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Theorem gsumzres 19866
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 18790 . . . . . . 7 ((𝐺 ∈ Mnd ∧ (𝐴 ∩ π‘Š) ∈ V) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
71, 4, 6syl2anc 582 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = 0 )
85gsumz 18790 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
91, 2, 8syl2anc 582 . . . . . 6 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )) = 0 )
107, 9eqtr4d 2768 . . . . 5 (πœ‘ β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
1110adantr 479 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 )) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
12 resres 5990 . . . . . . . 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 6715 . . . . . . . . . 10 (𝐹:𝐴⟢𝐡 β†’ 𝐹 Fn 𝐴)
15 fnresdm 6667 . . . . . . . . . 10 (𝐹 Fn 𝐴 β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1613, 14, 153syl 18 . . . . . . . . 9 (πœ‘ β†’ (𝐹 β†Ύ 𝐴) = 𝐹)
1716reseq1d 5976 . . . . . . . 8 (πœ‘ β†’ ((𝐹 β†Ύ 𝐴) β†Ύ π‘Š) = (𝐹 β†Ύ π‘Š))
1812, 17eqtr3id 2779 . . . . . . 7 (πœ‘ β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
1918adantr 479 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (𝐹 β†Ύ π‘Š))
205fvexi 6904 . . . . . . . . . 10 0 ∈ V
2120a1i 11 . . . . . . . . 9 (πœ‘ β†’ 0 ∈ V)
22 ssid 3994 . . . . . . . . . 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 19865 . . . . . . . 8 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ 𝐹 = (π‘˜ ∈ 𝐴 ↦ 0 ))
2524reseq1d 5976 . . . . . . 7 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = ((π‘˜ ∈ 𝐴 ↦ 0 ) β†Ύ (𝐴 ∩ π‘Š)))
26 inss1 4221 . . . . . . . 8 (𝐴 ∩ π‘Š) βŠ† 𝐴
27 resmpt 6034 . . . . . . . 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 2781 . . . . . 6 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
3019, 29eqtr3d 2767 . . . . 5 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐹 β†Ύ π‘Š) = (π‘˜ ∈ (𝐴 ∩ π‘Š) ↦ 0 ))
3130oveq2d 7430 . . . 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 7430 . . . 4 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g 𝐹) = (𝐺 Ξ£g (π‘˜ ∈ 𝐴 ↦ 0 )))
3311, 31, 323eqtr4d 2775 . . 3 ((πœ‘ ∧ (𝐹 supp 0 ) = βˆ…) β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹))
3433ex 411 . 2 (πœ‘ β†’ ((𝐹 supp 0 ) = βˆ… β†’ (𝐺 Ξ£g (𝐹 β†Ύ π‘Š)) = (𝐺 Ξ£g 𝐹)))
35 f1ofo 6839 . . . . . . . . . . . 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 6807 . . . . . . . . . . . 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 479 . . . . . . . . . 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 4010 . . . . . . . . 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 6246 . . . . . . . . 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 14004 . . . . . . 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 6892 . . . . . 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 2725 . . . . . . 7 (+gβ€˜πΊ) = (+gβ€˜πΊ)
48 gsumzcl.z . . . . . . 7 𝑍 = (Cntzβ€˜πΊ)
491adantr 479 . . . . . . 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 479 . . . . . . 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 479 . . . . . . . . 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 6756 . . . . . . . . 9 ((𝐹:𝐴⟢𝐡 ∧ (𝐴 ∩ π‘Š) βŠ† 𝐴) β†’ (𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡)
5351, 26, 52sylancl 584 . . . . . . . 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 6700 . . . . . . . . 9 (πœ‘ β†’ ((𝐹 β†Ύ (𝐴 ∩ π‘Š)):(𝐴 ∩ π‘Š)⟢𝐡 ↔ (𝐹 β†Ύ π‘Š):(𝐴 ∩ π‘Š)⟢𝐡))
5554biimpa 475 . . . . . . . 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 589 . . . . . . 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 5999 . . . . . . . . . 10 (𝐹 β†Ύ π‘Š) βŠ† 𝐹
5958rnssi 5934 . . . . . . . . 9 ran (𝐹 β†Ύ π‘Š) βŠ† ran 𝐹
6048cntzidss 19293 . . . . . . . . 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 584 . . . . . . . 8 (πœ‘ β†’ ran (𝐹 β†Ύ π‘Š) βŠ† (π‘β€˜ran (𝐹 β†Ύ π‘Š)))
6261adantr 479 . . . . . . 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 6831 . . . . . . . . 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 8178 . . . . . . . . . . 11 (𝐹 supp 0 ) βŠ† dom 𝐹
6766, 13fssdm 6735 . . . . . . . . . 10 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† 𝐴)
6867, 39ssind 4225 . . . . . . . . 9 (πœ‘ β†’ (𝐹 supp 0 ) βŠ† (𝐴 ∩ π‘Š))
6968adantr 479 . . . . . . . 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 6792 . . . . . . . 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 582 . . . . . . 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 7233 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹 ∈ V)
73 ressuppss 8184 . . . . . . . . . . . 12 ((𝐹 ∈ V ∧ 0 ∈ V) β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
7472, 20, 73sylancl 584 . . . . . . . . . . 11 (πœ‘ β†’ ((𝐹 β†Ύ π‘Š) supp 0 ) βŠ† (𝐹 supp 0 ))
75 sseq2 3998 . . . . . . . . . . 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 480 . . . . . . . 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 406 . . . . . . 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 2725 . . . . . . 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 19864 . . . . . 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 479 . . . . . . 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 479 . . . . . . 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 479 . . . . . . . 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 6792 . . . . . . . 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 582 . . . . . . 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 4025 . . . . . . 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 2725 . . . . . . 7 ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
8946, 5, 47, 48, 49, 82, 51, 83, 63, 86, 87, 88gsumval3 19864 . . . . . 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 2775 . . . . 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 455 . . . 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 452 . 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 9390 . . . 4 (𝐹 finSupp 0 β†’ (Fun 𝐹 ∧ (𝐹 supp 0 ) ∈ Fin))
9695simprd 494 . . 3 (𝐹 finSupp 0 β†’ (𝐹 supp 0 ) ∈ Fin)
97 fz1f1o 15686 . . 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 394   ∨ wo 845   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  Vcvv 3463   ∩ cin 3938   βŠ† wss 3939  βˆ…c0 4316   class class class wbr 5141   ↦ cmpt 5224  ran crn 5671   β†Ύ cres 5672   ∘ ccom 5674  Fun wfun 6535   Fn wfn 6536  βŸΆwf 6537  β€“1-1β†’wf1 6538  β€“ontoβ†’wfo 6539  β€“1-1-ontoβ†’wf1o 6540  β€˜cfv 6541  (class class class)co 7414   supp csupp 8161  Fincfn 8960   finSupp cfsupp 9383  1c1 11137  β„•cn 12240  ...cfz 13514  seqcseq 13996  β™―chash 14319  Basecbs 17177  +gcplusg 17230  0gc0g 17418   Ξ£g cgsu 17419  Mndcmnd 18691  Cntzccntz 19268
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 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-fsupp 9384  df-oi 9531  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-seq 13997  df-hash 14320  df-0g 17420  df-gsum 17421  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-cntz 19270
This theorem is referenced by:  gsumres  19870  gsumzsplit  19884  gsumpt  19919  dmdprdsplitlem  19996  dpjidcl  20017  mplcoe5  21983
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