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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > itgocn | Structured version Visualization version GIF version |
Description: All integral elements are complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
itgocn | β’ (IntgOverβπ) β β |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-itgo 42583 | . . . . 5 β’ IntgOver = (π β π« β β¦ {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
2 | 1 | dmmptss 6245 | . . . 4 β’ dom IntgOver β π« β |
3 | 2 | sseli 3976 | . . 3 β’ (π β dom IntgOver β π β π« β) |
4 | cnex 11220 | . . . . 5 β’ β β V | |
5 | 4 | elpw2 5347 | . . . 4 β’ (π β π« β β π β β) |
6 | itgoval 42585 | . . . . 5 β’ (π β β β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
7 | ssrab2 4075 | . . . . 5 β’ {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} β β | |
8 | 6, 7 | eqsstrdi 4034 | . . . 4 β’ (π β β β (IntgOverβπ) β β) |
9 | 5, 8 | sylbi 216 | . . 3 β’ (π β π« β β (IntgOverβπ) β β) |
10 | 3, 9 | syl 17 | . 2 β’ (π β dom IntgOver β (IntgOverβπ) β β) |
11 | ndmfv 6932 | . . 3 β’ (Β¬ π β dom IntgOver β (IntgOverβπ) = β ) | |
12 | 0ss 4397 | . . 3 β’ β β β | |
13 | 11, 12 | eqsstrdi 4034 | . 2 β’ (Β¬ π β dom IntgOver β (IntgOverβπ) β β) |
14 | 10, 13 | pm2.61i 182 | 1 β’ (IntgOverβπ) β β |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β§ wa 395 = wceq 1534 β wcel 2099 βwrex 3067 {crab 3429 β wss 3947 β c0 4323 π« cpw 4603 dom cdm 5678 βcfv 6548 βcc 11137 0cc0 11139 1c1 11140 Polycply 26131 coeffccoe 26133 degcdgr 26134 IntgOvercitgo 42581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-cnex 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-itgo 42583 |
This theorem is referenced by: (None) |
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