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Theorem breprexpnat 33944
Description: Express the 𝑆 th power of the finite series in terms of the number of representations of integers π‘š as sums of 𝑆 terms of elements of 𝐴, bounded by 𝑁. Proposition of [Nathanson] p. 123. (Contributed by Thierry Arnoux, 11-Dec-2021.)
Hypotheses
Ref Expression
breprexp.n (πœ‘ β†’ 𝑁 ∈ β„•0)
breprexp.s (πœ‘ β†’ 𝑆 ∈ β„•0)
breprexp.z (πœ‘ β†’ 𝑍 ∈ β„‚)
breprexpnat.a (πœ‘ β†’ 𝐴 βŠ† β„•)
breprexpnat.p 𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)
breprexpnat.r 𝑅 = (β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š))
Assertion
Ref Expression
breprexpnat (πœ‘ β†’ (𝑃↑𝑆) = Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))(𝑅 Β· (π‘β†‘π‘š)))
Distinct variable groups:   π‘š,𝑁   𝑆,π‘š   π‘š,𝑍   𝐴,𝑏,π‘š   𝑁,𝑏   𝑆,𝑏   𝑍,𝑏   πœ‘,𝑏,π‘š
Allowed substitution hints:   𝑃(π‘š,𝑏)   𝑅(π‘š,𝑏)

Proof of Theorem breprexpnat
Dummy variables 𝑐 π‘Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breprexp.n . . . 4 (πœ‘ β†’ 𝑁 ∈ β„•0)
2 breprexp.s . . . 4 (πœ‘ β†’ 𝑆 ∈ β„•0)
3 breprexp.z . . . 4 (πœ‘ β†’ 𝑍 ∈ β„‚)
4 fvex 6903 . . . . . 6 ((πŸ­β€˜β„•)β€˜π΄) ∈ V
54fconst 6776 . . . . 5 ((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)}):(0..^𝑆)⟢{((πŸ­β€˜β„•)β€˜π΄)}
6 nnex 12222 . . . . . . . . 9 β„• ∈ V
7 breprexpnat.a . . . . . . . . 9 (πœ‘ β†’ 𝐴 βŠ† β„•)
8 indf 33311 . . . . . . . . 9 ((β„• ∈ V ∧ 𝐴 βŠ† β„•) β†’ ((πŸ­β€˜β„•)β€˜π΄):β„•βŸΆ{0, 1})
96, 7, 8sylancr 585 . . . . . . . 8 (πœ‘ β†’ ((πŸ­β€˜β„•)β€˜π΄):β„•βŸΆ{0, 1})
10 0cn 11210 . . . . . . . . 9 0 ∈ β„‚
11 ax-1cn 11170 . . . . . . . . 9 1 ∈ β„‚
12 prssi 4823 . . . . . . . . 9 ((0 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ {0, 1} βŠ† β„‚)
1310, 11, 12mp2an 688 . . . . . . . 8 {0, 1} βŠ† β„‚
14 fss 6733 . . . . . . . 8 ((((πŸ­β€˜β„•)β€˜π΄):β„•βŸΆ{0, 1} ∧ {0, 1} βŠ† β„‚) β†’ ((πŸ­β€˜β„•)β€˜π΄):β„•βŸΆβ„‚)
159, 13, 14sylancl 584 . . . . . . 7 (πœ‘ β†’ ((πŸ­β€˜β„•)β€˜π΄):β„•βŸΆβ„‚)
16 cnex 11193 . . . . . . . 8 β„‚ ∈ V
1716, 6elmap 8867 . . . . . . 7 (((πŸ­β€˜β„•)β€˜π΄) ∈ (β„‚ ↑m β„•) ↔ ((πŸ­β€˜β„•)β€˜π΄):β„•βŸΆβ„‚)
1815, 17sylibr 233 . . . . . 6 (πœ‘ β†’ ((πŸ­β€˜β„•)β€˜π΄) ∈ (β„‚ ↑m β„•))
194snss 4788 . . . . . 6 (((πŸ­β€˜β„•)β€˜π΄) ∈ (β„‚ ↑m β„•) ↔ {((πŸ­β€˜β„•)β€˜π΄)} βŠ† (β„‚ ↑m β„•))
2018, 19sylib 217 . . . . 5 (πœ‘ β†’ {((πŸ­β€˜β„•)β€˜π΄)} βŠ† (β„‚ ↑m β„•))
21 fss 6733 . . . . 5 ((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)}):(0..^𝑆)⟢{((πŸ­β€˜β„•)β€˜π΄)} ∧ {((πŸ­β€˜β„•)β€˜π΄)} βŠ† (β„‚ ↑m β„•)) β†’ ((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)}):(0..^𝑆)⟢(β„‚ ↑m β„•))
225, 20, 21sylancr 585 . . . 4 (πœ‘ β†’ ((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)}):(0..^𝑆)⟢(β„‚ ↑m β„•))
231, 2, 3, 22breprexp 33943 . . 3 (πœ‘ β†’ βˆπ‘Ž ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜π‘) Β· (𝑍↑𝑏)) = Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)(βˆπ‘Ž ∈ (0..^𝑆)((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)))
244fvconst2 7206 . . . . . . . . . 10 (π‘Ž ∈ (0..^𝑆) β†’ (((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž) = ((πŸ­β€˜β„•)β€˜π΄))
2524ad2antlr 723 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) β†’ (((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž) = ((πŸ­β€˜β„•)β€˜π΄))
2625fveq1d 6892 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) β†’ ((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜π‘) = (((πŸ­β€˜β„•)β€˜π΄)β€˜π‘))
2726oveq1d 7426 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) β†’ (((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜π‘) Β· (𝑍↑𝑏)) = ((((πŸ­β€˜β„•)β€˜π΄)β€˜π‘) Β· (𝑍↑𝑏)))
2827sumeq2dv 15653 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜π‘) Β· (𝑍↑𝑏)) = Σ𝑏 ∈ (1...𝑁)((((πŸ­β€˜β„•)β€˜π΄)β€˜π‘) Β· (𝑍↑𝑏)))
296a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ β„• ∈ V)
30 fzfi 13941 . . . . . . . 8 (1...𝑁) ∈ Fin
3130a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ (1...𝑁) ∈ Fin)
32 fz1ssnn 13536 . . . . . . . 8 (1...𝑁) βŠ† β„•
3332a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ (1...𝑁) βŠ† β„•)
347adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ 𝐴 βŠ† β„•)
353ad2antrr 722 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) β†’ 𝑍 ∈ β„‚)
36 nnssnn0 12479 . . . . . . . . . 10 β„• βŠ† β„•0
3732, 36sstri 3990 . . . . . . . . 9 (1...𝑁) βŠ† β„•0
38 simpr 483 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) β†’ 𝑏 ∈ (1...𝑁))
3937, 38sselid 3979 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) β†’ 𝑏 ∈ β„•0)
4035, 39expcld 14115 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) ∧ 𝑏 ∈ (1...𝑁)) β†’ (𝑍↑𝑏) ∈ β„‚)
4129, 31, 33, 34, 40indsumin 33318 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ Σ𝑏 ∈ (1...𝑁)((((πŸ­β€˜β„•)β€˜π΄)β€˜π‘) Β· (𝑍↑𝑏)) = Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏))
42 incom 4200 . . . . . . . 8 ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁))
4342a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((1...𝑁) ∩ 𝐴) = (𝐴 ∩ (1...𝑁)))
4443sumeq1d 15651 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ Σ𝑏 ∈ ((1...𝑁) ∩ 𝐴)(𝑍↑𝑏) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
4528, 41, 443eqtrd 2774 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (0..^𝑆)) β†’ Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜π‘) Β· (𝑍↑𝑏)) = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
4645prodeq2dv 15871 . . . 4 (πœ‘ β†’ βˆπ‘Ž ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜π‘) Β· (𝑍↑𝑏)) = βˆπ‘Ž ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏))
47 fzofi 13943 . . . . . 6 (0..^𝑆) ∈ Fin
4847a1i 11 . . . . 5 (πœ‘ β†’ (0..^𝑆) ∈ Fin)
49 inss2 4228 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) βŠ† (1...𝑁)
50 ssfi 9175 . . . . . . . 8 (((1...𝑁) ∈ Fin ∧ (𝐴 ∩ (1...𝑁)) βŠ† (1...𝑁)) β†’ (𝐴 ∩ (1...𝑁)) ∈ Fin)
5130, 49, 50mp2an 688 . . . . . . 7 (𝐴 ∩ (1...𝑁)) ∈ Fin
5251a1i 11 . . . . . 6 (πœ‘ β†’ (𝐴 ∩ (1...𝑁)) ∈ Fin)
533adantr 479 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) β†’ 𝑍 ∈ β„‚)
5449, 37sstri 3990 . . . . . . . 8 (𝐴 ∩ (1...𝑁)) βŠ† β„•0
55 simpr 483 . . . . . . . 8 ((πœ‘ ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) β†’ 𝑏 ∈ (𝐴 ∩ (1...𝑁)))
5654, 55sselid 3979 . . . . . . 7 ((πœ‘ ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) β†’ 𝑏 ∈ β„•0)
5753, 56expcld 14115 . . . . . 6 ((πœ‘ ∧ 𝑏 ∈ (𝐴 ∩ (1...𝑁))) β†’ (𝑍↑𝑏) ∈ β„‚)
5852, 57fsumcl 15683 . . . . 5 (πœ‘ β†’ Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ β„‚)
59 fprodconst 15926 . . . . 5 (((0..^𝑆) ∈ Fin ∧ Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) ∈ β„‚) β†’ βˆπ‘Ž ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(β™―β€˜(0..^𝑆))))
6048, 58, 59syl2anc 582 . . . 4 (πœ‘ β†’ βˆπ‘Ž ∈ (0..^𝑆)Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(β™―β€˜(0..^𝑆))))
61 hashfzo0 14394 . . . . . 6 (𝑆 ∈ β„•0 β†’ (β™―β€˜(0..^𝑆)) = 𝑆)
622, 61syl 17 . . . . 5 (πœ‘ β†’ (β™―β€˜(0..^𝑆)) = 𝑆)
6362oveq2d 7427 . . . 4 (πœ‘ β†’ (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑(β™―β€˜(0..^𝑆))) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
6446, 60, 633eqtrd 2774 . . 3 (πœ‘ β†’ βˆπ‘Ž ∈ (0..^𝑆)Σ𝑏 ∈ (1...𝑁)(((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜π‘) Β· (𝑍↑𝑏)) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆))
6532a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ (1...𝑁) βŠ† β„•)
66 fzssz 13507 . . . . . . . 8 (0...(𝑆 Β· 𝑁)) βŠ† β„€
67 simpr 483 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ π‘š ∈ (0...(𝑆 Β· 𝑁)))
6866, 67sselid 3979 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ π‘š ∈ β„€)
692adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ 𝑆 ∈ β„•0)
7030a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ (1...𝑁) ∈ Fin)
7165, 68, 69, 70reprfi 33926 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ ((1...𝑁)(reprβ€˜π‘†)π‘š) ∈ Fin)
723adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ 𝑍 ∈ β„‚)
73 fz0ssnn0 13600 . . . . . . . 8 (0...(𝑆 Β· 𝑁)) βŠ† β„•0
7473, 67sselid 3979 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ π‘š ∈ β„•0)
7572, 74expcld 14115 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ (π‘β†‘π‘š) ∈ β„‚)
7647a1i 11 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ (0..^𝑆) ∈ Fin)
779ad3antrrr 726 . . . . . . . . 9 ((((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((πŸ­β€˜β„•)β€˜π΄):β„•βŸΆ{0, 1})
7832a1i 11 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ (1...𝑁) βŠ† β„•)
7968adantr 479 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ π‘š ∈ β„€)
8069adantr 479 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ 𝑆 ∈ β„•0)
81 simpr 483 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š))
8278, 79, 80, 81reprf 33922 . . . . . . . . . . 11 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ 𝑐:(0..^𝑆)⟢(1...𝑁))
8382ffvelcdmda 7085 . . . . . . . . . 10 ((((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) ∧ π‘Ž ∈ (0..^𝑆)) β†’ (π‘β€˜π‘Ž) ∈ (1...𝑁))
8432, 83sselid 3979 . . . . . . . . 9 ((((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) ∧ π‘Ž ∈ (0..^𝑆)) β†’ (π‘β€˜π‘Ž) ∈ β„•)
8577, 84ffvelcdmd 7086 . . . . . . . 8 ((((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) ∧ π‘Ž ∈ (0..^𝑆)) β†’ (((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) ∈ {0, 1})
8613, 85sselid 3979 . . . . . . 7 ((((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) ∧ π‘Ž ∈ (0..^𝑆)) β†’ (((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) ∈ β„‚)
8776, 86fprodcl 15900 . . . . . 6 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) ∈ β„‚)
8871, 75, 87fsummulc1 15735 . . . . 5 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ (Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)) = Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)(βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)))
897adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ 𝐴 βŠ† β„•)
9089, 68, 69, 70, 65hashreprin 33930 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ (β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) = Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)))
9190oveq1d 7426 . . . . 5 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ ((β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) Β· (π‘β†‘π‘š)) = (Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)))
9224fveq1d 6892 . . . . . . . . . 10 (π‘Ž ∈ (0..^𝑆) β†’ ((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) = (((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)))
9392adantl 480 . . . . . . . . 9 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ π‘Ž ∈ (0..^𝑆)) β†’ ((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) = (((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)))
9493prodeq2dv 15871 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ βˆπ‘Ž ∈ (0..^𝑆)((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) = βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)))
9594adantr 479 . . . . . . 7 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ βˆπ‘Ž ∈ (0..^𝑆)((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) = βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)))
9695oveq1d 7426 . . . . . 6 (((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) ∧ 𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)) β†’ (βˆπ‘Ž ∈ (0..^𝑆)((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)) = (βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)))
9796sumeq2dv 15653 . . . . 5 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)(βˆπ‘Ž ∈ (0..^𝑆)((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)) = Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)(βˆπ‘Ž ∈ (0..^𝑆)(((πŸ­β€˜β„•)β€˜π΄)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)))
9888, 91, 973eqtr4rd 2781 . . . 4 ((πœ‘ ∧ π‘š ∈ (0...(𝑆 Β· 𝑁))) β†’ Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)(βˆπ‘Ž ∈ (0..^𝑆)((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)) = ((β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) Β· (π‘β†‘π‘š)))
9998sumeq2dv 15653 . . 3 (πœ‘ β†’ Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))Σ𝑐 ∈ ((1...𝑁)(reprβ€˜π‘†)π‘š)(βˆπ‘Ž ∈ (0..^𝑆)((((0..^𝑆) Γ— {((πŸ­β€˜β„•)β€˜π΄)})β€˜π‘Ž)β€˜(π‘β€˜π‘Ž)) Β· (π‘β†‘π‘š)) = Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))((β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) Β· (π‘β†‘π‘š)))
10023, 64, 993eqtr3d 2778 . 2 (πœ‘ β†’ (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆) = Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))((β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) Β· (π‘β†‘π‘š)))
101 breprexpnat.p . . 3 𝑃 = Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)
102101oveq1i 7421 . 2 (𝑃↑𝑆) = (Σ𝑏 ∈ (𝐴 ∩ (1...𝑁))(𝑍↑𝑏)↑𝑆)
103 breprexpnat.r . . . . 5 𝑅 = (β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š))
104103oveq1i 7421 . . . 4 (𝑅 Β· (π‘β†‘π‘š)) = ((β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) Β· (π‘β†‘π‘š))
105104a1i 11 . . 3 (π‘š ∈ (0...(𝑆 Β· 𝑁)) β†’ (𝑅 Β· (π‘β†‘π‘š)) = ((β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) Β· (π‘β†‘π‘š)))
106105sumeq2i 15649 . 2 Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))(𝑅 Β· (π‘β†‘π‘š)) = Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))((β™―β€˜((𝐴 ∩ (1...𝑁))(reprβ€˜π‘†)π‘š)) Β· (π‘β†‘π‘š))
107100, 102, 1063eqtr4g 2795 1 (πœ‘ β†’ (𝑃↑𝑆) = Ξ£π‘š ∈ (0...(𝑆 Β· 𝑁))(𝑅 Β· (π‘β†‘π‘š)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  {csn 4627  {cpr 4629   Γ— cxp 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  Fincfn 8941  β„‚cc 11110  0cc0 11112  1c1 11113   Β· cmul 11117  β„•cn 12216  β„•0cn0 12476  β„€cz 12562  ...cfz 13488  ..^cfzo 13631  β†‘cexp 14031  β™―chash 14294  Ξ£csu 15636  βˆcprod 15853  πŸ­cind 33306  reprcrepr 33918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  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  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  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-disj 5113  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  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-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-ico 13334  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-prod 15854  df-ind 33307  df-repr 33919
This theorem is referenced by: (None)
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