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Theorem gsumzres 19829
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 5312 . . . . . . . 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 18761 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴 ∩ π‘Š) ∈ V) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
71, 4, 6syl2anc 583 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
85gsumz 18761 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
91, 2, 8syl2anc 583 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
107, 9eqtr4d 2769 . . . . 5 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
1110adantr 480 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
12 resres 5988 . . . . . . . 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 6711 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
15 fnresdm 6663 . . . . . . . . . 10 (𝐹 Fn 𝐴 β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1613, 14, 153syl 18 . . . . . . . . 9 (πœ‘ β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1716reseq1d 5974 . . . . . . . 8 (πœ‘ β†’ ((𝐹 β†Ύ 𝐴) β†Ύ π‘Š) = (𝐹 β†Ύ π‘Š))
1812, 17eqtr3id 2780 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
1918adantr 480 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
205fvexi 6899 . . . . . . . . . 10 0 ∈ V
2120a1i 11 . . . . . . . . 9 (πœ‘ β†’ 0 ∈ V)
22 ssid 3999 . . . . . . . . . 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 19828 . . . . . . . 8 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ 0 ))
2524reseq1d 5974 . . . . . . 7 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = ((π‘˜ ∈ 𝐴 ↦ 0 ) β†Ύ (𝐴 ∩ π‘Š)))
26 inss1 4223 . . . . . . . 8 (𝐴 ∩ π‘Š) βŠ† 𝐴
27 resmpt 6031 . . . . . . . 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 2782 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
3019, 29eqtr3d 2768 . . . . 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 2776 . . 3 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹))
3433ex 412 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹)))
35 f1ofo 6834 . . . . . . . . . . . 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 6802 . . . . . . . . . . . 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 726 . . . . . . . . . 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 480 . . . . . . . . . 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 4015 . . . . . . . . 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 6242 . . . . . . . . 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 13980 . . . . . . 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 6887 . . . . . 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 2726 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
48 gsumzcl.z . . . . . . 7 𝑍 = (Cntzβ€˜πΊ)
491adantr 480 . . . . . . 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 480 . . . . . . 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 480 . . . . . . . . 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 6751 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ (𝐴 ∩ π‘Š) βŠ† 𝐴) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡)
5351, 26, 52sylancl 585 . . . . . . . 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 6696 . . . . . . . . 9 (πœ‘ β†’ ((𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡 ↔ (𝐹 β†Ύ π‘Š):(𝐴 ∩ π‘Š)⟢𝐡))
5554biimpa 476 . . . . . . . 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 590 . . . . . . 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 6000 . . . . . . . . . 10 (𝐹 β†Ύ π‘Š) βŠ† 𝐹
5958rnssi 5933 . . . . . . . . 9 ran (𝐹 β†Ύ π‘Š) βŠ† ran 𝐹
6048cntzidss 19256 . . . . . . . . 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 585 . . . . . . . 8 (πœ‘ β†’ ran (𝐹 β†Ύ π‘Š) βŠ† (π‘β€˜ran (𝐹 β†Ύ π‘Š)))
6261adantr 480 . . . . . . 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 768 . . . . . . 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 6826 . . . . . . . . 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 726 . . . . . . . 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 8162 . . . . . . . . . . 11 (𝐹 supp 0 ) βŠ† dom 𝐹
6766, 13fssdm 6731 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝐴)
6867, 39ssind 4227 . . . . . . . . 9 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐴 ∩ π‘Š))
6968adantr 480 . . . . . . . 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 6787 . . . . . . . 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 583 . . . . . . 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 7224 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹 ∈ V)
73 ressuppss 8168 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
7472, 20, 73sylancl 585 . . . . . . . . . . 11 (πœ‘ β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
75 sseq2 4003 . . . . . . . . . . 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 481 . . . . . . . 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 407 . . . . . . 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 2726 . . . . . . 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 19827 . . . . . 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 480 . . . . . . 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 480 . . . . . . 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 480 . . . . . . . 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 6787 . . . . . . . 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 583 . . . . . . 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 4030 . . . . . . 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 2726 . . . . . . 7 ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
8946, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88gsumval3 19827 . . . . . 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 2776 . . . . 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 456 . . . 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 1928 . . 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 453 . 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 9370 . . . 4 (𝐹 finSupp 0 β†’ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
9695simprd 495 . . 3 (𝐹 finSupp 0 β†’ (𝐹 supp 0 ) ∈ Fin)
97 fz1f1o 15662 . . 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 857 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 395   ∨ wo 844   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317   class class class wbr 5141   ↦ cmpt 5224  ran crn 5670   β†Ύ cres 5671   ∘ ccom 5673  Fun wfun 6531   Fn wfn 6532  βŸΆwf 6533  β€“1-1β†’wf1 6534  β€“ontoβ†’wfo 6535  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405   supp csupp 8146  Fincfn 8941   finSupp cfsupp 9363  1c1 11113  β„•cn 12216  ...cfz 13490  seqcseq 13972  β™―chash 14295  Basecbs 17153  +gcplusg 17206  0gc0g 17394   Ξ£g cgsu 17395  Mndcmnd 18667  Cntzccntz 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-supp 8147  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-0g 17396  df-gsum 17397  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-cntz 19233
This theorem is referenced by:  gsumres  19833  gsumzsplit  19847  gsumpt  19882  dmdprdsplitlem  19959  dpjidcl  19980  mplcoe5  21937
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