10, 11fsuppmptif 8188 . . 3 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) finSupp 0 )
137, 6, 10, 11fsuppmptif 8188 . . 3 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) finSupp 0 )
141submacs 17188 . . . . 5 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
15 acsmre 16136 . . . . 5 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
165, 14, 153syl 18 . . . 4 (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
17 frn 5966 . . . . 5 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
187, 17syl 17 . . . 4 (𝜑 → ran 𝐹𝐵)
19 eqid 2610 . . . . 5 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
2019mrccl 16094 . . . 4 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
2116, 18, 20syl2anc 691 . . 3 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
22 gsumzsplit.c . . . . 5 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
23 eqid 2610 . . . . . 6 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
244, 19, 23cntzspan 18070 . . . . 5 ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
255, 22, 24syl2anc 691 . . . 4 (𝜑 → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
2623, 4submcmn2 18067 . . . . 5 (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → ((𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
2721, 26syl 17 . . . 4 (𝜑 → ((𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
2825, 27mpbid 221 . . 3 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
2916, 19, 18mrcssidd 16108 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
3029adantr 480 . . . . . 6 ((𝜑𝑘𝐴) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
31 ffn 5958 . . . . . . . 8 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
327, 31syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
33 fnfvelrn 6264 . . . . . . 7 ((𝐹 Fn 𝐴𝑘𝐴) → (𝐹𝑘) ∈ ran 𝐹)
3432, 33sylan 487 . . . . . 6 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ ran 𝐹)
3530, 34sseldd 3569 . . . . 5 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
362subm0cl 17175 . . . . . . 7 (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
3721, 36syl 17 . . . . . 6 (𝜑0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
3837adantr 480 . . . . 5 ((𝜑𝑘𝐴) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
3935, 38ifcld 4081 . . . 4 ((𝜑𝑘𝐴) → if(𝑘𝐶, (𝐹𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
40 eqid 2610 . . . 4 (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))
4139, 40fmptd 6292 . . 3 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
4235, 38ifcld 4081 . . . 4 ((𝜑𝑘𝐴) → if(𝑘𝐷, (𝐹𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
43 eqid 2610 . . . 4 (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))
4442, 43fmptd 6292 . . 3 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
451, 2, 3, 4, 5, 6, 12, 13, 21, 28, 41, 44gsumzadd 18145 . 2 (𝜑 → (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ∘𝑓 + (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))) = ((𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))) + (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))))
467feqmptd 6159 . . . . 5 (𝜑𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
47 iftrue 4042 . . . . . . . . . 10 (𝑘𝐶 → if(𝑘𝐶, (𝐹𝑘), 0 ) = (𝐹𝑘))
4847adantl 481 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → if(𝑘𝐶, (𝐹𝑘), 0 ) = (𝐹𝑘))
49 gsumzsplit.i . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐷) = ∅)
50 noel 3878 . . . . . . . . . . . . . . . 16 ¬ 𝑘 ∈ ∅
51 eleq2 2677 . . . . . . . . . . . . . . . 16 ((𝐶𝐷) = ∅ → (𝑘 ∈ (𝐶𝐷) ↔ 𝑘 ∈ ∅))
5250, 51mtbiri 316 . . . . . . . . . . . . . . 15 ((𝐶𝐷) = ∅ → ¬ 𝑘 ∈ (𝐶𝐷))
5349, 52syl 17 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑘 ∈ (𝐶𝐷))
5453adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ¬ 𝑘 ∈ (𝐶𝐷))
55 elin 3758 . . . . . . . . . . . . 13 (𝑘 ∈ (𝐶𝐷) ↔ (𝑘𝐶𝑘𝐷))
5654, 55sylnib 317 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ¬ (𝑘𝐶𝑘𝐷))
57 imnan 437 . . . . . . . . . . . 12 ((𝑘𝐶 → ¬ 𝑘𝐷) ↔ ¬ (𝑘𝐶𝑘𝐷))
5856, 57sylibr 223 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝑘𝐶 → ¬ 𝑘𝐷))
5958imp 444 . . . . . . . . . 10 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → ¬ 𝑘𝐷)
6059iffalsed 4047 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → if(𝑘𝐷, (𝐹𝑘), 0 ) = 0 )
6148, 60oveq12d 6567 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = ((𝐹𝑘) + 0 ))
627ffvelrnda 6267 . . . . . . . . . 10 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ 𝐵)
631, 3, 2mndrid 17135 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
645, 63sylan 487 . . . . . . . . . 10 ((𝜑 ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
6562, 64syldan 486 . . . . . . . . 9 ((𝜑𝑘𝐴) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
6665adantr 480 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
6761, 66eqtrd 2644 . . . . . . 7 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝐹𝑘))
6858con2d 128 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝑘𝐷 → ¬ 𝑘𝐶))
6968imp 444 . . . . . . . . . 10 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → ¬ 𝑘𝐶)
7069iffalsed 4047 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → if(𝑘𝐶, (𝐹𝑘), 0 ) = 0 )
71 iftrue 4042 . . . . . . . . . 10 (𝑘𝐷 → if(𝑘𝐷, (𝐹𝑘), 0 ) = (𝐹𝑘))
7271adantl 481 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → if(𝑘𝐷, (𝐹𝑘), 0 ) = (𝐹𝑘))
7370, 72oveq12d 6567 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = ( 0 + (𝐹𝑘)))
741, 3, 2mndlid 17134 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ (𝐹𝑘) ∈ 𝐵) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
755, 74sylan 487 . . . . . . . . . 10 ((𝜑 ∧ (𝐹𝑘) ∈ 𝐵) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
7662, 75syldan 486 . . . . . . . . 9 ((𝜑𝑘𝐴) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
7776adantr 480 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
7873, 77eqtrd 2644 . . . . . . 7 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝐹𝑘))
79 gsumzsplit.u . . . . . . . . . 10 (𝜑𝐴 = (𝐶𝐷))
8079eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑘𝐴𝑘 ∈ (𝐶𝐷)))
81 elun 3715 . . . . . . . . 9 (𝑘 ∈ (𝐶𝐷) ↔ (𝑘𝐶𝑘𝐷))
8280, 81syl6bb 275 . . . . . . . 8 (𝜑 → (𝑘𝐴 ↔ (𝑘𝐶𝑘𝐷)))
8382biimpa 500 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑘𝐶𝑘𝐷))
8467, 78, 83mpjaodan 823 . . . . . 6 ((𝜑𝑘𝐴) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝐹𝑘))
8584mpteq2dva 4672 . . . . 5 (𝜑 → (𝑘𝐴 ↦ (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 ))) = (𝑘𝐴 ↦ (𝐹𝑘)))
8646, 85eqtr4d 2647 . . . 4 (𝜑𝐹 = (𝑘𝐴 ↦ (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 ))))
871, 2mndidcl 17131 . . . . . . . 8 (𝐺 ∈ Mnd → 0𝐵)
885, 87syl 17 . . . . . . 7 (𝜑0𝐵)
8988adantr 480 . . . . . 6 ((𝜑𝑘𝐴) → 0𝐵)
9062, 89ifcld 4081 . . . . 5 ((𝜑𝑘𝐴) → if(𝑘𝐶, (𝐹𝑘), 0 ) ∈ 𝐵)
9162, 89ifcld 4081 . . . . 5 ((𝜑𝑘𝐴) → if(𝑘𝐷, (𝐹𝑘), 0 ) ∈ 𝐵)
92 eqidd 2611 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )))
93 eqidd 2611 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))
946, 90, 91, 92, 93offval2 6812 . . . 4 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ∘𝑓 + (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))) = (𝑘𝐴 ↦ (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 ))))
9586, 94eqtr4d 2647 . . 3 (𝜑𝐹 = ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ∘𝑓 + (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
9695oveq2d 6565 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ∘𝑓 + (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))))
9746reseq1d 5316 . . . . . 6 (𝜑 → (𝐹𝐶) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐶))
98 ssun1 3738 . . . . . . . 8 𝐶 ⊆ (𝐶𝐷)
9998, 79syl5sseqr 3617 . . . . . . 7 (𝜑𝐶𝐴)
10047mpteq2ia 4668 . . . . . . . 8 (𝑘𝐶 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) = (𝑘𝐶 ↦ (𝐹𝑘))
101 resmpt 5369 . . . . . . . 8 (𝐶𝐴 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶) = (𝑘𝐶 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )))
102 resmpt 5369 . . . . . . . 8 (𝐶𝐴 → ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐶) = (𝑘𝐶 ↦ (𝐹𝑘)))
103100, 101, 1023eqtr4a 2670 . . . . . . 7 (𝐶𝐴 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐶))
10499, 103syl 17 . . . . . 6 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐶))
10597, 104eqtr4d 2647 . . . . 5 (𝜑 → (𝐹𝐶) = ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶))
106105oveq2d 6565 . . . 4 (𝜑 → (𝐺 Σg (𝐹𝐶)) = (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶)))
10790, 40fmptd 6292 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )):𝐴𝐵)
108 frn 5966 . . . . . . 7 ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) → ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
10941, 108syl 17 . . . . . 6 (𝜑 → ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
1104cntzidss 17593 . . . . . 6 ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ⊆ (𝑍‘ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))))
11128, 109, 110syl2anc 691 . . . . 5 (𝜑 → ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ⊆ (𝑍‘ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))))
112 eldifn 3695 . . . . . . . 8 (𝑘 ∈ (𝐴𝐶) → ¬ 𝑘𝐶)
113112adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (𝐴𝐶)) → ¬ 𝑘𝐶)
114113iffalsed 4047 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝐶)) → if(𝑘𝐶, (𝐹𝑘), 0 ) = 0 )
115114, 6suppss2 7216 . . . . 5 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) supp 0 ) ⊆ 𝐶)
1161, 2, 4, 5, 6, 107, 111, 115, 12gsumzres 18133 . . . 4 (𝜑 → (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))))
117106, 116eqtrd 2644 . . 3 (𝜑 → (𝐺 Σg (𝐹𝐶)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))))
11846reseq1d 5316 . . . . . 6 (𝜑 → (𝐹𝐷) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐷))
119 ssun2 3739 . . . . . . . 8 𝐷 ⊆ (𝐶𝐷)
120119, 79syl5sseqr 3617 . . . . . . 7 (𝜑𝐷𝐴)
12171mpteq2ia 4668 . . . . . . . 8 (𝑘𝐷 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝑘𝐷 ↦ (𝐹𝑘))
122 resmpt 5369 . . . . . . . 8 (𝐷𝐴 → ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷) = (𝑘𝐷 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))
123 resmpt 5369 . . . . . . . 8 (𝐷𝐴 → ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐷) = (𝑘𝐷 ↦ (𝐹𝑘)))
124121, 122, 1233eqtr4a 2670 . . . . . . 7 (𝐷𝐴 → ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐷))
125120, 124syl 17 . . . . . 6 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐷))
126118, 125eqtr4d 2647 . . . . 5 (𝜑 → (𝐹𝐷) = ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷))
127126oveq2d 6565 . . . 4 (𝜑 → (𝐺 Σg (𝐹𝐷)) = (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷)))
12891, 43fmptd 6292 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )):𝐴𝐵)
129 frn 5966 . . . . . . 7 ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) → ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
13044, 129syl 17 . . . . . 6 (𝜑 → ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
1314cntzidss 17593 . . . . . 6 ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ⊆ (𝑍‘ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
13228, 130, 131syl2anc 691 . . . . 5 (𝜑 → ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ⊆ (𝑍‘ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
133 eldifn 3695 . . . . . . . 8 (𝑘 ∈ (𝐴𝐷) → ¬ 𝑘𝐷)
134133adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (𝐴𝐷)) → ¬ 𝑘𝐷)
135134iffalsed 4047 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝐷)) → if(𝑘𝐷, (𝐹𝑘), 0 ) = 0 )
136135, 6suppss2 7216 . . . . 5 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) supp 0 ) ⊆ 𝐷)
1371, 2, 4, 5, 6, 128, 132, 136, 13gsumzres 18133 . . . 4 (𝜑 → (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
138127, 137eqtrd 2644 . . 3 (𝜑 → (𝐺 Σg (𝐹𝐷)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
139117, 138oveq12d 6567 . 2 (𝜑 → ((𝐺 Σg (𝐹𝐶)) + (𝐺 Σg (𝐹𝐷))) = ((𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))) + (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))))
14045, 96, 1393eqtr4d 2654 1 (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹𝐶)) + (𝐺 Σg (𝐹𝐷))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   finSupp cfsupp 8158  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068  Mndcmnd 17117  SubMndcsubmnd 17157  Cntzccntz 17571  CMndccmn 18016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-cntz 17573  df-cmn 18018 This theorem is referenced by:  gsumsplit  18151  gsumzunsnd  18178  dpjidcl  18280
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