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Theorem fsummulc2 15676
Description: A finite sum multiplied by a constant. (Contributed by NM, 12-Nov-2005.) (Revised by Mario Carneiro, 24-Apr-2014.)
Hypotheses
Ref Expression
fsummulc2.1 (πœ‘ β†’ 𝐴 ∈ Fin)
fsummulc2.2 (πœ‘ β†’ 𝐢 ∈ β„‚)
fsummulc2.3 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)
Assertion
Ref Expression
fsummulc2 (πœ‘ β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡))
Distinct variable groups:   𝐴,π‘˜   𝐢,π‘˜   πœ‘,π‘˜
Allowed substitution hint:   𝐡(π‘˜)

Proof of Theorem fsummulc2
Dummy variables 𝑓 π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsummulc2.2 . . . 4 (πœ‘ β†’ 𝐢 ∈ β„‚)
21mul01d 11361 . . 3 (πœ‘ β†’ (𝐢 Β· 0) = 0)
3 sumeq1 15580 . . . . . 6 (𝐴 = βˆ… β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 = Ξ£π‘˜ ∈ βˆ… 𝐡)
4 sum0 15613 . . . . . 6 Ξ£π‘˜ ∈ βˆ… 𝐡 = 0
53, 4eqtrdi 2793 . . . . 5 (𝐴 = βˆ… β†’ Ξ£π‘˜ ∈ 𝐴 𝐡 = 0)
65oveq2d 7378 . . . 4 (𝐴 = βˆ… β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = (𝐢 Β· 0))
7 sumeq1 15580 . . . . 5 (𝐴 = βˆ… β†’ Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡) = Ξ£π‘˜ ∈ βˆ… (𝐢 Β· 𝐡))
8 sum0 15613 . . . . 5 Ξ£π‘˜ ∈ βˆ… (𝐢 Β· 𝐡) = 0
97, 8eqtrdi 2793 . . . 4 (𝐴 = βˆ… β†’ Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡) = 0)
106, 9eqeq12d 2753 . . 3 (𝐴 = βˆ… β†’ ((𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡) ↔ (𝐢 Β· 0) = 0))
112, 10syl5ibrcom 247 . 2 (πœ‘ β†’ (𝐴 = βˆ… β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡)))
12 addcl 11140 . . . . . . . . 9 ((𝑛 ∈ β„‚ ∧ π‘š ∈ β„‚) β†’ (𝑛 + π‘š) ∈ β„‚)
1312adantl 483 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ (𝑛 ∈ β„‚ ∧ π‘š ∈ β„‚)) β†’ (𝑛 + π‘š) ∈ β„‚)
141adantr 482 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝐢 ∈ β„‚)
15 adddi 11147 . . . . . . . . . 10 ((𝐢 ∈ β„‚ ∧ 𝑛 ∈ β„‚ ∧ π‘š ∈ β„‚) β†’ (𝐢 Β· (𝑛 + π‘š)) = ((𝐢 Β· 𝑛) + (𝐢 Β· π‘š)))
16153expb 1121 . . . . . . . . 9 ((𝐢 ∈ β„‚ ∧ (𝑛 ∈ β„‚ ∧ π‘š ∈ β„‚)) β†’ (𝐢 Β· (𝑛 + π‘š)) = ((𝐢 Β· 𝑛) + (𝐢 Β· π‘š)))
1714, 16sylan 581 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ (𝑛 ∈ β„‚ ∧ π‘š ∈ β„‚)) β†’ (𝐢 Β· (𝑛 + π‘š)) = ((𝐢 Β· 𝑛) + (𝐢 Β· π‘š)))
18 simprl 770 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (β™―β€˜π΄) ∈ β„•)
19 nnuz 12813 . . . . . . . . 9 β„• = (β„€β‰₯β€˜1)
2018, 19eleqtrdi 2848 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (β™―β€˜π΄) ∈ (β„€β‰₯β€˜1))
21 fsummulc2.3 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)
2221fmpttd 7068 . . . . . . . . . . 11 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
2322ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
24 simprr 772 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)
2524adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)
26 f1of 6789 . . . . . . . . . . 11 (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ 𝑓:(1...(β™―β€˜π΄))⟢𝐴)
2725, 26syl 17 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ 𝑓:(1...(β™―β€˜π΄))⟢𝐴)
28 fco 6697 . . . . . . . . . 10 (((π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚ ∧ 𝑓:(1...(β™―β€˜π΄))⟢𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓):(1...(β™―β€˜π΄))βŸΆβ„‚)
2923, 27, 28syl2anc 585 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓):(1...(β™―β€˜π΄))βŸΆβ„‚)
30 simpr 486 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ 𝑛 ∈ (1...(β™―β€˜π΄)))
3129, 30ffvelcdmd 7041 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓)β€˜π‘›) ∈ β„‚)
3227, 30ffvelcdmd 7041 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (π‘“β€˜π‘›) ∈ 𝐴)
33 simpr 486 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ π‘˜ ∈ 𝐴)
341adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
3534, 21mulcld 11182 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (𝐢 Β· 𝐡) ∈ β„‚)
36 eqid 2737 . . . . . . . . . . . . . . 15 (π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)) = (π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))
3736fvmpt2 6964 . . . . . . . . . . . . . 14 ((π‘˜ ∈ 𝐴 ∧ (𝐢 Β· 𝐡) ∈ β„‚) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = (𝐢 Β· 𝐡))
3833, 35, 37syl2anc 585 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = (𝐢 Β· 𝐡))
39 eqid 2737 . . . . . . . . . . . . . . . 16 (π‘˜ ∈ 𝐴 ↦ 𝐡) = (π‘˜ ∈ 𝐴 ↦ 𝐡)
4039fvmpt2 6964 . . . . . . . . . . . . . . 15 ((π‘˜ ∈ 𝐴 ∧ 𝐡 ∈ β„‚) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜) = 𝐡)
4133, 21, 40syl2anc 585 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜) = 𝐡)
4241oveq2d 7378 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜)) = (𝐢 Β· 𝐡))
4338, 42eqtr4d 2780 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜)))
4443ralrimiva 3144 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜)))
4544ad2antrr 725 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ βˆ€π‘˜ ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜)))
46 nffvmpt1 6858 . . . . . . . . . . . 12 β„²π‘˜((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›))
47 nfcv 2908 . . . . . . . . . . . . 13 β„²π‘˜πΆ
48 nfcv 2908 . . . . . . . . . . . . 13 β„²π‘˜ Β·
49 nffvmpt1 6858 . . . . . . . . . . . . 13 β„²π‘˜((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›))
5047, 48, 49nfov 7392 . . . . . . . . . . . 12 β„²π‘˜(𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))
5146, 50nfeq 2921 . . . . . . . . . . 11 β„²π‘˜((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))
52 fveq2 6847 . . . . . . . . . . . 12 (π‘˜ = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)))
53 fveq2 6847 . . . . . . . . . . . . 13 (π‘˜ = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))
5453oveq2d 7378 . . . . . . . . . . . 12 (π‘˜ = (π‘“β€˜π‘›) β†’ (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜)) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›))))
5552, 54eqeq12d 2753 . . . . . . . . . . 11 (π‘˜ = (π‘“β€˜π‘›) β†’ (((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜)) ↔ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))))
5651, 55rspc 3572 . . . . . . . . . 10 ((π‘“β€˜π‘›) ∈ 𝐴 β†’ (βˆ€π‘˜ ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘˜) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘˜)) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))))
5732, 45, 56sylc 65 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›))))
5826ad2antll 728 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ 𝑓:(1...(β™―β€˜π΄))⟢𝐴)
59 fvco3 6945 . . . . . . . . . 10 ((𝑓:(1...(β™―β€˜π΄))⟢𝐴 ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)) ∘ 𝑓)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)))
6058, 59sylan 581 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)) ∘ 𝑓)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)))
61 fvco3 6945 . . . . . . . . . . 11 ((𝑓:(1...(β™―β€˜π΄))⟢𝐴 ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))
6258, 61sylan 581 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))
6362oveq2d 7378 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (𝐢 Β· (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓)β€˜π‘›)) = (𝐢 Β· ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›))))
6457, 60, 633eqtr4d 2787 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ 𝑛 ∈ (1...(β™―β€˜π΄))) β†’ (((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)) ∘ 𝑓)β€˜π‘›) = (𝐢 Β· (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓)β€˜π‘›)))
6513, 17, 20, 31, 64seqdistr 13966 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (seq1( + , ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)) ∘ 𝑓))β€˜(β™―β€˜π΄)) = (𝐢 Β· (seq1( + , ((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓))β€˜(β™―β€˜π΄))))
66 fveq2 6847 . . . . . . . 8 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜(π‘“β€˜π‘›)))
6735fmpttd 7068 . . . . . . . . . 10 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)):π΄βŸΆβ„‚)
6867adantr 482 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)):π΄βŸΆβ„‚)
6968ffvelcdmda 7040 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘š) ∈ β„‚)
7066, 18, 24, 69, 60fsum 15612 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘š) = (seq1( + , ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡)) ∘ 𝑓))β€˜(β™―β€˜π΄)))
71 fveq2 6847 . . . . . . . . 9 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(π‘“β€˜π‘›)))
7222adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
7372ffvelcdmda 7040 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) ∈ β„‚)
7471, 18, 24, 73, 62fsum 15612 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) = (seq1( + , ((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓))β€˜(β™―β€˜π΄)))
7574oveq2d 7378 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (𝐢 Β· Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š)) = (𝐢 Β· (seq1( + , ((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ 𝑓))β€˜(β™―β€˜π΄))))
7665, 70, 753eqtr4rd 2788 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (𝐢 Β· Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š)) = Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘š))
77 sumfc 15601 . . . . . . 7 Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐴 𝐡
7877oveq2i 7373 . . . . . 6 (𝐢 Β· Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š)) = (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡)
79 sumfc 15601 . . . . . 6 Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ (𝐢 Β· 𝐡))β€˜π‘š) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡)
8076, 78, 793eqtr3g 2800 . . . . 5 ((πœ‘ ∧ ((β™―β€˜π΄) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)) β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡))
8180expr 458 . . . 4 ((πœ‘ ∧ (β™―β€˜π΄) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡)))
8281exlimdv 1937 . . 3 ((πœ‘ ∧ (β™―β€˜π΄) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴 β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡)))
8382expimpd 455 . 2 (πœ‘ β†’ (((β™―β€˜π΄) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴) β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡)))
84 fsummulc2.1 . . 3 (πœ‘ β†’ 𝐴 ∈ Fin)
85 fz1f1o 15602 . . 3 (𝐴 ∈ Fin β†’ (𝐴 = βˆ… ∨ ((β™―β€˜π΄) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)))
8684, 85syl 17 . 2 (πœ‘ β†’ (𝐴 = βˆ… ∨ ((β™―β€˜π΄) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΄))–1-1-onto→𝐴)))
8711, 83, 86mpjaod 859 1 (πœ‘ β†’ (𝐢 Β· Ξ£π‘˜ ∈ 𝐴 𝐡) = Ξ£π‘˜ ∈ 𝐴 (𝐢 Β· 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3065  βˆ…c0 4287   ↦ cmpt 5193   ∘ ccom 5642  βŸΆwf 6497  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  Fincfn 8890  β„‚cc 11056  0cc0 11058  1c1 11059   + caddc 11061   Β· cmul 11063  β„•cn 12160  β„€β‰₯cuz 12770  ...cfz 13431  seqcseq 13913  β™―chash 14237  Ξ£csu 15577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-oi 9453  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-rp 12923  df-fz 13432  df-fzo 13575  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-sum 15578
This theorem is referenced by:  fsummulc1  15677  fsumneg  15679  fsum2mul  15681  incexc2  15730  pwdif  15760  mertens  15778  binomrisefac  15932  fsumkthpow  15946  eirrlem  16093  pwp1fsum  16280  csbren  24779  trirn  24780  itg1addlem4  25079  itg1addlem4OLD  25080  itg1addlem5  25081  itg1mulc  25085  elqaalem3  25697  advlogexp  26026  fsumharmonic  26377  basellem8  26453  muinv  26558  fsumdvdsmul  26560  logfaclbnd  26586  dchrsum2  26632  sumdchr2  26634  rplogsumlem2  26849  rpvmasumlem  26851  dchrmusum2  26858  dchrvmasumlem1  26859  dchrvmasum2lem  26860  dchrvmasumlem2  26862  dchrvmasumiflem1  26865  rpvmasum2  26876  dchrisum0lem2  26882  mudivsum  26894  mulogsum  26896  mulog2sumlem1  26898  mulog2sumlem2  26899  mulog2sumlem3  26900  vmalogdivsum2  26902  logsqvma  26906  selberglem1  26909  selberglem2  26910  selberg  26912  selberg3lem1  26921  selberg4lem1  26924  selberg4  26925  selbergr  26932  selberg3r  26933  selberg34r  26935  pntsval2  26940  pntrlog2bndlem2  26942  pntrlog2bndlem3  26943  pntrlog2bndlem4  26944  pntrlog2bndlem6  26947  pntpbnd2  26951  pntlemk  26970  axsegconlem9  27916  ax5seglem1  27919  ax5seglem2  27920  ax5seglem9  27928  hgt750lemf  33306  hgt750lemb  33309  knoppndvlem11  35014  3factsumint4  40510  lcmineqlem6  40520  jm2.22  41348  dvnprodlem2  44262  stoweidlem26  44341  stirlinglem12  44400  fourierdlem83  44504  etransclem46  44595  altgsumbcALT  46503  aacllem  47322
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