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Theorem fprodf1o 15896
Description: Re-index a finite product using a bijection. (Contributed by Scott Fenton, 7-Dec-2017.)
Hypotheses
Ref Expression
fprodf1o.1 (π‘˜ = 𝐺 β†’ 𝐡 = 𝐷)
fprodf1o.2 (πœ‘ β†’ 𝐢 ∈ Fin)
fprodf1o.3 (πœ‘ β†’ 𝐹:𝐢–1-1-onto→𝐴)
fprodf1o.4 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ (πΉβ€˜π‘›) = 𝐺)
fprodf1o.5 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)
Assertion
Ref Expression
fprodf1o (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷)
Distinct variable groups:   𝐴,π‘˜,𝑛   𝐡,𝑛   𝐢,𝑛   𝐷,π‘˜   𝑛,𝐹   π‘˜,𝐺   π‘˜,𝑛,πœ‘
Allowed substitution hints:   𝐡(π‘˜)   𝐢(π‘˜)   𝐷(𝑛)   𝐹(π‘˜)   𝐺(𝑛)

Proof of Theorem fprodf1o
Dummy variables 𝑓 π‘š are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prod0 15893 . . . 4 βˆπ‘˜ ∈ βˆ… 𝐡 = 1
2 fprodf1o.3 . . . . . . . . 9 (πœ‘ β†’ 𝐹:𝐢–1-1-onto→𝐴)
32adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ 𝐹:𝐢–1-1-onto→𝐴)
4 f1oeq2 6816 . . . . . . . . 9 (𝐢 = βˆ… β†’ (𝐹:𝐢–1-1-onto→𝐴 ↔ 𝐹:βˆ…β€“1-1-onto→𝐴))
54adantl 481 . . . . . . . 8 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ (𝐹:𝐢–1-1-onto→𝐴 ↔ 𝐹:βˆ…β€“1-1-onto→𝐴))
63, 5mpbid 231 . . . . . . 7 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ 𝐹:βˆ…β€“1-1-onto→𝐴)
7 f1ofo 6834 . . . . . . 7 (𝐹:βˆ…β€“1-1-onto→𝐴 β†’ 𝐹:βˆ…β€“onto→𝐴)
86, 7syl 17 . . . . . 6 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ 𝐹:βˆ…β€“onto→𝐴)
9 fo00 6863 . . . . . . 7 (𝐹:βˆ…β€“onto→𝐴 ↔ (𝐹 = βˆ… ∧ 𝐴 = βˆ…))
109simprbi 496 . . . . . 6 (𝐹:βˆ…β€“onto→𝐴 β†’ 𝐴 = βˆ…)
118, 10syl 17 . . . . 5 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ 𝐴 = βˆ…)
1211prodeq1d 15871 . . . 4 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘˜ ∈ βˆ… 𝐡)
13 prodeq1 15859 . . . . . 6 (𝐢 = βˆ… β†’ βˆπ‘› ∈ 𝐢 𝐷 = βˆπ‘› ∈ βˆ… 𝐷)
14 prod0 15893 . . . . . 6 βˆπ‘› ∈ βˆ… 𝐷 = 1
1513, 14eqtrdi 2782 . . . . 5 (𝐢 = βˆ… β†’ βˆπ‘› ∈ 𝐢 𝐷 = 1)
1615adantl 481 . . . 4 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ βˆπ‘› ∈ 𝐢 𝐷 = 1)
171, 12, 163eqtr4a 2792 . . 3 ((πœ‘ ∧ 𝐢 = βˆ…) β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷)
1817ex 412 . 2 (πœ‘ β†’ (𝐢 = βˆ… β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷))
19 2fveq3 6890 . . . . . . . 8 (π‘š = (π‘“β€˜π‘›) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜(π‘“β€˜π‘›))))
20 simprl 768 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ (β™―β€˜πΆ) ∈ β„•)
21 simprr 770 . . . . . . . 8 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)
22 f1of 6827 . . . . . . . . . . . 12 (𝐹:𝐢–1-1-onto→𝐴 β†’ 𝐹:𝐢⟢𝐴)
232, 22syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:𝐢⟢𝐴)
2423ffvelcdmda 7080 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ 𝐢) β†’ (πΉβ€˜π‘š) ∈ 𝐴)
25 fprodf1o.5 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ 𝐴) β†’ 𝐡 ∈ β„‚)
2625fmpttd 7110 . . . . . . . . . . 11 (πœ‘ β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
2726ffvelcdmda 7080 . . . . . . . . . 10 ((πœ‘ ∧ (πΉβ€˜π‘š) ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)) ∈ β„‚)
2824, 27syldan 590 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ 𝐢) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)) ∈ β„‚)
2928adantlr 712 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) ∧ π‘š ∈ 𝐢) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)) ∈ β„‚)
30 simpr 484 . . . . . . . . . . . 12 (((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢) β†’ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)
31 f1oco 6850 . . . . . . . . . . . 12 ((𝐹:𝐢–1-1-onto→𝐴 ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢) β†’ (𝐹 ∘ 𝑓):(1...(β™―β€˜πΆ))–1-1-onto→𝐴)
322, 30, 31syl2an 595 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ (𝐹 ∘ 𝑓):(1...(β™―β€˜πΆ))–1-1-onto→𝐴)
33 f1of 6827 . . . . . . . . . . 11 ((𝐹 ∘ 𝑓):(1...(β™―β€˜πΆ))–1-1-onto→𝐴 β†’ (𝐹 ∘ 𝑓):(1...(β™―β€˜πΆ))⟢𝐴)
3432, 33syl 17 . . . . . . . . . 10 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ (𝐹 ∘ 𝑓):(1...(β™―β€˜πΆ))⟢𝐴)
35 fvco3 6984 . . . . . . . . . 10 (((𝐹 ∘ 𝑓):(1...(β™―β€˜πΆ))⟢𝐴 ∧ 𝑛 ∈ (1...(β™―β€˜πΆ))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ (𝐹 ∘ 𝑓))β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜((𝐹 ∘ 𝑓)β€˜π‘›)))
3634, 35sylan 579 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) ∧ 𝑛 ∈ (1...(β™―β€˜πΆ))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ (𝐹 ∘ 𝑓))β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜((𝐹 ∘ 𝑓)β€˜π‘›)))
37 f1of 6827 . . . . . . . . . . . . 13 (𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢 β†’ 𝑓:(1...(β™―β€˜πΆ))⟢𝐢)
3837adantl 481 . . . . . . . . . . . 12 (((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢) β†’ 𝑓:(1...(β™―β€˜πΆ))⟢𝐢)
3938adantl 481 . . . . . . . . . . 11 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ 𝑓:(1...(β™―β€˜πΆ))⟢𝐢)
40 fvco3 6984 . . . . . . . . . . 11 ((𝑓:(1...(β™―β€˜πΆ))⟢𝐢 ∧ 𝑛 ∈ (1...(β™―β€˜πΆ))) β†’ ((𝐹 ∘ 𝑓)β€˜π‘›) = (πΉβ€˜(π‘“β€˜π‘›)))
4139, 40sylan 579 . . . . . . . . . 10 (((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) ∧ 𝑛 ∈ (1...(β™―β€˜πΆ))) β†’ ((𝐹 ∘ 𝑓)β€˜π‘›) = (πΉβ€˜(π‘“β€˜π‘›)))
4241fveq2d 6889 . . . . . . . . 9 (((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) ∧ 𝑛 ∈ (1...(β™―β€˜πΆ))) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜((𝐹 ∘ 𝑓)β€˜π‘›)) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜(π‘“β€˜π‘›))))
4336, 42eqtrd 2766 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) ∧ 𝑛 ∈ (1...(β™―β€˜πΆ))) β†’ (((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ (𝐹 ∘ 𝑓))β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜(π‘“β€˜π‘›))))
4419, 20, 21, 29, 43fprod 15891 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ βˆπ‘š ∈ 𝐢 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)) = (seq1( Β· , ((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ (𝐹 ∘ 𝑓)))β€˜(β™―β€˜πΆ)))
45 fprodf1o.4 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ (πΉβ€˜π‘›) = 𝐺)
4623ffvelcdmda 7080 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ (πΉβ€˜π‘›) ∈ 𝐴)
4745, 46eqeltrrd 2828 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ 𝐺 ∈ 𝐴)
48 fprodf1o.1 . . . . . . . . . . . . . 14 (π‘˜ = 𝐺 β†’ 𝐡 = 𝐷)
49 eqid 2726 . . . . . . . . . . . . . 14 (π‘˜ ∈ 𝐴 ↦ 𝐡) = (π‘˜ ∈ 𝐴 ↦ 𝐡)
5048, 49fvmpti 6991 . . . . . . . . . . . . 13 (𝐺 ∈ 𝐴 β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜πΊ) = ( I β€˜π·))
5147, 50syl 17 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜πΊ) = ( I β€˜π·))
5245fveq2d 6889 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘›)) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜πΊ))
53 eqid 2726 . . . . . . . . . . . . . 14 (𝑛 ∈ 𝐢 ↦ 𝐷) = (𝑛 ∈ 𝐢 ↦ 𝐷)
5453fvmpt2i 7002 . . . . . . . . . . . . 13 (𝑛 ∈ 𝐢 β†’ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘›) = ( I β€˜π·))
5554adantl 481 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘›) = ( I β€˜π·))
5651, 52, 553eqtr4rd 2777 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ 𝐢) β†’ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘›)))
5756ralrimiva 3140 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘› ∈ 𝐢 ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘›)))
58 nffvmpt1 6896 . . . . . . . . . . . 12 Ⅎ𝑛((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š)
5958nfeq1 2912 . . . . . . . . . . 11 Ⅎ𝑛((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š))
60 fveq2 6885 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘›) = ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š))
61 2fveq3 6890 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘›)) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)))
6260, 61eqeq12d 2742 . . . . . . . . . . 11 (𝑛 = π‘š β†’ (((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘›)) ↔ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š))))
6359, 62rspc 3594 . . . . . . . . . 10 (π‘š ∈ 𝐢 β†’ (βˆ€π‘› ∈ 𝐢 ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘›) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘›)) β†’ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š))))
6457, 63mpan9 506 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ 𝐢) β†’ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)))
6564adantlr 712 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) ∧ π‘š ∈ 𝐢) β†’ ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)))
6665prodeq2dv 15873 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ βˆπ‘š ∈ 𝐢 ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š) = βˆπ‘š ∈ 𝐢 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜(πΉβ€˜π‘š)))
67 fveq2 6885 . . . . . . . 8 (π‘š = ((𝐹 ∘ 𝑓)β€˜π‘›) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) = ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜((𝐹 ∘ 𝑓)β€˜π‘›)))
6826adantr 480 . . . . . . . . 9 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ (π‘˜ ∈ 𝐴 ↦ 𝐡):π΄βŸΆβ„‚)
6968ffvelcdmda 7080 . . . . . . . 8 (((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) ∧ π‘š ∈ 𝐴) β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) ∈ β„‚)
7067, 20, 32, 69, 36fprod 15891 . . . . . . 7 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) = (seq1( Β· , ((π‘˜ ∈ 𝐴 ↦ 𝐡) ∘ (𝐹 ∘ 𝑓)))β€˜(β™―β€˜πΆ)))
7144, 66, 703eqtr4rd 2777 . . . . . 6 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) = βˆπ‘š ∈ 𝐢 ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š))
72 prodfc 15895 . . . . . 6 βˆπ‘š ∈ 𝐴 ((π‘˜ ∈ 𝐴 ↦ 𝐡)β€˜π‘š) = βˆπ‘˜ ∈ 𝐴 𝐡
73 prodfc 15895 . . . . . 6 βˆπ‘š ∈ 𝐢 ((𝑛 ∈ 𝐢 ↦ 𝐷)β€˜π‘š) = βˆπ‘› ∈ 𝐢 𝐷
7471, 72, 733eqtr3g 2789 . . . . 5 ((πœ‘ ∧ ((β™―β€˜πΆ) ∈ β„• ∧ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)) β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷)
7574expr 456 . . . 4 ((πœ‘ ∧ (β™―β€˜πΆ) ∈ β„•) β†’ (𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢 β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷))
7675exlimdv 1928 . . 3 ((πœ‘ ∧ (β™―β€˜πΆ) ∈ β„•) β†’ (βˆƒπ‘“ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢 β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷))
7776expimpd 453 . 2 (πœ‘ β†’ (((β™―β€˜πΆ) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢) β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷))
78 fprodf1o.2 . . 3 (πœ‘ β†’ 𝐢 ∈ Fin)
79 fz1f1o 15662 . . 3 (𝐢 ∈ Fin β†’ (𝐢 = βˆ… ∨ ((β™―β€˜πΆ) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)))
8078, 79syl 17 . 2 (πœ‘ β†’ (𝐢 = βˆ… ∨ ((β™―β€˜πΆ) ∈ β„• ∧ βˆƒπ‘“ 𝑓:(1...(β™―β€˜πΆ))–1-1-onto→𝐢)))
8118, 77, 80mpjaod 857 1 (πœ‘ β†’ βˆπ‘˜ ∈ 𝐴 𝐡 = βˆπ‘› ∈ 𝐢 𝐷)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3055  βˆ…c0 4317   ↦ cmpt 5224   I cid 5566   ∘ ccom 5673  βŸΆwf 6533  β€“ontoβ†’wfo 6535  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405  Fincfn 8941  β„‚cc 11110  1c1 11113   Β· cmul 11117  β„•cn 12216  ...cfz 13490  seqcseq 13972  β™―chash 14295  βˆcprod 15855
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  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 12981  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-prod 15856
This theorem is referenced by:  fprodss  15898  fprodshft  15926  fprodrev  15927  fprod2dlem  15930  fprodcnv  15933  gausslemma2dlem1  27254  hgt750lemg  34195
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