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Theorem gsumzres 19693
Description: Extend a finite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 31-May-2019.)
Hypotheses
Ref Expression
gsumzcl.b 𝐡 = (Baseβ€˜πΊ)
gsumzcl.0 0 = (0gβ€˜πΊ)
gsumzcl.z 𝑍 = (Cntzβ€˜πΊ)
gsumzcl.g (πœ‘ β†’ 𝐺 ∈ Mnd)
gsumzcl.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
gsumzcl.f (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
gsumzcl.c (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
gsumzres.s (πœ‘ β†’ (𝐹 supp 0 ) βŠ† π‘Š)
gsumzres.w (πœ‘ β†’ 𝐹 finSupp 0 )
Assertion
Ref Expression
gsumzres (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹))

Proof of Theorem gsumzres
Dummy variables 𝑓 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzcl.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ Mnd)
2 gsumzcl.a . . . . . . . 8 (πœ‘ β†’ 𝐴 ∈ 𝑉)
3 inex1g 5281 . . . . . . . 8 (𝐴 ∈ 𝑉 β†’ (𝐴 ∩ π‘Š) ∈ V)
42, 3syl 17 . . . . . . 7 (πœ‘ β†’ (𝐴 ∩ π‘Š) ∈ V)
5 gsumzcl.0 . . . . . . . 8 0 = (0gβ€˜πΊ)
65gsumz 18653 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴 ∩ π‘Š) ∈ V) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
71, 4, 6syl2anc 585 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
85gsumz 18653 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
91, 2, 8syl2anc 585 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
107, 9eqtr4d 2780 . . . . 5 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
1110adantr 482 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
12 resres 5955 . . . . . . . 8 ((𝐹 β†Ύ 𝐴) β†Ύ π‘Š) = (𝐹 β†Ύ (𝐴 ∩ π‘Š))
13 gsumzcl.f . . . . . . . . . 10 (πœ‘ β†’ 𝐹:𝐴⟢𝐡)
14 ffn 6673 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
15 fnresdm 6625 . . . . . . . . . 10 (𝐹 Fn 𝐴 β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1613, 14, 153syl 18 . . . . . . . . 9 (πœ‘ β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1716reseq1d 5941 . . . . . . . 8 (πœ‘ β†’ ((𝐹 β†Ύ 𝐴) β†Ύ π‘Š) = (𝐹 β†Ύ π‘Š))
1812, 17eqtr3id 2791 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
1918adantr 482 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
205fvexi 6861 . . . . . . . . . 10 0 ∈ V
2120a1i 11 . . . . . . . . 9 (πœ‘ β†’ 0 ∈ V)
22 ssid 3971 . . . . . . . . . 10 (𝐹 supp 0 ) βŠ† (𝐹 supp 0 )
2322a1i 11 . . . . . . . . 9 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐹 supp 0 ))
2413, 2, 21, 23gsumcllem 19692 . . . . . . . 8 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ 0 ))
2524reseq1d 5941 . . . . . . 7 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = ((π‘˜ ∈ 𝐴 ↦ 0 ) β†Ύ (𝐴 ∩ π‘Š)))
26 inss1 4193 . . . . . . . 8 (𝐴 ∩ π‘Š) βŠ† 𝐴
27 resmpt 5996 . . . . . . . 8 ((𝐴 ∩ π‘Š) βŠ† 𝐴 β†’ ((π‘˜ ∈ 𝐴 ↦ 0 ) β†Ύ (𝐴 ∩ π‘Š)) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
2826, 27ax-mp 5 . . . . . . 7 ((π‘˜ ∈ 𝐴 ↦ 0 ) β†Ύ (𝐴 ∩ π‘Š)) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )
2925, 28eqtrdi 2793 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
3019, 29eqtr3d 2779 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ π‘Š) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
3130oveq2d 7378 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )))
3224oveq2d 7378 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
3311, 31, 323eqtr4d 2787 . . 3 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹))
3433ex 414 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹)))
35 f1ofo 6796 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–ontoβ†’(𝐹 supp 0 ))
36 forn 6764 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–ontoβ†’(𝐹 supp 0 ) β†’ ran 𝑓 = (𝐹 supp 0 ))
3735, 36syl 17 . . . . . . . . . . 11 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ ran 𝑓 = (𝐹 supp 0 ))
3837ad2antll 728 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran 𝑓 = (𝐹 supp 0 ))
39 gsumzres.s . . . . . . . . . . 11 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† π‘Š)
4039adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† π‘Š)
4138, 40eqsstrd 3987 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran 𝑓 βŠ† π‘Š)
42 cores 6206 . . . . . . . . 9 (ran 𝑓 βŠ† π‘Š β†’ ((𝐹 β†Ύ π‘Š) ∘ 𝑓) = (𝐹 ∘ 𝑓))
4341, 42syl 17 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ((𝐹 β†Ύ π‘Š) ∘ 𝑓) = (𝐹 ∘ 𝑓))
4443seqeq3d 13921 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ seq1((+gβ€˜πΊ), ((𝐹 β†Ύ π‘Š) ∘ 𝑓)) = seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓)))
4544fveq1d 6849 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (seq1((+gβ€˜πΊ), ((𝐹 β†Ύ π‘Š) ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))) = (seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))))
46 gsumzcl.b . . . . . . 7 𝐡 = (Baseβ€˜πΊ)
47 eqid 2737 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
48 gsumzcl.z . . . . . . 7 𝑍 = (Cntzβ€˜πΊ)
491adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐺 ∈ Mnd)
504adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐴 ∩ π‘Š) ∈ V)
5113adantr 482 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐹:𝐴⟢𝐡)
52 fssres 6713 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ (𝐴 ∩ π‘Š) βŠ† 𝐴) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡)
5351, 26, 52sylancl 587 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡)
5418feq1d 6658 . . . . . . . . 9 (πœ‘ β†’ ((𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡 ↔ (𝐹 β†Ύ π‘Š):(𝐴 ∩ π‘Š)⟢𝐡))
5554biimpa 478 . . . . . . . 8 ((πœ‘ ∧ (𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡) β†’ (𝐹 β†Ύ π‘Š):(𝐴 ∩ π‘Š)⟢𝐡)
5653, 55syldan 592 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 β†Ύ π‘Š):(𝐴 ∩ π‘Š)⟢𝐡)
57 gsumzcl.c . . . . . . . . 9 (πœ‘ β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
58 resss 5967 . . . . . . . . . 10 (𝐹 β†Ύ π‘Š) βŠ† 𝐹
5958rnssi 5900 . . . . . . . . 9 ran (𝐹 β†Ύ π‘Š) βŠ† ran 𝐹
6048cntzidss 19125 . . . . . . . . 9 ((ran 𝐹 βŠ† (π‘β€˜ran 𝐹) ∧ ran (𝐹 β†Ύ π‘Š) βŠ† ran 𝐹) β†’ ran (𝐹 β†Ύ π‘Š) βŠ† (π‘β€˜ran (𝐹 β†Ύ π‘Š)))
6157, 59, 60sylancl 587 . . . . . . . 8 (πœ‘ β†’ ran (𝐹 β†Ύ π‘Š) βŠ† (π‘β€˜ran (𝐹 β†Ύ π‘Š)))
6261adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran (𝐹 β†Ύ π‘Š) βŠ† (π‘β€˜ran (𝐹 β†Ύ π‘Š)))
63 simprl 770 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (β™―β€˜(𝐹 supp 0 )) ∈ β„•)
64 f1of1 6788 . . . . . . . . 9 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ))
6564ad2antll 728 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ))
66 suppssdm 8113 . . . . . . . . . . 11 (𝐹 supp 0 ) βŠ† dom 𝐹
6766, 13fssdm 6693 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝐴)
6867, 39ssind 4197 . . . . . . . . 9 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐴 ∩ π‘Š))
6968adantr 482 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† (𝐴 ∩ π‘Š))
70 f1ss 6749 . . . . . . . 8 ((𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) βŠ† (𝐴 ∩ π‘Š)) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐴 ∩ π‘Š))
7165, 69, 70syl2anc 585 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐴 ∩ π‘Š))
7213, 2fexd 7182 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹 ∈ V)
73 ressuppss 8119 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
7472, 20, 73sylancl 587 . . . . . . . . . . 11 (πœ‘ β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
75 sseq2 3975 . . . . . . . . . . 11 (ran 𝑓 = (𝐹 supp 0 ) β†’ (((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† ran 𝑓 ↔ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 )))
7674, 75syl5ibr 246 . . . . . . . . . 10 (ran 𝑓 = (𝐹 supp 0 ) β†’ (πœ‘ β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† ran 𝑓))
7735, 36, 763syl 18 . . . . . . . . 9 (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ (πœ‘ β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† ran 𝑓))
7877adantl 483 . . . . . . . 8 (((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 )) β†’ (πœ‘ β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† ran 𝑓))
7978impcom 409 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† ran 𝑓)
80 eqid 2737 . . . . . . 7 (((𝐹 β†Ύ π‘Š) ∘ 𝑓) supp 0 ) = (((𝐹 β†Ύ π‘Š) ∘ 𝑓) supp 0 )
8146, 5, 47, 48, 49, 50, 56, 62, 63, 71, 79, 80gsumval3 19691 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (seq1((+gβ€˜πΊ), ((𝐹 β†Ύ π‘Š) ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))))
822adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝐴 ∈ 𝑉)
8357adantr 482 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ ran 𝐹 βŠ† (π‘β€˜ran 𝐹))
8467adantr 482 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† 𝐴)
85 f1ss 6749 . . . . . . . 8 ((𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1β†’(𝐹 supp 0 ) ∧ (𝐹 supp 0 ) βŠ† 𝐴) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴)
8665, 84, 85syl2anc 585 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1→𝐴)
8722, 38sseqtrrid 4002 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐹 supp 0 ) βŠ† ran 𝑓)
88 eqid 2737 . . . . . . 7 ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
8946, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88gsumval3 19691 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g 𝐹) = (seq1((+gβ€˜πΊ), (𝐹 ∘ 𝑓))β€˜(β™―β€˜(𝐹 supp 0 ))))
9045, 81, 893eqtr4d 2787 . . . . 5 ((πœ‘ ∧ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹))
9190expr 458 . . . 4 ((πœ‘ ∧ (β™―β€˜(𝐹 supp 0 )) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹)))
9291exlimdv 1937 . . 3 ((πœ‘ ∧ (β™―β€˜(𝐹 supp 0 )) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹)))
9392expimpd 455 . 2 (πœ‘ β†’ (((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 )) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹)))
94 gsumzres.w . . 3 (πœ‘ β†’ 𝐹 finSupp 0 )
95 fsuppimp 9318 . . . 4 (𝐹 finSupp 0 β†’ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
9695simprd 497 . . 3 (𝐹 finSupp 0 β†’ (𝐹 supp 0 ) ∈ Fin)
97 fz1f1o 15602 . . 3 ((𝐹 supp 0 ) ∈ Fin β†’ ((𝐹 supp 0 ) = βˆ… ∨ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))))
9894, 96, 973syl 18 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… ∨ ((β™―β€˜(𝐹 supp 0 )) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜(𝐹 supp 0 )))–1-1-ontoβ†’(𝐹 supp 0 ))))
9934, 93, 98mpjaod 859 1 (πœ‘ β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287   class class class wbr 5110   ↦ cmpt 5193  ran crn 5639   β†Ύ cres 5640   ∘ ccom 5642  Fun wfun 6495   Fn wfn 6496  βŸΆwf 6497  β€“1-1β†’wf1 6498  β€“ontoβ†’wfo 6499  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362   supp csupp 8097  Fincfn 8890   finSupp cfsupp 9312  1c1 11059  β„•cn 12160  ...cfz 13431  seqcseq 13913  β™―chash 14237  Basecbs 17090  +gcplusg 17140  0gc0g 17328   Ξ£g cgsu 17329  Mndcmnd 18563  Cntzccntz 19102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-seq 13914  df-hash 14238  df-0g 17330  df-gsum 17331  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-cntz 19104
This theorem is referenced by:  gsumres  19697  gsumzsplit  19711  gsumpt  19746  dmdprdsplitlem  19823  dpjidcl  19844  mplcoe5  21457
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