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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 42453 | . . . . 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 6231 | . . . 4 β’ dom IntgOver β π« β |
3 | 2 | sseli 3971 | . . 3 β’ (π β dom IntgOver β π β π« β) |
4 | cnex 11188 | . . . . 5 β’ β β V | |
5 | 4 | elpw2 5336 | . . . 4 β’ (π β π« β β π β β) |
6 | itgoval 42455 | . . . . 5 β’ (π β β β (IntgOverβπ) = {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)}) | |
7 | ssrab2 4070 | . . . . 5 β’ {π β β β£ βπ β (Polyβπ)((πβπ) = 0 β§ ((coeffβπ)β(degβπ)) = 1)} β β | |
8 | 6, 7 | eqsstrdi 4029 | . . . 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 6917 | . . 3 β’ (Β¬ π β dom IntgOver β (IntgOverβπ) = β ) | |
12 | 0ss 4389 | . . 3 β’ β β β | |
13 | 11, 12 | eqsstrdi 4029 | . 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 1533 β wcel 2098 βwrex 3062 {crab 3424 β wss 3941 β c0 4315 π« cpw 4595 dom cdm 5667 βcfv 6534 βcc 11105 0cc0 11107 1c1 11108 Polycply 26062 coeffccoe 26064 degcdgr 26065 IntgOvercitgo 42451 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-cnex 11163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fv 6542 df-itgo 42453 |
This theorem is referenced by: (None) |
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