MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumss Structured version   Visualization version   GIF version

Theorem fsumss 15615
Description: Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.)
Hypotheses
Ref Expression
sumss.1 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
sumss.2 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
sumss.3 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 0)
fsumss.4 (πœ‘ β†’ 𝐡 ∈ Fin)
Assertion
Ref Expression
fsumss (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
Distinct variable groups:   𝐴,π‘˜   𝐡,π‘˜   πœ‘,π‘˜
Allowed substitution hint:   𝐢(π‘˜)

Proof of Theorem fsumss
Dummy variables 𝑓 π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sumss.1 . . . . 5 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
21adantr 482 . . . 4 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐴 βŠ† 𝐡)
3 sumss.2 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
43adantlr 714 . . . 4 (((πœ‘ ∧ 𝐡 = βˆ…) ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
5 sumss.3 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 0)
65adantlr 714 . . . 4 (((πœ‘ ∧ 𝐡 = βˆ…) ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 0)
7 simpr 486 . . . . 5 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐡 = βˆ…)
8 0ss 4357 . . . . 5 βˆ… βŠ† (β„€β‰₯β€˜0)
97, 8eqsstrdi 3999 . . . 4 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐡 βŠ† (β„€β‰₯β€˜0))
102, 4, 6, 9sumss 15614 . . 3 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
1110ex 414 . 2 (πœ‘ β†’ (𝐡 = βˆ… β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢))
12 cnvimass 6034 . . . . . . . . 9 (◑𝑓 β€œ 𝐴) βŠ† dom 𝑓
13 simprr 772 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)
14 f1of 6785 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
1513, 14syl 17 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
1612, 15fssdm 6689 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (◑𝑓 β€œ 𝐴) βŠ† (1...(β™―β€˜π΅)))
1715ffnd 6670 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓 Fn (1...(β™―β€˜π΅)))
18 elpreima 7009 . . . . . . . . . . . 12 (𝑓 Fn (1...(β™―β€˜π΅)) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
1917, 18syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
2015ffvelcdmda 7036 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (1...(β™―β€˜π΅))) β†’ (π‘“β€˜π‘›) ∈ 𝐡)
2120ex 414 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑛 ∈ (1...(β™―β€˜π΅)) β†’ (π‘“β€˜π‘›) ∈ 𝐡))
2221adantrd 493 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ ((𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴) β†’ (π‘“β€˜π‘›) ∈ 𝐡))
2319, 22sylbid 239 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) β†’ (π‘“β€˜π‘›) ∈ 𝐡))
2423imp 408 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (◑𝑓 β€œ 𝐴)) β†’ (π‘“β€˜π‘›) ∈ 𝐡)
253ex 414 . . . . . . . . . . . . . 14 (πœ‘ β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
2625adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
27 eldif 3921 . . . . . . . . . . . . . . 15 (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↔ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴))
28 0cn 11152 . . . . . . . . . . . . . . . 16 0 ∈ β„‚
295, 28eqeltrdi 2842 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ β„‚)
3027, 29sylan2br 596 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴)) β†’ 𝐢 ∈ β„‚)
3130expr 458 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (Β¬ π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
3226, 31pm2.61d 179 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
3332fmpttd 7064 . . . . . . . . . . 11 (πœ‘ β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
3433adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
3534ffvelcdmda 7036 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ (π‘“β€˜π‘›) ∈ 𝐡) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ β„‚)
3624, 35syldan 592 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (◑𝑓 β€œ 𝐴)) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ β„‚)
37 eldifi 4087 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ 𝑛 ∈ (1...(β™―β€˜π΅)))
3837, 20sylan2 594 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (π‘“β€˜π‘›) ∈ 𝐡)
39 eldifn 4088 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ Β¬ 𝑛 ∈ (◑𝑓 β€œ 𝐴))
4039adantl 483 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ Β¬ 𝑛 ∈ (◑𝑓 β€œ 𝐴))
4137adantl 483 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ 𝑛 ∈ (1...(β™―β€˜π΅)))
4219adantr 482 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
4341, 42mpbirand 706 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (π‘“β€˜π‘›) ∈ 𝐴))
4440, 43mtbid 324 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ Β¬ (π‘“β€˜π‘›) ∈ 𝐴)
4538, 44eldifd 3922 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴))
46 difss 4092 . . . . . . . . . . . . 13 (𝐡 βˆ– 𝐴) βŠ† 𝐡
47 resmpt 5992 . . . . . . . . . . . . 13 ((𝐡 βˆ– 𝐴) βŠ† 𝐡 β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢))
4846, 47ax-mp 5 . . . . . . . . . . . 12 ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)
4948fveq1i 6844 . . . . . . . . . . 11 (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴))β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›))
50 fvres 6862 . . . . . . . . . . 11 ((π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴) β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴))β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
5149, 50eqtr3id 2787 . . . . . . . . . 10 ((π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
5245, 51syl 17 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
53 c0ex 11154 . . . . . . . . . . . . . . 15 0 ∈ V
5453elsn2 4626 . . . . . . . . . . . . . 14 (𝐢 ∈ {0} ↔ 𝐢 = 0)
555, 54sylibr 233 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ {0})
5655fmpttd 7064 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢):(𝐡 βˆ– 𝐴)⟢{0})
5756ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢):(𝐡 βˆ– 𝐴)⟢{0})
5857, 45ffvelcdmd 7037 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ {0})
59 elsni 4604 . . . . . . . . . 10 (((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ {0} β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 0)
6058, 59syl 17 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 0)
6152, 60eqtr3d 2775 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴))) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 0)
62 fzssuz 13488 . . . . . . . . 9 (1...(β™―β€˜π΅)) βŠ† (β„€β‰₯β€˜1)
6362a1i 11 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) βŠ† (β„€β‰₯β€˜1))
6416, 36, 61, 63sumss 15614 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ Σ𝑛 ∈ (◑𝑓 β€œ 𝐴)((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = Σ𝑛 ∈ (1...(β™―β€˜π΅))((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
651ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ 𝐴 βŠ† 𝐡)
6665resmptd 5995 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴) = (π‘˜ ∈ 𝐴 ↦ 𝐢))
6766fveq1d 6845 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š))
68 fvres 6862 . . . . . . . . . . 11 (π‘š ∈ 𝐴 β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
6968adantl 483 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
7067, 69eqtr3d 2775 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
7170sumeq2dv 15593 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
72 fveq2 6843 . . . . . . . . 9 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
73 fzfid 13884 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) ∈ Fin)
7473, 15fisuppfi 9317 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (◑𝑓 β€œ 𝐴) ∈ Fin)
75 f1of1 6784 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–1-1→𝐡)
7613, 75syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–1-1→𝐡)
77 f1ores 6799 . . . . . . . . . . 11 ((𝑓:(1...(β™―β€˜π΅))–1-1→𝐡 ∧ (◑𝑓 β€œ 𝐴) βŠ† (1...(β™―β€˜π΅))) β†’ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-ontoβ†’(𝑓 β€œ (◑𝑓 β€œ 𝐴)))
7876, 16, 77syl2anc 585 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-ontoβ†’(𝑓 β€œ (◑𝑓 β€œ 𝐴)))
79 f1ofo 6792 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
8013, 79syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
811adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐴 βŠ† 𝐡)
82 foimacnv 6802 . . . . . . . . . . . 12 ((𝑓:(1...(β™―β€˜π΅))–onto→𝐡 ∧ 𝐴 βŠ† 𝐡) β†’ (𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴)
8380, 81, 82syl2anc 585 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴)
8483f1oeq3d 6782 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-ontoβ†’(𝑓 β€œ (◑𝑓 β€œ 𝐴)) ↔ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-onto→𝐴))
8578, 84mpbid 231 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (𝑓 β†Ύ (◑𝑓 β€œ 𝐴)):(◑𝑓 β€œ 𝐴)–1-1-onto→𝐴)
86 fvres 6862 . . . . . . . . . 10 (𝑛 ∈ (◑𝑓 β€œ 𝐴) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴))β€˜π‘›) = (π‘“β€˜π‘›))
8786adantl 483 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (◑𝑓 β€œ 𝐴)) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴))β€˜π‘›) = (π‘“β€˜π‘›))
8881sselda 3945 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ π‘š ∈ 𝐡)
8934ffvelcdmda 7036 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐡) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) ∈ β„‚)
9088, 89syldan 592 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) ∈ β„‚)
9172, 74, 85, 87, 90fsumf1o 15613 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = Σ𝑛 ∈ (◑𝑓 β€œ 𝐴)((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
9271, 91eqtrd 2773 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = Σ𝑛 ∈ (◑𝑓 β€œ 𝐴)((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
93 eqidd 2734 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑛 ∈ (1...(β™―β€˜π΅))) β†’ (π‘“β€˜π‘›) = (π‘“β€˜π‘›))
9472, 73, 13, 93, 89fsumf1o 15613 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ Ξ£π‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = Σ𝑛 ∈ (1...(β™―β€˜π΅))((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
9564, 92, 943eqtr4d 2783 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = Ξ£π‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
96 sumfc 15599 . . . . . 6 Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐴 𝐢
97 sumfc 15599 . . . . . 6 Ξ£π‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐡 𝐢
9895, 96, 973eqtr3g 2796 . . . . 5 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
9998expr 458 . . . 4 ((πœ‘ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢))
10099exlimdv 1937 . . 3 ((πœ‘ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢))
101100expimpd 455 . 2 (πœ‘ β†’ (((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡) β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢))
102 fsumss.4 . . 3 (πœ‘ β†’ 𝐡 ∈ Fin)
103 fz1f1o 15600 . . 3 (𝐡 ∈ Fin β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
104102, 103syl 17 . 2 (πœ‘ β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
10511, 101, 104mpjaod 859 1 (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   βˆ– cdif 3908   βŠ† wss 3911  βˆ…c0 4283  {csn 4587   ↦ cmpt 5189  β—‘ccnv 5633   β†Ύ cres 5636   β€œ cima 5637   Fn wfn 6492  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€“ontoβ†’wfo 6495  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Fincfn 8886  β„‚cc 11054  0cc0 11056  1c1 11057  β„•cn 12158  β„€β‰₯cuz 12768  ...cfz 13430  β™―chash 14236  Ξ£csu 15576
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 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577
This theorem is referenced by:  sumss2  15616  rrxmval  24785  rrxmetlem  24787  itg1val2  25064  itg1addlem4  25079  itg1addlem4OLD  25080  itg1addlem5  25081  ply1termlem  25580  plyaddlem1  25590  plymullem1  25591  coeeulem  25601  coeidlem  25614  coeid3  25617  coefv0  25625  coemulhi  25631  coemulc  25632  dvply1  25660  vieta1lem2  25687  dvtaylp  25745  pserdvlem2  25803  basellem3  26448  musum  26556  muinv  26558  fsumvma  26577  chpub  26584  logexprlim  26589  dchrsum  26633  chebbnd1lem1  26833  rpvmasumlem  26851  dchrisum0fno1  26875  rplogsum  26891  indsum  32677  eulerpartlemgs2  33037  flcidc  41544  fsumsupp0  43905  elaa2lem  44560
  Copyright terms: Public domain W3C validator