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Theorem fisumss 11414
Description: Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)
Hypotheses
Ref Expression
fsumss.1 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
fsumss.2 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
fsumss.3 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 0)
fisumss.adc (πœ‘ β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
fsumss.4 (πœ‘ β†’ 𝐡 ∈ Fin)
Assertion
Ref Expression
fisumss (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
Distinct variable groups:   𝐴,𝑗,π‘˜   𝐡,𝑗,π‘˜   πœ‘,π‘˜
Allowed substitution hints:   πœ‘(𝑗)   𝐢(𝑗,π‘˜)

Proof of Theorem fisumss
Dummy variables 𝑓 𝑒 π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsumss.1 . . . . . 6 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
2 sseq0 3476 . . . . . 6 ((𝐴 βŠ† 𝐡 ∧ 𝐡 = βˆ…) β†’ 𝐴 = βˆ…)
31, 2sylan 283 . . . . 5 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐴 = βˆ…)
43sumeq1d 11388 . . . 4 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ βˆ… 𝐢)
5 simpr 110 . . . . 5 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ 𝐡 = βˆ…)
65sumeq1d 11388 . . . 4 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ Ξ£π‘˜ ∈ 𝐡 𝐢 = Ξ£π‘˜ ∈ βˆ… 𝐢)
74, 6eqtr4d 2223 . . 3 ((πœ‘ ∧ 𝐡 = βˆ…) β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
87ex 115 . 2 (πœ‘ β†’ (𝐡 = βˆ… β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢))
9 cnvimass 5003 . . . . . . . . 9 (◑𝑓 β€œ 𝐴) βŠ† dom 𝑓
10 simprr 531 . . . . . . . . . 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)
11 f1of 5473 . . . . . . . . . 10 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
1210, 11syl 14 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
139, 12fssdm 5392 . . . . . . . 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΅)))
1412ffnd 5378 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓 Fn (1...(β™―β€˜π΅)))
15 elpreima 5648 . . . . . . . . . . . 12 (𝑓 Fn (1...(β™―β€˜π΅)) β†’ (𝑛 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
1614, 15syl 14 . . . . . . . . . . 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)))
1712ffvelcdmda 5664 . . . . . . . . . . . . 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)
1817ex 115 . . . . . . . . . . . 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))
1918adantrd 279 . . . . . . . . . . 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))
2016, 19sylbid 150 . . . . . . . . . 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))
2120imp 124 . . . . . . . . 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)
22 fsumss.2 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
2322ex 115 . . . . . . . . . . . . . 14 (πœ‘ β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
2423adantr 276 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
25 eldif 3150 . . . . . . . . . . . . . . 15 (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↔ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴))
26 fsumss.3 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 = 0)
27 0cn 7963 . . . . . . . . . . . . . . . 16 0 ∈ β„‚
2826, 27eqeltrdi 2278 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ β„‚)
2925, 28sylan2br 288 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘˜ ∈ 𝐡 ∧ Β¬ π‘˜ ∈ 𝐴)) β†’ 𝐢 ∈ β„‚)
3029expr 375 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (Β¬ π‘˜ ∈ 𝐴 β†’ 𝐢 ∈ β„‚))
31 eleq1w 2248 . . . . . . . . . . . . . . . 16 (𝑗 = π‘˜ β†’ (𝑗 ∈ 𝐴 ↔ π‘˜ ∈ 𝐴))
3231dcbid 839 . . . . . . . . . . . . . . 15 (𝑗 = π‘˜ β†’ (DECID 𝑗 ∈ 𝐴 ↔ DECID π‘˜ ∈ 𝐴))
33 fisumss.adc . . . . . . . . . . . . . . . 16 (πœ‘ β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
3433adantr 276 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
35 simpr 110 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ π‘˜ ∈ 𝐡)
3632, 34, 35rspcdva 2858 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ DECID π‘˜ ∈ 𝐴)
37 exmiddc 837 . . . . . . . . . . . . . 14 (DECID π‘˜ ∈ 𝐴 β†’ (π‘˜ ∈ 𝐴 ∨ Β¬ π‘˜ ∈ 𝐴))
3836, 37syl 14 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ (π‘˜ ∈ 𝐴 ∨ Β¬ π‘˜ ∈ 𝐴))
3924, 30, 38mpjaod 719 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
4039fmpttd 5684 . . . . . . . . . . 11 (πœ‘ β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
4140adantr 276 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (π‘˜ ∈ 𝐡 ↦ 𝐢):π΅βŸΆβ„‚)
4241ffvelcdmda 5664 . . . . . . . . 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„‚)
4321, 42syldan 282 . . . . . . . 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„‚)
44 eldifi 3269 . . . . . . . . . . . 12 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ 𝑛 ∈ (1...(β™―β€˜π΅)))
4544, 17sylan2 286 . . . . . . . . . . 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)
46 eldifn 3270 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...(β™―β€˜π΅)) βˆ– (◑𝑓 β€œ 𝐴)) β†’ Β¬ 𝑛 ∈ (◑𝑓 β€œ 𝐴))
4746adantl 277 . . . . . . . . . . . 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))
4816adantr 276 . . . . . . . . . . . . 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)))
4944adantl 277 . . . . . . . . . . . . . 14 (((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΅)))
5049biantrurd 305 . . . . . . . . . . . . 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†’ ((π‘“β€˜π‘›) ∈ 𝐴 ↔ (𝑛 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘›) ∈ 𝐴)))
5148, 50bitr4d 191 . . . . . . . . . . . 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))
5247, 51mtbid 673 . . . . . . . . . . 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)
5345, 52eldifd 3151 . . . . . . . . . 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))
54 difss 3273 . . . . . . . . . . . . 13 (𝐡 βˆ– 𝐴) βŠ† 𝐡
55 resmpt 4967 . . . . . . . . . . . . 13 ((𝐡 βˆ– 𝐴) βŠ† 𝐡 β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢))
5654, 55ax-mp 5 . . . . . . . . . . . 12 ((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴)) = (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)
5756fveq1i 5528 . . . . . . . . . . 11 (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ (𝐡 βˆ– 𝐴))β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›))
58 fvres 5551 . . . . . . . . . . 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‘›)))
5957, 58eqtr3id 2234 . . . . . . . . . 10 ((π‘“β€˜π‘›) ∈ (𝐡 βˆ– 𝐴) β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
6053, 59syl 14 . . . . . . . . 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‘›)))
61 c0ex 7965 . . . . . . . . . . . . . . 15 0 ∈ V
6261elsn2 3638 . . . . . . . . . . . . . 14 (𝐢 ∈ {0} ↔ 𝐢 = 0)
6326, 62sylibr 134 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ (𝐡 βˆ– 𝐴)) β†’ 𝐢 ∈ {0})
6463fmpttd 5684 . . . . . . . . . . . 12 (πœ‘ β†’ (π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢):(𝐡 βˆ– 𝐴)⟢{0})
6564ad2antrr 488 . . . . . . . . . . 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})
6665, 53ffvelcdmd 5665 . . . . . . . . . 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})
67 elsni 3622 . . . . . . . . . 10 (((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) ∈ {0} β†’ ((π‘˜ ∈ (𝐡 βˆ– 𝐴) ↦ 𝐢)β€˜(π‘“β€˜π‘›)) = 0)
6866, 67syl 14 . . . . . . . . 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)
6960, 68eqtr3d 2222 . . . . . . . 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)
70 eleq1 2250 . . . . . . . . . . . . 13 (𝑗 = (π‘“β€˜π‘’) β†’ (𝑗 ∈ 𝐴 ↔ (π‘“β€˜π‘’) ∈ 𝐴))
7170dcbid 839 . . . . . . . . . . . 12 (𝑗 = (π‘“β€˜π‘’) β†’ (DECID 𝑗 ∈ 𝐴 ↔ DECID (π‘“β€˜π‘’) ∈ 𝐴))
7233ad3antrrr 492 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴)
7312ad2antrr 488 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ 𝑓:(1...(β™―β€˜π΅))⟢𝐡)
74 simpr 110 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ 𝑒 ∈ (1...(β™―β€˜π΅)))
7573, 74ffvelcdmd 5665 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ (π‘“β€˜π‘’) ∈ 𝐡)
7671, 72, 75rspcdva 2858 . . . . . . . . . . 11 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ DECID (π‘“β€˜π‘’) ∈ 𝐴)
7710ad2antrr 488 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)
78 f1ofun 5475 . . . . . . . . . . . . . 14 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ Fun 𝑓)
7977, 78syl 14 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ Fun 𝑓)
80 f1odm 5477 . . . . . . . . . . . . . . 15 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ dom 𝑓 = (1...(β™―β€˜π΅)))
8177, 80syl 14 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ dom 𝑓 = (1...(β™―β€˜π΅)))
8274, 81eleqtrrd 2267 . . . . . . . . . . . . 13 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ 𝑒 ∈ dom 𝑓)
83 fvimacnv 5644 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ 𝑒 ∈ dom 𝑓) β†’ ((π‘“β€˜π‘’) ∈ 𝐴 ↔ 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
8479, 82, 83syl2anc 411 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ ((π‘“β€˜π‘’) ∈ 𝐴 ↔ 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
8584dcbid 839 . . . . . . . . . . 11 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ (DECID (π‘“β€˜π‘’) ∈ 𝐴 ↔ DECID 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
8676, 85mpbid 147 . . . . . . . . . 10 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ DECID 𝑒 ∈ (◑𝑓 β€œ 𝐴))
87 elpreima 5648 . . . . . . . . . . . . . . . . 17 (𝑓 Fn (1...(β™―β€˜π΅)) β†’ (𝑒 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑒 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘’) ∈ 𝐴)))
88 simpl 109 . . . . . . . . . . . . . . . . 17 ((𝑒 ∈ (1...(β™―β€˜π΅)) ∧ (π‘“β€˜π‘’) ∈ 𝐴) β†’ 𝑒 ∈ (1...(β™―β€˜π΅)))
8987, 88biimtrdi 163 . . . . . . . . . . . . . . . 16 (𝑓 Fn (1...(β™―β€˜π΅)) β†’ (𝑒 ∈ (◑𝑓 β€œ 𝐴) β†’ 𝑒 ∈ (1...(β™―β€˜π΅))))
9089con3d 632 . . . . . . . . . . . . . . 15 (𝑓 Fn (1...(β™―β€˜π΅)) β†’ (Β¬ 𝑒 ∈ (1...(β™―β€˜π΅)) β†’ Β¬ 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
9114, 90syl 14 . . . . . . . . . . . . . 14 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (Β¬ 𝑒 ∈ (1...(β™―β€˜π΅)) β†’ Β¬ 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
9291adantr 276 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ (Β¬ 𝑒 ∈ (1...(β™―β€˜π΅)) β†’ Β¬ 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
9392imp 124 . . . . . . . . . . . 12 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ Β¬ 𝑒 ∈ (◑𝑓 β€œ 𝐴))
9493olcd 735 . . . . . . . . . . 11 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ (𝑒 ∈ (◑𝑓 β€œ 𝐴) ∨ Β¬ 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
95 df-dc 836 . . . . . . . . . . 11 (DECID 𝑒 ∈ (◑𝑓 β€œ 𝐴) ↔ (𝑒 ∈ (◑𝑓 β€œ 𝐴) ∨ Β¬ 𝑒 ∈ (◑𝑓 β€œ 𝐴)))
9694, 95sylibr 134 . . . . . . . . . 10 ((((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) ∧ Β¬ 𝑒 ∈ (1...(β™―β€˜π΅))) β†’ DECID 𝑒 ∈ (◑𝑓 β€œ 𝐴))
97 eluzelz 9551 . . . . . . . . . . . . 13 (𝑒 ∈ (β„€β‰₯β€˜1) β†’ 𝑒 ∈ β„€)
9897adantl 277 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ 𝑒 ∈ β„€)
99 1zzd 9294 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ 1 ∈ β„€)
100 simplrl 535 . . . . . . . . . . . . 13 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ (β™―β€˜π΅) ∈ β„•)
101100nnzd 9388 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ (β™―β€˜π΅) ∈ β„€)
102 fzdcel 10054 . . . . . . . . . . . 12 ((𝑒 ∈ β„€ ∧ 1 ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„€) β†’ DECID 𝑒 ∈ (1...(β™―β€˜π΅)))
10398, 99, 101, 102syl3anc 1248 . . . . . . . . . . 11 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ DECID 𝑒 ∈ (1...(β™―β€˜π΅)))
104 exmiddc 837 . . . . . . . . . . 11 (DECID 𝑒 ∈ (1...(β™―β€˜π΅)) β†’ (𝑒 ∈ (1...(β™―β€˜π΅)) ∨ Β¬ 𝑒 ∈ (1...(β™―β€˜π΅))))
105103, 104syl 14 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ (𝑒 ∈ (1...(β™―β€˜π΅)) ∨ Β¬ 𝑒 ∈ (1...(β™―β€˜π΅))))
10686, 96, 105mpjaodan 799 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ 𝑒 ∈ (β„€β‰₯β€˜1)) β†’ DECID 𝑒 ∈ (◑𝑓 β€œ 𝐴))
107106ralrimiva 2560 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆ€π‘’ ∈ (β„€β‰₯β€˜1)DECID 𝑒 ∈ (◑𝑓 β€œ 𝐴))
108 1zzd 9294 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 1 ∈ β„€)
109 fzssuz 10079 . . . . . . . . 9 (1...(β™―β€˜π΅)) βŠ† (β„€β‰₯β€˜1)
110109a1i 9 . . . . . . . 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))
111103ralrimiva 2560 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ βˆ€π‘’ ∈ (β„€β‰₯β€˜1)DECID 𝑒 ∈ (1...(β™―β€˜π΅)))
11213, 43, 69, 107, 108, 110, 111isumss 11413 . . . . . . 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‘›)))
1131ad2antrr 488 . . . . . . . . . . . 12 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ 𝐴 βŠ† 𝐡)
114113resmptd 4970 . . . . . . . . . . 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))
115114fveq1d 5529 . . . . . . . . . 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))
116 fvres 5551 . . . . . . . . . . 11 (π‘š ∈ 𝐴 β†’ (((π‘˜ ∈ 𝐡 ↦ 𝐢) β†Ύ 𝐴)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š))
117116adantl 277 . . . . . . . . . 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))
118115, 117eqtr3d 2222 . . . . . . . . 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))
119118sumeq2dv 11390 . . . . . . . 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))
120 fveq2 5527 . . . . . . . . 9 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜(π‘“β€˜π‘›)))
1211adantr 276 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐴 βŠ† 𝐡)
122 fsumss.4 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐡 ∈ Fin)
123 ssfidc 6948 . . . . . . . . . . . 12 ((𝐡 ∈ Fin ∧ 𝐴 βŠ† 𝐡 ∧ βˆ€π‘— ∈ 𝐡 DECID 𝑗 ∈ 𝐴) β†’ 𝐴 ∈ Fin)
124122, 1, 33, 123syl3anc 1248 . . . . . . . . . . 11 (πœ‘ β†’ 𝐴 ∈ Fin)
125124adantr 276 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝐴 ∈ Fin)
126121, 10, 125preimaf1ofi 6964 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (◑𝑓 β€œ 𝐴) ∈ Fin)
127 f1of1 5472 . . . . . . . . . . . 12 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–1-1→𝐡)
12810, 127syl 14 . . . . . . . . . . 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)
129 f1ores 5488 . . . . . . . . . . 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)))
130128, 13, 129syl2anc 411 . . . . . . . . . 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)))
131 f1ofo 5480 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡 β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
13210, 131syl 14 . . . . . . . . . . . 12 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ 𝑓:(1...(β™―β€˜π΅))–onto→𝐡)
133 foimacnv 5491 . . . . . . . . . . . 12 ((𝑓:(1...(β™―β€˜π΅))–onto→𝐡 ∧ 𝐴 βŠ† 𝐡) β†’ (𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴)
134132, 121, 133syl2anc 411 . . . . . . . . . . 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)
135 f1oeq3 5463 . . . . . . . . . . 11 ((𝑓 β€œ (◑𝑓 β€œ 𝐴)) = 𝐴 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))
136134, 135syl 14 . . . . . . . . . 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))
137130, 136mpbid 147 . . . . . . . . 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)
138 fvres 5551 . . . . . . . . . 10 (𝑛 ∈ (◑𝑓 β€œ 𝐴) β†’ ((𝑓 β†Ύ (◑𝑓 β€œ 𝐴))β€˜π‘›) = (π‘“β€˜π‘›))
139138adantl 277 . . . . . . . . 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‘›))
140121sselda 3167 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) ∧ π‘š ∈ 𝐴) β†’ π‘š ∈ 𝐡)
14141ffvelcdmda 5664 . . . . . . . . . 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„‚)
142140, 141syldan 282 . . . . . . . . 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„‚)
143120, 126, 137, 139, 142fsumf1o 11412 . . . . . . . 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‘›)))
144119, 143eqtrd 2220 . . . . . . 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‘›)))
145 simprl 529 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (β™―β€˜π΅) ∈ β„•)
146145nnzd 9388 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (β™―β€˜π΅) ∈ β„€)
147108, 146fzfigd 10445 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜π΅) ∈ β„• ∧ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)) β†’ (1...(β™―β€˜π΅)) ∈ Fin)
148 eqidd 2188 . . . . . . . 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‘›))
149120, 147, 10, 148, 141fsumf1o 11412 . . . . . . 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‘›)))
150112, 144, 1493eqtr4d 2230 . . . . . 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))
15122ralrimiva 2560 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝐴 𝐢 ∈ β„‚)
152 sumfct 11396 . . . . . . . 8 (βˆ€π‘˜ ∈ 𝐴 𝐢 ∈ β„‚ β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐴 𝐢)
153151, 152syl 14 . . . . . . 7 (πœ‘ β†’ Ξ£π‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐴 𝐢)
154153adantr 276 . . . . . 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‘˜ ∈ 𝐴 𝐢)
15522adantlr 477 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
156 simpll 527 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ Β¬ π‘˜ ∈ 𝐴) β†’ πœ‘)
157 simplr 528 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ Β¬ π‘˜ ∈ 𝐴) β†’ π‘˜ ∈ 𝐡)
158 simpr 110 . . . . . . . . . . . . 13 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ Β¬ π‘˜ ∈ 𝐴) β†’ Β¬ π‘˜ ∈ 𝐴)
159157, 158eldifd 3151 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ Β¬ π‘˜ ∈ 𝐴) β†’ π‘˜ ∈ (𝐡 βˆ– 𝐴))
160156, 159, 26syl2anc 411 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ Β¬ π‘˜ ∈ 𝐴) β†’ 𝐢 = 0)
161 0cnd 7964 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ Β¬ π‘˜ ∈ 𝐴) β†’ 0 ∈ β„‚)
162160, 161eqeltrd 2264 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ 𝐡) ∧ Β¬ π‘˜ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
163155, 162, 38mpjaodan 799 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
164163ralrimiva 2560 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘˜ ∈ 𝐡 𝐢 ∈ β„‚)
165 sumfct 11396 . . . . . . . 8 (βˆ€π‘˜ ∈ 𝐡 𝐢 ∈ β„‚ β†’ Ξ£π‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐡 𝐢)
166164, 165syl 14 . . . . . . 7 (πœ‘ β†’ Ξ£π‘š ∈ 𝐡 ((π‘˜ ∈ 𝐡 ↦ 𝐢)β€˜π‘š) = Ξ£π‘˜ ∈ 𝐡 𝐢)
167166adantr 276 . . . . . 6 ((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) = Ξ£π‘˜ ∈ 𝐡 𝐢)
168150, 154, 1673eqtr3d 2228 . . . . 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)
169168expr 375 . . . 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))
170169exlimdv 1829 . . 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))
171170expimpd 363 . 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))
172 fz1f1o 11397 . . 3 (𝐡 ∈ Fin β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
173122, 172syl 14 . 2 (πœ‘ β†’ (𝐡 = βˆ… ∨ ((β™―β€˜π΅) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜π΅))–1-1-onto→𝐡)))
1748, 171, 173mpjaod 719 1 (πœ‘ β†’ Ξ£π‘˜ ∈ 𝐴 𝐢 = Ξ£π‘˜ ∈ 𝐡 𝐢)
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1363  βˆƒwex 1502   ∈ wcel 2158  βˆ€wral 2465   βˆ– cdif 3138   βŠ† wss 3141  βˆ…c0 3434  {csn 3604   ↦ cmpt 4076  β—‘ccnv 4637  dom cdm 4638   β†Ύ cres 4640   β€œ cima 4641  Fun wfun 5222   Fn wfn 5223  βŸΆwf 5224  β€“1-1β†’wf1 5225  β€“ontoβ†’wfo 5226  β€“1-1-ontoβ†’wf1o 5227  β€˜cfv 5228  (class class class)co 5888  Fincfn 6754  β„‚cc 7823  0cc0 7825  1c1 7826  β„•cn 8933  β„€cz 9267  β„€β‰₯cuz 9542  ...cfz 10022  β™―chash 10769  Ξ£csu 11375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-en 6755  df-dom 6756  df-fin 6757  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-ihash 10770  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376
This theorem is referenced by:  isumss2  11415
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