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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 41529 | . . . . 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 6194 | . . . 4 β’ dom IntgOver β π« β |
3 | 2 | sseli 3941 | . . 3 β’ (π β dom IntgOver β π β π« β) |
4 | cnex 11137 | . . . . 5 β’ β β V | |
5 | 4 | elpw2 5303 | . . . 4 β’ (π β π« β β π β β) |
6 | itgoval 41531 | . . . . 5 β’ (π β β β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
7 | ssrab2 4038 | . . . . 5 β’ {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} β β | |
8 | 6, 7 | eqsstrdi 3999 | . . . 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 6878 | . . 3 β’ (Β¬ π β dom IntgOver β (IntgOverβπ) = β ) | |
12 | 0ss 4357 | . . 3 β’ β β β | |
13 | 11, 12 | eqsstrdi 3999 | . 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 397 = wceq 1542 β wcel 2107 βwrex 3070 {crab 3406 β wss 3911 β c0 4283 π« cpw 4561 dom cdm 5634 βcfv 6497 βcc 11054 0cc0 11056 1c1 11057 Polycply 25561 coeffccoe 25563 degcdgr 25564 IntgOvercitgo 41527 |
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-sep 5257 ax-nul 5264 ax-pr 5385 ax-cnex 11112 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 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-iota 6449 df-fun 6499 df-fv 6505 df-itgo 41529 |
This theorem is referenced by: (None) |
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