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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mzpf | Structured version Visualization version GIF version |
Description: A polynomial function is a function from the coordinate space to the integers. (Contributed by Stefan O'Rear, 5-Oct-2014.) |
Ref | Expression |
---|---|
mzpf | β’ (πΉ β (mzPolyβπ) β πΉ:(β€ βm π)βΆβ€) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6922 | . . . . 5 β’ (πΉ β (mzPolyβπ) β π β V) | |
2 | mzpval 42030 | . . . . . 6 β’ (π β V β (mzPolyβπ) = β© (mzPolyCldβπ)) | |
3 | mzpclall 42025 | . . . . . . 7 β’ (π β V β (β€ βm (β€ βm π)) β (mzPolyCldβπ)) | |
4 | intss1 4960 | . . . . . . 7 β’ ((β€ βm (β€ βm π)) β (mzPolyCldβπ) β β© (mzPolyCldβπ) β (β€ βm (β€ βm π))) | |
5 | 3, 4 | syl 17 | . . . . . 6 β’ (π β V β β© (mzPolyCldβπ) β (β€ βm (β€ βm π))) |
6 | 2, 5 | eqsstrd 4015 | . . . . 5 β’ (π β V β (mzPolyβπ) β (β€ βm (β€ βm π))) |
7 | 1, 6 | syl 17 | . . . 4 β’ (πΉ β (mzPolyβπ) β (mzPolyβπ) β (β€ βm (β€ βm π))) |
8 | 7 | sselda 3977 | . . 3 β’ ((πΉ β (mzPolyβπ) β§ πΉ β (mzPolyβπ)) β πΉ β (β€ βm (β€ βm π))) |
9 | 8 | anidms 566 | . 2 β’ (πΉ β (mzPolyβπ) β πΉ β (β€ βm (β€ βm π))) |
10 | zex 12568 | . . 3 β’ β€ β V | |
11 | ovex 7437 | . . 3 β’ (β€ βm π) β V | |
12 | 10, 11 | elmap 8864 | . 2 β’ (πΉ β (β€ βm (β€ βm π)) β πΉ:(β€ βm π)βΆβ€) |
13 | 9, 12 | sylib 217 | 1 β’ (πΉ β (mzPolyβπ) β πΉ:(β€ βm π)βΆβ€) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 Vcvv 3468 β wss 3943 β© cint 4943 βΆwf 6532 βcfv 6536 (class class class)co 7404 βm cmap 8819 β€cz 12559 mzPolyCldcmzpcl 42019 mzPolycmzp 42020 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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-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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-mzpcl 42021 df-mzp 42022 |
This theorem is referenced by: mzpaddmpt 42039 mzpmulmpt 42040 mzpsubmpt 42041 mzpexpmpt 42043 mzpsubst 42046 mzpcompact2lem 42049 diophin 42070 diophun 42071 eq0rabdioph 42074 eqrabdioph 42075 rabdiophlem1 42099 rabdiophlem2 42100 |
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