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Mirrors > Home > MPE Home > Th. List > pf1addcl | Structured version Visualization version GIF version |
Description: The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.) |
Ref | Expression |
---|---|
pf1rcl.q | β’ π = ran (eval1βπ ) |
pf1addcl.a | β’ + = (+gβπ ) |
Ref | Expression |
---|---|
pf1addcl | β’ ((πΉ β π β§ πΊ β π) β (πΉ βf + πΊ) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 β’ (π βs (Baseβπ )) = (π βs (Baseβπ )) | |
2 | eqid 2737 | . . 3 β’ (Baseβ(π βs (Baseβπ ))) = (Baseβ(π βs (Baseβπ ))) | |
3 | pf1rcl.q | . . . . 5 β’ π = ran (eval1βπ ) | |
4 | 3 | pf1rcl 21731 | . . . 4 β’ (πΉ β π β π β CRing) |
5 | 4 | adantr 482 | . . 3 β’ ((πΉ β π β§ πΊ β π) β π β CRing) |
6 | fvexd 6862 | . . 3 β’ ((πΉ β π β§ πΊ β π) β (Baseβπ ) β V) | |
7 | eqid 2737 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
8 | 3, 7 | pf1f 21732 | . . . . 5 β’ (πΉ β π β πΉ:(Baseβπ )βΆ(Baseβπ )) |
9 | 8 | adantr 482 | . . . 4 β’ ((πΉ β π β§ πΊ β π) β πΉ:(Baseβπ )βΆ(Baseβπ )) |
10 | fvex 6860 | . . . . 5 β’ (Baseβπ ) β V | |
11 | 1, 7, 2 | pwselbasb 17377 | . . . . 5 β’ ((π β CRing β§ (Baseβπ ) β V) β (πΉ β (Baseβ(π βs (Baseβπ ))) β πΉ:(Baseβπ )βΆ(Baseβπ ))) |
12 | 5, 10, 11 | sylancl 587 | . . . 4 β’ ((πΉ β π β§ πΊ β π) β (πΉ β (Baseβ(π βs (Baseβπ ))) β πΉ:(Baseβπ )βΆ(Baseβπ ))) |
13 | 9, 12 | mpbird 257 | . . 3 β’ ((πΉ β π β§ πΊ β π) β πΉ β (Baseβ(π βs (Baseβπ )))) |
14 | 3, 7 | pf1f 21732 | . . . . 5 β’ (πΊ β π β πΊ:(Baseβπ )βΆ(Baseβπ )) |
15 | 14 | adantl 483 | . . . 4 β’ ((πΉ β π β§ πΊ β π) β πΊ:(Baseβπ )βΆ(Baseβπ )) |
16 | 1, 7, 2 | pwselbasb 17377 | . . . . 5 β’ ((π β CRing β§ (Baseβπ ) β V) β (πΊ β (Baseβ(π βs (Baseβπ ))) β πΊ:(Baseβπ )βΆ(Baseβπ ))) |
17 | 5, 10, 16 | sylancl 587 | . . . 4 β’ ((πΉ β π β§ πΊ β π) β (πΊ β (Baseβ(π βs (Baseβπ ))) β πΊ:(Baseβπ )βΆ(Baseβπ ))) |
18 | 15, 17 | mpbird 257 | . . 3 β’ ((πΉ β π β§ πΊ β π) β πΊ β (Baseβ(π βs (Baseβπ )))) |
19 | pf1addcl.a | . . 3 β’ + = (+gβπ ) | |
20 | eqid 2737 | . . 3 β’ (+gβ(π βs (Baseβπ ))) = (+gβ(π βs (Baseβπ ))) | |
21 | 1, 2, 5, 6, 13, 18, 19, 20 | pwsplusgval 17379 | . 2 β’ ((πΉ β π β§ πΊ β π) β (πΉ(+gβ(π βs (Baseβπ )))πΊ) = (πΉ βf + πΊ)) |
22 | 7, 3 | pf1subrg 21730 | . . . 4 β’ (π β CRing β π β (SubRingβ(π βs (Baseβπ )))) |
23 | 5, 22 | syl 17 | . . 3 β’ ((πΉ β π β§ πΊ β π) β π β (SubRingβ(π βs (Baseβπ )))) |
24 | 20 | subrgacl 20249 | . . . 4 β’ ((π β (SubRingβ(π βs (Baseβπ ))) β§ πΉ β π β§ πΊ β π) β (πΉ(+gβ(π βs (Baseβπ )))πΊ) β π) |
25 | 24 | 3expib 1123 | . . 3 β’ (π β (SubRingβ(π βs (Baseβπ ))) β ((πΉ β π β§ πΊ β π) β (πΉ(+gβ(π βs (Baseβπ )))πΊ) β π)) |
26 | 23, 25 | mpcom 38 | . 2 β’ ((πΉ β π β§ πΊ β π) β (πΉ(+gβ(π βs (Baseβπ )))πΊ) β π) |
27 | 21, 26 | eqeltrrd 2839 | 1 β’ ((πΉ β π β§ πΊ β π) β (πΉ βf + πΊ) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3448 ran crn 5639 βΆwf 6497 βcfv 6501 (class class class)co 7362 βf cof 7620 Basecbs 17090 +gcplusg 17140 βs cpws 17335 CRingccrg 19972 SubRingcsubrg 20234 eval1ce1 21696 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-ofr 7623 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-fzo 13575 df-seq 13914 df-hash 14238 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-hom 17164 df-cco 17165 df-0g 17330 df-gsum 17331 df-prds 17336 df-pws 17338 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-ghm 19013 df-cntz 19104 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-srg 19925 df-ring 19973 df-cring 19974 df-rnghom 20155 df-subrg 20236 df-lmod 20340 df-lss 20409 df-lsp 20449 df-assa 21275 df-asp 21276 df-ascl 21277 df-psr 21327 df-mvr 21328 df-mpl 21329 df-opsr 21331 df-evls 21498 df-evl 21499 df-psr1 21567 df-ply1 21569 df-evl1 21698 |
This theorem is referenced by: (None) |
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