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Theorem gsumzsplit 18096
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 6098 . . . . . 6 (0g𝐺) ∈ V
92, 8eqeltri 2683 . . . . 5 0 ∈ V
109a1i 11 . . . 4 (𝜑0 ∈ V)
11 gsumzsplit.w . . . 4 (𝜑𝐹 finSupp 0 )
127, 6, 10, 11fsuppmptif 8165 . . 3 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) finSupp 0 )
137, 6, 10, 11fsuppmptif 8165 . . 3 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) finSupp 0 )
141submacs 17134 . . . . 5 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
15 acsmre 16082 . . . . 5 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
165, 14, 153syl 18 . . . 4 (𝜑 → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
17 frn 5952 . . . . 5 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
187, 17syl 17 . . . 4 (𝜑 → ran 𝐹𝐵)
19 eqid 2609 . . . . 5 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
2019mrccl 16040 . . . 4 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
2116, 18, 20syl2anc 690 . . 3 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
22 gsumzsplit.c . . . . 5 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
23 eqid 2609 . . . . . 6 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
244, 19, 23cntzspan 18016 . . . . 5 ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
255, 22, 24syl2anc 690 . . . 4 (𝜑 → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
2623, 4submcmn2 18013 . . . . 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 220 . . 3 (𝜑 → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
2916, 19, 18mrcssidd 16054 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
3029adantr 479 . . . . . 6 ((𝜑𝑘𝐴) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
31 ffn 5944 . . . . . . . 8 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
327, 31syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
33 fnfvelrn 6249 . . . . . . 7 ((𝐹 Fn 𝐴𝑘𝐴) → (𝐹𝑘) ∈ ran 𝐹)
3432, 33sylan 486 . . . . . 6 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ ran 𝐹)
3530, 34sseldd 3568 . . . . 5 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
362subm0cl 17121 . . . . . . 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 479 . . . . 5 ((𝜑𝑘𝐴) → 0 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
3935, 38ifcld 4080 . . . 4 ((𝜑𝑘𝐴) → if(𝑘𝐶, (𝐹𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
40 eqid 2609 . . . 4 (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))
4139, 40fmptd 6277 . . 3 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )):𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
4235, 38ifcld 4080 . . . 4 ((𝜑𝑘𝐴) → if(𝑘𝐷, (𝐹𝑘), 0 ) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
43 eqid 2609 . . . 4 (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))
4442, 43fmptd 6277 . . 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 18091 . 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 6144 . . . . 5 (𝜑𝐹 = (𝑘𝐴 ↦ (𝐹𝑘)))
47 iftrue 4041 . . . . . . . . . 10 (𝑘𝐶 → if(𝑘𝐶, (𝐹𝑘), 0 ) = (𝐹𝑘))
4847adantl 480 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → if(𝑘𝐶, (𝐹𝑘), 0 ) = (𝐹𝑘))
49 gsumzsplit.i . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐷) = ∅)
50 noel 3877 . . . . . . . . . . . . . . . 16 ¬ 𝑘 ∈ ∅
51 eleq2 2676 . . . . . . . . . . . . . . . 16 ((𝐶𝐷) = ∅ → (𝑘 ∈ (𝐶𝐷) ↔ 𝑘 ∈ ∅))
5250, 51mtbiri 315 . . . . . . . . . . . . . . 15 ((𝐶𝐷) = ∅ → ¬ 𝑘 ∈ (𝐶𝐷))
5349, 52syl 17 . . . . . . . . . . . . . 14 (𝜑 → ¬ 𝑘 ∈ (𝐶𝐷))
5453adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ¬ 𝑘 ∈ (𝐶𝐷))
55 elin 3757 . . . . . . . . . . . . 13 (𝑘 ∈ (𝐶𝐷) ↔ (𝑘𝐶𝑘𝐷))
5654, 55sylnib 316 . . . . . . . . . . . 12 ((𝜑𝑘𝐴) → ¬ (𝑘𝐶𝑘𝐷))
57 imnan 436 . . . . . . . . . . . 12 ((𝑘𝐶 → ¬ 𝑘𝐷) ↔ ¬ (𝑘𝐶𝑘𝐷))
5856, 57sylibr 222 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝑘𝐶 → ¬ 𝑘𝐷))
5958imp 443 . . . . . . . . . 10 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → ¬ 𝑘𝐷)
6059iffalsed 4046 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → if(𝑘𝐷, (𝐹𝑘), 0 ) = 0 )
6148, 60oveq12d 6545 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = ((𝐹𝑘) + 0 ))
627ffvelrnda 6252 . . . . . . . . . 10 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ 𝐵)
631, 3, 2mndrid 17081 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
645, 63sylan 486 . . . . . . . . . 10 ((𝜑 ∧ (𝐹𝑘) ∈ 𝐵) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
6562, 64syldan 485 . . . . . . . . 9 ((𝜑𝑘𝐴) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
6665adantr 479 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → ((𝐹𝑘) + 0 ) = (𝐹𝑘))
6761, 66eqtrd 2643 . . . . . . 7 (((𝜑𝑘𝐴) ∧ 𝑘𝐶) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝐹𝑘))
6858con2d 127 . . . . . . . . . . 11 ((𝜑𝑘𝐴) → (𝑘𝐷 → ¬ 𝑘𝐶))
6968imp 443 . . . . . . . . . 10 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → ¬ 𝑘𝐶)
7069iffalsed 4046 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → if(𝑘𝐶, (𝐹𝑘), 0 ) = 0 )
71 iftrue 4041 . . . . . . . . . 10 (𝑘𝐷 → if(𝑘𝐷, (𝐹𝑘), 0 ) = (𝐹𝑘))
7271adantl 480 . . . . . . . . 9 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → if(𝑘𝐷, (𝐹𝑘), 0 ) = (𝐹𝑘))
7370, 72oveq12d 6545 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = ( 0 + (𝐹𝑘)))
741, 3, 2mndlid 17080 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ (𝐹𝑘) ∈ 𝐵) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
755, 74sylan 486 . . . . . . . . . 10 ((𝜑 ∧ (𝐹𝑘) ∈ 𝐵) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
7662, 75syldan 485 . . . . . . . . 9 ((𝜑𝑘𝐴) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
7776adantr 479 . . . . . . . 8 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → ( 0 + (𝐹𝑘)) = (𝐹𝑘))
7873, 77eqtrd 2643 . . . . . . 7 (((𝜑𝑘𝐴) ∧ 𝑘𝐷) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝐹𝑘))
79 gsumzsplit.u . . . . . . . . . 10 (𝜑𝐴 = (𝐶𝐷))
8079eleq2d 2672 . . . . . . . . 9 (𝜑 → (𝑘𝐴𝑘 ∈ (𝐶𝐷)))
81 elun 3714 . . . . . . . . 9 (𝑘 ∈ (𝐶𝐷) ↔ (𝑘𝐶𝑘𝐷))
8280, 81syl6bb 274 . . . . . . . 8 (𝜑 → (𝑘𝐴 ↔ (𝑘𝐶𝑘𝐷)))
8382biimpa 499 . . . . . . 7 ((𝜑𝑘𝐴) → (𝑘𝐶𝑘𝐷))
8467, 78, 83mpjaodan 822 . . . . . 6 ((𝜑𝑘𝐴) → (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝐹𝑘))
8584mpteq2dva 4666 . . . . 5 (𝜑 → (𝑘𝐴 ↦ (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 ))) = (𝑘𝐴 ↦ (𝐹𝑘)))
8646, 85eqtr4d 2646 . . . 4 (𝜑𝐹 = (𝑘𝐴 ↦ (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 ))))
871, 2mndidcl 17077 . . . . . . . 8 (𝐺 ∈ Mnd → 0𝐵)
885, 87syl 17 . . . . . . 7 (𝜑0𝐵)
8988adantr 479 . . . . . 6 ((𝜑𝑘𝐴) → 0𝐵)
9062, 89ifcld 4080 . . . . 5 ((𝜑𝑘𝐴) → if(𝑘𝐶, (𝐹𝑘), 0 ) ∈ 𝐵)
9162, 89ifcld 4080 . . . . 5 ((𝜑𝑘𝐴) → if(𝑘𝐷, (𝐹𝑘), 0 ) ∈ 𝐵)
92 eqidd 2610 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )))
93 eqidd 2610 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))
946, 90, 91, 92, 93offval2 6789 . . . 4 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ∘𝑓 + (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))) = (𝑘𝐴 ↦ (if(𝑘𝐶, (𝐹𝑘), 0 ) + if(𝑘𝐷, (𝐹𝑘), 0 ))))
9586, 94eqtr4d 2646 . . 3 (𝜑𝐹 = ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ∘𝑓 + (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
9695oveq2d 6543 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ∘𝑓 + (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))))
9746reseq1d 5303 . . . . . 6 (𝜑 → (𝐹𝐶) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐶))
98 ssun1 3737 . . . . . . . 8 𝐶 ⊆ (𝐶𝐷)
9998, 79syl5sseqr 3616 . . . . . . 7 (𝜑𝐶𝐴)
10047mpteq2ia 4662 . . . . . . . 8 (𝑘𝐶 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) = (𝑘𝐶 ↦ (𝐹𝑘))
101 resmpt 5356 . . . . . . . 8 (𝐶𝐴 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶) = (𝑘𝐶 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )))
102 resmpt 5356 . . . . . . . 8 (𝐶𝐴 → ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐶) = (𝑘𝐶 ↦ (𝐹𝑘)))
103100, 101, 1023eqtr4a 2669 . . . . . . 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 2646 . . . . 5 (𝜑 → (𝐹𝐶) = ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶))
106105oveq2d 6543 . . . 4 (𝜑 → (𝐺 Σg (𝐹𝐶)) = (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶)))
10790, 40fmptd 6277 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )):𝐴𝐵)
108 frn 5952 . . . . . . 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 17539 . . . . . 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 690 . . . . 5 (𝜑 → ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ⊆ (𝑍‘ran (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))))
112 eldifn 3694 . . . . . . . 8 (𝑘 ∈ (𝐴𝐶) → ¬ 𝑘𝐶)
113112adantl 480 . . . . . . 7 ((𝜑𝑘 ∈ (𝐴𝐶)) → ¬ 𝑘𝐶)
114113iffalsed 4046 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝐶)) → if(𝑘𝐶, (𝐹𝑘), 0 ) = 0 )
115114, 6suppss2 7193 . . . . 5 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) supp 0 ) ⊆ 𝐶)
1161, 2, 4, 5, 6, 107, 111, 115, 12gsumzres 18079 . . . 4 (𝜑 → (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 )) ↾ 𝐶)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))))
117106, 116eqtrd 2643 . . 3 (𝜑 → (𝐺 Σg (𝐹𝐶)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐶, (𝐹𝑘), 0 ))))
11846reseq1d 5303 . . . . . 6 (𝜑 → (𝐹𝐷) = ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐷))
119 ssun2 3738 . . . . . . . 8 𝐷 ⊆ (𝐶𝐷)
120119, 79syl5sseqr 3616 . . . . . . 7 (𝜑𝐷𝐴)
12171mpteq2ia 4662 . . . . . . . 8 (𝑘𝐷 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) = (𝑘𝐷 ↦ (𝐹𝑘))
122 resmpt 5356 . . . . . . . 8 (𝐷𝐴 → ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷) = (𝑘𝐷 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )))
123 resmpt 5356 . . . . . . . 8 (𝐷𝐴 → ((𝑘𝐴 ↦ (𝐹𝑘)) ↾ 𝐷) = (𝑘𝐷 ↦ (𝐹𝑘)))
124121, 122, 1233eqtr4a 2669 . . . . . . 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 2646 . . . . 5 (𝜑 → (𝐹𝐷) = ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷))
127126oveq2d 6543 . . . 4 (𝜑 → (𝐺 Σg (𝐹𝐷)) = (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷)))
12891, 43fmptd 6277 . . . . 5 (𝜑 → (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )):𝐴𝐵)
129 frn 5952 . . . . . . 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 17539 . . . . . 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 690 . . . . 5 (𝜑 → ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ⊆ (𝑍‘ran (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
133 eldifn 3694 . . . . . . . 8 (𝑘 ∈ (𝐴𝐷) → ¬ 𝑘𝐷)
134133adantl 480 . . . . . . 7 ((𝜑𝑘 ∈ (𝐴𝐷)) → ¬ 𝑘𝐷)
135134iffalsed 4046 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝐷)) → if(𝑘𝐷, (𝐹𝑘), 0 ) = 0 )
136135, 6suppss2 7193 . . . . 5 (𝜑 → ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) supp 0 ) ⊆ 𝐷)
1371, 2, 4, 5, 6, 128, 132, 136, 13gsumzres 18079 . . . 4 (𝜑 → (𝐺 Σg ((𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 )) ↾ 𝐷)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
138127, 137eqtrd 2643 . . 3 (𝜑 → (𝐺 Σg (𝐹𝐷)) = (𝐺 Σg (𝑘𝐴 ↦ if(𝑘𝐷, (𝐹𝑘), 0 ))))
139117, 138oveq12d 6545 . 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 2653 1 (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹𝐶)) + (𝐺 Σg (𝐹𝐷))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  ifcif 4035   class class class wbr 4577  cmpt 4637  ran crn 5029  cres 5030   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  𝑓 cof 6770   finSupp cfsupp 8135  Basecbs 15641  s cress 15642  +gcplusg 15714  0gc0g 15869   Σg cgsu 15870  Moorecmre 16011  mrClscmrc 16012  ACScacs 16014  Mndcmnd 17063  SubMndcsubmnd 17103  Cntzccntz 17517  CMndccmn 17962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-oi 8275  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-seq 12619  df-hash 12935  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-0g 15871  df-gsum 15872  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-submnd 17105  df-cntz 17519  df-cmn 17964
This theorem is referenced by:  gsumsplit  18097  gsumzunsnd  18124  dpjidcl  18226
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