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Theorem gsumzres 19771
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 5318 . . . . . . . 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 18713 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴 ∩ π‘Š) ∈ V) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
71, 4, 6syl2anc 584 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
85gsumz 18713 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
91, 2, 8syl2anc 584 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
107, 9eqtr4d 2775 . . . . 5 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
1110adantr 481 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
12 resres 5992 . . . . . . . 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 6714 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
15 fnresdm 6666 . . . . . . . . . 10 (𝐹 Fn 𝐴 β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1613, 14, 153syl 18 . . . . . . . . 9 (πœ‘ β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1716reseq1d 5978 . . . . . . . 8 (πœ‘ β†’ ((𝐹 β†Ύ 𝐴) β†Ύ π‘Š) = (𝐹 β†Ύ π‘Š))
1812, 17eqtr3id 2786 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
1918adantr 481 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
205fvexi 6902 . . . . . . . . . 10 0 ∈ V
2120a1i 11 . . . . . . . . 9 (πœ‘ β†’ 0 ∈ V)
22 ssid 4003 . . . . . . . . . 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 19770 . . . . . . . 8 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ 0 ))
2524reseq1d 5978 . . . . . . 7 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = ((π‘˜ ∈ 𝐴 ↦ 0 ) β†Ύ (𝐴 ∩ π‘Š)))
26 inss1 4227 . . . . . . . 8 (𝐴 ∩ π‘Š) βŠ† 𝐴
27 resmpt 6035 . . . . . . . 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 2788 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
3019, 29eqtr3d 2774 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ π‘Š) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
3130oveq2d 7421 . . . 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 7421 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
3311, 31, 323eqtr4d 2782 . . 3 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹))
3433ex 413 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹)))
35 f1ofo 6837 . . . . . . . . . . . 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 6805 . . . . . . . . . . . 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 727 . . . . . . . . . 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 481 . . . . . . . . . 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 4019 . . . . . . . . 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 6245 . . . . . . . . 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 13970 . . . . . . 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 6890 . . . . . 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 2732 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
48 gsumzcl.z . . . . . . 7 𝑍 = (Cntzβ€˜πΊ)
491adantr 481 . . . . . . 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 481 . . . . . . 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 481 . . . . . . . . 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 6754 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ (𝐴 ∩ π‘Š) βŠ† 𝐴) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡)
5351, 26, 52sylancl 586 . . . . . . . 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 6699 . . . . . . . . 9 (πœ‘ β†’ ((𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡 ↔ (𝐹 β†Ύ π‘Š):(𝐴 ∩ π‘Š)⟢𝐡))
5554biimpa 477 . . . . . . . 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 591 . . . . . . 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 6004 . . . . . . . . . 10 (𝐹 β†Ύ π‘Š) βŠ† 𝐹
5958rnssi 5937 . . . . . . . . 9 ran (𝐹 β†Ύ π‘Š) βŠ† ran 𝐹
6048cntzidss 19198 . . . . . . . . 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 586 . . . . . . . 8 (πœ‘ β†’ ran (𝐹 β†Ύ π‘Š) βŠ† (π‘β€˜ran (𝐹 β†Ύ π‘Š)))
6261adantr 481 . . . . . . 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 769 . . . . . . 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 6829 . . . . . . . . 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 727 . . . . . . . 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 8158 . . . . . . . . . . 11 (𝐹 supp 0 ) βŠ† dom 𝐹
6766, 13fssdm 6734 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝐴)
6867, 39ssind 4231 . . . . . . . . 9 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐴 ∩ π‘Š))
6968adantr 481 . . . . . . . 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 6790 . . . . . . . 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 584 . . . . . . 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 7225 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹 ∈ V)
73 ressuppss 8164 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
7472, 20, 73sylancl 586 . . . . . . . . . . 11 (πœ‘ β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
75 sseq2 4007 . . . . . . . . . . 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, 75imbitrrid 245 . . . . . . . . . 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 482 . . . . . . . 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 408 . . . . . . 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 2732 . . . . . . 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 19769 . . . . . 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 481 . . . . . . 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 481 . . . . . . 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 481 . . . . . . . 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 6790 . . . . . . . 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 584 . . . . . . 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 4034 . . . . . . 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 2732 . . . . . . 7 ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
8946, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88gsumval3 19769 . . . . . 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 2782 . . . . 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 457 . . . 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 1936 . . 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 454 . 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 9364 . . . 4 (𝐹 finSupp 0 β†’ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
9695simprd 496 . . 3 (𝐹 finSupp 0 β†’ (𝐹 supp 0 ) ∈ Fin)
97 fz1f1o 15652 . . 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 858 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 396   ∨ wo 845   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   supp csupp 8142  Fincfn 8935   finSupp cfsupp 9357  1c1 11107  β„•cn 12208  ...cfz 13480  seqcseq 13962  β™―chash 14286  Basecbs 17140  +gcplusg 17193  0gc0g 17381   Ξ£g cgsu 17382  Mndcmnd 18621  Cntzccntz 19173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-0g 17383  df-gsum 17384  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-cntz 19175
This theorem is referenced by:  gsumres  19775  gsumzsplit  19789  gsumpt  19824  dmdprdsplitlem  19901  dpjidcl  19922  mplcoe5  21586
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