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Theorem fsumss 15711
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 479 . . . 4 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐴 βŠ† 𝐡)
3 sumss.2 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
43adantlr 713 . . . 4 (((πœ‘ ∧ 𝐡 = βˆ…) ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
5 sumss.3 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 0)
65adantlr 713 . . . 4 (((πœ‘ ∧ 𝐡 = βˆ…) ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 0)
7 simpr 483 . . . . 5 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐡 = βˆ…)
8 0ss 4400 . . . . 5 βˆ… βŠ† (β„€β‰₯β€˜0)
97, 8eqsstrdi 4036 . . . 4 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐡 βŠ† (β„€β‰₯β€˜0))
102, 4, 6, 9sumss 15710 . . 3 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
1110ex 411 . 2 (πœ‘ β†’ (𝐡 = βˆ… β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢))
12 cnvimass 6090 . . . . . . . . 9 (◑𝑓 β€œ 𝐴) βŠ† dom 𝑓
13 simprr 771 . . . . . . . . . 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 6844 . . . . . . . . . 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 6747 . . . . . . . 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 6728 . . . . . . . . . . . 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 7072 . . . . . . . . . . . 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 7099 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 490 . . . . . . . . . . 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 405 . . . . . . . . 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 411 . . . . . . . . . . . . . 14 (πœ‘ β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
2625adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
27 eldif 3959 . . . . . . . . . . . . . . 15 (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↔ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴))
28 0cn 11244 . . . . . . . . . . . . . . . 16 0 ∈ β„‚
295, 28eqeltrdi 2837 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ β„‚)
3027, 29sylan2br 593 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴)) β†’ 𝐢 ∈ β„‚)
3130expr 455 . . . . . . . . . . . . 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 7130 . . . . . . . . . . 11 (πœ‘ β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
3433adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
3534ffvelcdmda 7099 . . . . . . . . 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 589 . . . . . . . 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 4127 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ 𝑛 ∈ (1...(β™―β€˜π΅)))
3837, 20sylan2 591 . . . . . . . . . . 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 4128 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ Β¬ 𝑛 ∈ (◑𝑓 β€œ 𝐴))
4039adantl 480 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 705 . . . . . . . . . . . 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 323 . . . . . . . . . . 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 3960 . . . . . . . . . 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 4132 . . . . . . . . . . . . 13 (𝐡 βˆ– 𝐴) βŠ† 𝐡
47 resmpt 6046 . . . . . . . . . . . . 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 6903 . . . . . . . . . . 11 (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴))β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›))
50 fvres 6921 . . . . . . . . . . 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 2782 . . . . . . . . . 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 11246 . . . . . . . . . . . . . . 15 0 ∈ V
5453elsn2 4672 . . . . . . . . . . . . . 14 (𝐢 ∈ {0} ↔ 𝐢 = 0)
555, 54sylibr 233 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ {0})
5655fmpttd 7130 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢):(𝐡 βˆ– 𝐴)⟢{0})
5756ad2antrr 724 . . . . . . . . . . 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 7100 . . . . . . . . . 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 4649 . . . . . . . . . 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 2770 . . . . . . . 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 13582 . . . . . . . . 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 15710 . . . . . . 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 724 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ 𝐴 βŠ† 𝐡)
6665resmptd 6049 . . . . . . . . . . 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 6904 . . . . . . . . . 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 6921 . . . . . . . . . . 11 (π‘š ∈ 𝐴 β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
6968adantl 480 . . . . . . . . . 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 2770 . . . . . . . . 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 15689 . . . . . . . 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 6902 . . . . . . . . 9 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
73 fzfid 13978 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) ∈ Fin)
7473, 15fisuppfi 9403 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (◑𝑓 β€œ 𝐴) ∈ Fin)
75 f1of1 6843 . . . . . . . . . . . 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 6858 . . . . . . . . . . 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 582 . . . . . . . . . 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 6851 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐴 βŠ† 𝐡)
82 foimacnv 6861 . . . . . . . . . . . 12 ((𝑓:(1...(β™―β€˜π΅))–onto→𝐡 ∧ 𝐴 βŠ† 𝐡) β†’ (𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴)
8380, 81, 82syl2anc 582 . . . . . . . . . . 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 6841 . . . . . . . . . 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 6921 . . . . . . . . . 10 (𝑛 ∈ (◑𝑓 β€œ 𝐴) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴))β€˜π‘›) = (π‘“β€˜π‘›))
8786adantl 480 . . . . . . . . 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 3982 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ π‘š ∈ 𝐡)
8934ffvelcdmda 7099 . . . . . . . . . 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 589 . . . . . . . . 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 15709 . . . . . . . 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 2768 . . . . . . 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 2729 . . . . . . . 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 15709 . . . . . . 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 2778 . . . . . 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 15695 . . . . . 6 Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐴 𝐢
97 sumfc 15695 . . . . . 6 Ξ£π‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐡 𝐢
9895, 96, 973eqtr3g 2791 . . . . 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 455 . . . 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 1928 . . 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 452 . 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 15696 . . 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 858 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 394   ∨ wo 845   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   βˆ– cdif 3946   βŠ† wss 3949  βˆ…c0 4326  {csn 4632   ↦ cmpt 5235  β—‘ccnv 5681   β†Ύ cres 5684   β€œ cima 5685   Fn wfn 6548  βŸΆwf 6549  β€“1-1β†’wf1 6550  β€“ontoβ†’wfo 6551  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  (class class class)co 7426  Fincfn 8970  β„‚cc 11144  0cc0 11146  1c1 11147  β„•cn 12250  β„€β‰₯cuz 12860  ...cfz 13524  β™―chash 14329  Ξ£csu 15672
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-fz 13525  df-fzo 13668  df-seq 14007  df-exp 14067  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-sum 15673
This theorem is referenced by:  sumss2  15712  rrxmval  25353  rrxmetlem  25355  itg1val2  25633  itg1addlem4  25648  itg1addlem4OLD  25649  itg1addlem5  25650  ply1termlem  26157  plyaddlem1  26167  plymullem1  26168  coeeulem  26178  coeidlem  26191  coeid3  26194  coefv0  26202  coemulhi  26208  coemulc  26209  dvply1  26238  vieta1lem2  26266  dvtaylp  26325  pserdvlem2  26385  basellem3  27035  musum  27143  muinv  27145  fsumvma  27166  chpub  27173  logexprlim  27178  dchrsum  27222  chebbnd1lem1  27422  rpvmasumlem  27440  dchrisum0fno1  27464  rplogsum  27480  indsum  33673  eulerpartlemgs2  34033  flcidc  42629  fsumsupp0  44995  elaa2lem  45650
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