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Theorem gsumval2a 18606
Description: Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumval2.b 𝐡 = (Baseβ€˜πΊ)
gsumval2.p + = (+gβ€˜πΊ)
gsumval2.g (πœ‘ β†’ 𝐺 ∈ 𝑉)
gsumval2.n (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
gsumval2.f (πœ‘ β†’ 𝐹:(𝑀...𝑁)⟢𝐡)
gsumval2a.o 𝑂 = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ + 𝑦) = 𝑦 ∧ (𝑦 + π‘₯) = 𝑦)}
gsumval2a.f (πœ‘ β†’ Β¬ ran 𝐹 βŠ† 𝑂)
Assertion
Ref Expression
gsumval2a (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (seq𝑀( + , 𝐹)β€˜π‘))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐺,𝑦   π‘₯,𝑉   π‘₯, + ,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝑀(π‘₯,𝑦)   𝑁(π‘₯,𝑦)   𝑂(π‘₯,𝑦)   𝑉(𝑦)

Proof of Theorem gsumval2a
Dummy variables 𝑧 𝑓 π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumval2.b . . . 4 𝐡 = (Baseβ€˜πΊ)
2 eqid 2732 . . . 4 (0gβ€˜πΊ) = (0gβ€˜πΊ)
3 gsumval2.p . . . 4 + = (+gβ€˜πΊ)
4 gsumval2a.o . . . 4 𝑂 = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ + 𝑦) = 𝑦 ∧ (𝑦 + π‘₯) = 𝑦)}
5 eqidd 2733 . . . 4 (πœ‘ β†’ (◑𝐹 β€œ (V βˆ– 𝑂)) = (◑𝐹 β€œ (V βˆ– 𝑂)))
6 gsumval2.g . . . 4 (πœ‘ β†’ 𝐺 ∈ 𝑉)
7 ovexd 7446 . . . 4 (πœ‘ β†’ (𝑀...𝑁) ∈ V)
8 gsumval2.f . . . 4 (πœ‘ β†’ 𝐹:(𝑀...𝑁)⟢𝐡)
91, 2, 3, 4, 5, 6, 7, 8gsumval 18598 . . 3 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = if(ran 𝐹 βŠ† 𝑂, (0gβ€˜πΊ), if((𝑀...𝑁) ∈ ran ..., (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘§βˆƒπ‘“(𝑓:(1...(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂))))–1-1-ontoβ†’(◑𝐹 β€œ (V βˆ– 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂)))))))))
10 gsumval2a.f . . . . 5 (πœ‘ β†’ Β¬ ran 𝐹 βŠ† 𝑂)
1110iffalsed 4539 . . . 4 (πœ‘ β†’ if(ran 𝐹 βŠ† 𝑂, (0gβ€˜πΊ), if((𝑀...𝑁) ∈ ran ..., (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘§βˆƒπ‘“(𝑓:(1...(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂))))–1-1-ontoβ†’(◑𝐹 β€œ (V βˆ– 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂)))))))) = if((𝑀...𝑁) ∈ ran ..., (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘§βˆƒπ‘“(𝑓:(1...(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂))))–1-1-ontoβ†’(◑𝐹 β€œ (V βˆ– 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂))))))))
12 fzf 13490 . . . . . . 7 ...:(β„€ Γ— β„€)βŸΆπ’« β„€
13 ffn 6717 . . . . . . 7 (...:(β„€ Γ— β„€)βŸΆπ’« β„€ β†’ ... Fn (β„€ Γ— β„€))
1412, 13ax-mp 5 . . . . . 6 ... Fn (β„€ Γ— β„€)
15 gsumval2.n . . . . . . 7 (πœ‘ β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
16 eluzel2 12829 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑀 ∈ β„€)
1715, 16syl 17 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ β„€)
18 eluzelz 12834 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ 𝑁 ∈ β„€)
1915, 18syl 17 . . . . . 6 (πœ‘ β†’ 𝑁 ∈ β„€)
20 fnovrn 7584 . . . . . 6 ((... Fn (β„€ Γ— β„€) ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀...𝑁) ∈ ran ...)
2114, 17, 19, 20mp3an2i 1466 . . . . 5 (πœ‘ β†’ (𝑀...𝑁) ∈ ran ...)
2221iftrued 4536 . . . 4 (πœ‘ β†’ if((𝑀...𝑁) ∈ ran ..., (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘§βˆƒπ‘“(𝑓:(1...(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂))))–1-1-ontoβ†’(◑𝐹 β€œ (V βˆ– 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂))))))) = (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))))
2311, 22eqtrd 2772 . . 3 (πœ‘ β†’ if(ran 𝐹 βŠ† 𝑂, (0gβ€˜πΊ), if((𝑀...𝑁) ∈ ran ..., (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))), (β„©π‘§βˆƒπ‘“(𝑓:(1...(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂))))–1-1-ontoβ†’(◑𝐹 β€œ (V βˆ– 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))β€˜(β™―β€˜(◑𝐹 β€œ (V βˆ– 𝑂)))))))) = (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))))
249, 23eqtrd 2772 . 2 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))))
25 fvex 6904 . . 3 (seq𝑀( + , 𝐹)β€˜π‘) ∈ V
26 fzopth 13540 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ ((𝑀...𝑁) = (π‘š...𝑛) ↔ (𝑀 = π‘š ∧ 𝑁 = 𝑛)))
2715, 26syl 17 . . . . . . . . . 10 (πœ‘ β†’ ((𝑀...𝑁) = (π‘š...𝑛) ↔ (𝑀 = π‘š ∧ 𝑁 = 𝑛)))
28 simpl 483 . . . . . . . . . . . . . 14 ((𝑀 = π‘š ∧ 𝑁 = 𝑛) β†’ 𝑀 = π‘š)
2928seqeq1d 13974 . . . . . . . . . . . . 13 ((𝑀 = π‘š ∧ 𝑁 = 𝑛) β†’ seq𝑀( + , 𝐹) = seqπ‘š( + , 𝐹))
30 simpr 485 . . . . . . . . . . . . 13 ((𝑀 = π‘š ∧ 𝑁 = 𝑛) β†’ 𝑁 = 𝑛)
3129, 30fveq12d 6898 . . . . . . . . . . . 12 ((𝑀 = π‘š ∧ 𝑁 = 𝑛) β†’ (seq𝑀( + , 𝐹)β€˜π‘) = (seqπ‘š( + , 𝐹)β€˜π‘›))
3231eqcomd 2738 . . . . . . . . . . 11 ((𝑀 = π‘š ∧ 𝑁 = 𝑛) β†’ (seqπ‘š( + , 𝐹)β€˜π‘›) = (seq𝑀( + , 𝐹)β€˜π‘))
33 eqeq1 2736 . . . . . . . . . . 11 (𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›) β†’ (𝑧 = (seq𝑀( + , 𝐹)β€˜π‘) ↔ (seqπ‘š( + , 𝐹)β€˜π‘›) = (seq𝑀( + , 𝐹)β€˜π‘)))
3432, 33syl5ibrcom 246 . . . . . . . . . 10 ((𝑀 = π‘š ∧ 𝑁 = 𝑛) β†’ (𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›) β†’ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
3527, 34syl6bi 252 . . . . . . . . 9 (πœ‘ β†’ ((𝑀...𝑁) = (π‘š...𝑛) β†’ (𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›) β†’ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘))))
3635impd 411 . . . . . . . 8 (πœ‘ β†’ (((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)) β†’ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
3736rexlimdvw 3160 . . . . . . 7 (πœ‘ β†’ (βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)) β†’ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
3837exlimdv 1936 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)) β†’ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
3917adantr 481 . . . . . . . 8 ((πœ‘ ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)) β†’ 𝑀 ∈ β„€)
40 oveq2 7419 . . . . . . . . . . . . 13 (𝑛 = 𝑁 β†’ (𝑀...𝑛) = (𝑀...𝑁))
4140eqcomd 2738 . . . . . . . . . . . 12 (𝑛 = 𝑁 β†’ (𝑀...𝑁) = (𝑀...𝑛))
4241biantrurd 533 . . . . . . . . . . 11 (𝑛 = 𝑁 β†’ (𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›))))
43 fveq2 6891 . . . . . . . . . . . 12 (𝑛 = 𝑁 β†’ (seq𝑀( + , 𝐹)β€˜π‘›) = (seq𝑀( + , 𝐹)β€˜π‘))
4443eqeq2d 2743 . . . . . . . . . . 11 (𝑛 = 𝑁 β†’ (𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›) ↔ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
4542, 44bitr3d 280 . . . . . . . . . 10 (𝑛 = 𝑁 β†’ (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›)) ↔ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
4645rspcev 3612 . . . . . . . . 9 ((𝑁 ∈ (β„€β‰₯β€˜π‘€) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)) β†’ βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘€)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›)))
4715, 46sylan 580 . . . . . . . 8 ((πœ‘ ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)) β†’ βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘€)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›)))
48 fveq2 6891 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (β„€β‰₯β€˜π‘š) = (β„€β‰₯β€˜π‘€))
49 oveq1 7418 . . . . . . . . . . . 12 (π‘š = 𝑀 β†’ (π‘š...𝑛) = (𝑀...𝑛))
5049eqeq2d 2743 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ ((𝑀...𝑁) = (π‘š...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛)))
51 seqeq1 13971 . . . . . . . . . . . . 13 (π‘š = 𝑀 β†’ seqπ‘š( + , 𝐹) = seq𝑀( + , 𝐹))
5251fveq1d 6893 . . . . . . . . . . . 12 (π‘š = 𝑀 β†’ (seqπ‘š( + , 𝐹)β€˜π‘›) = (seq𝑀( + , 𝐹)β€˜π‘›))
5352eqeq2d 2743 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ (𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›) ↔ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›)))
5450, 53anbi12d 631 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›))))
5548, 54rexeqbidv 3343 . . . . . . . . 9 (π‘š = 𝑀 β†’ (βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)) ↔ βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘€)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›))))
5655spcegv 3587 . . . . . . . 8 (𝑀 ∈ β„€ β†’ (βˆƒπ‘› ∈ (β„€β‰₯β€˜π‘€)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘›)) β†’ βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))))
5739, 47, 56sylc 65 . . . . . . 7 ((πœ‘ ∧ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)) β†’ βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)))
5857ex 413 . . . . . 6 (πœ‘ β†’ (𝑧 = (seq𝑀( + , 𝐹)β€˜π‘) β†’ βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))))
5938, 58impbid 211 . . . . 5 (πœ‘ β†’ (βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)) ↔ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
6059adantr 481 . . . 4 ((πœ‘ ∧ (seq𝑀( + , 𝐹)β€˜π‘) ∈ V) β†’ (βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›)) ↔ 𝑧 = (seq𝑀( + , 𝐹)β€˜π‘)))
6160iota5 6526 . . 3 ((πœ‘ ∧ (seq𝑀( + , 𝐹)β€˜π‘) ∈ V) β†’ (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))) = (seq𝑀( + , 𝐹)β€˜π‘))
6225, 61mpan2 689 . 2 (πœ‘ β†’ (β„©π‘§βˆƒπ‘šβˆƒπ‘› ∈ (β„€β‰₯β€˜π‘š)((𝑀...𝑁) = (π‘š...𝑛) ∧ 𝑧 = (seqπ‘š( + , 𝐹)β€˜π‘›))) = (seq𝑀( + , 𝐹)β€˜π‘))
6324, 62eqtrd 2772 1 (πœ‘ β†’ (𝐺 Ξ£g 𝐹) = (seq𝑀( + , 𝐹)β€˜π‘))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆ– cdif 3945   βŠ† wss 3948  ifcif 4528  π’« cpw 4602   Γ— cxp 5674  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679   ∘ ccom 5680  β„©cio 6493   Fn wfn 6538  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411  1c1 11113  β„€cz 12560  β„€β‰₯cuz 12824  ...cfz 13486  seqcseq 13968  β™―chash 14292  Basecbs 17146  +gcplusg 17199  0gc0g 17387   Ξ£g cgsu 17388
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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-pre-lttri 11186  ax-pre-lttrn 11187
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-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-neg 11449  df-z 12561  df-uz 12825  df-fz 13487  df-seq 13969  df-gsum 17390
This theorem is referenced by:  gsumval2  18607
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