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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 2728 | . . 3 β’ (π βs (Baseβπ )) = (π βs (Baseβπ )) | |
2 | eqid 2728 | . . 3 β’ (Baseβ(π βs (Baseβπ ))) = (Baseβ(π βs (Baseβπ ))) | |
3 | pf1rcl.q | . . . . 5 β’ π = ran (eval1βπ ) | |
4 | 3 | pf1rcl 22287 | . . . 4 β’ (πΉ β π β π β CRing) |
5 | 4 | adantr 479 | . . 3 β’ ((πΉ β π β§ πΊ β π) β π β CRing) |
6 | fvexd 6917 | . . 3 β’ ((πΉ β π β§ πΊ β π) β (Baseβπ ) β V) | |
7 | eqid 2728 | . . . . . 6 β’ (Baseβπ ) = (Baseβπ ) | |
8 | 3, 7 | pf1f 22288 | . . . . 5 β’ (πΉ β π β πΉ:(Baseβπ )βΆ(Baseβπ )) |
9 | 8 | adantr 479 | . . . 4 β’ ((πΉ β π β§ πΊ β π) β πΉ:(Baseβπ )βΆ(Baseβπ )) |
10 | fvex 6915 | . . . . 5 β’ (Baseβπ ) β V | |
11 | 1, 7, 2 | pwselbasb 17479 | . . . . 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 584 | . . . 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 256 | . . 3 β’ ((πΉ β π β§ πΊ β π) β πΉ β (Baseβ(π βs (Baseβπ )))) |
14 | 3, 7 | pf1f 22288 | . . . . 5 β’ (πΊ β π β πΊ:(Baseβπ )βΆ(Baseβπ )) |
15 | 14 | adantl 480 | . . . 4 β’ ((πΉ β π β§ πΊ β π) β πΊ:(Baseβπ )βΆ(Baseβπ )) |
16 | 1, 7, 2 | pwselbasb 17479 | . . . . 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 584 | . . . 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 256 | . . 3 β’ ((πΉ β π β§ πΊ β π) β πΊ β (Baseβ(π βs (Baseβπ )))) |
19 | pf1addcl.a | . . 3 β’ + = (+gβπ ) | |
20 | eqid 2728 | . . 3 β’ (+gβ(π βs (Baseβπ ))) = (+gβ(π βs (Baseβπ ))) | |
21 | 1, 2, 5, 6, 13, 18, 19, 20 | pwsplusgval 17481 | . 2 β’ ((πΉ β π β§ πΊ β π) β (πΉ(+gβ(π βs (Baseβπ )))πΊ) = (πΉ βf + πΊ)) |
22 | 7, 3 | pf1subrg 22286 | . . . 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 20536 | . . . 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 1119 | . . 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 2830 | 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 394 = wceq 1533 β wcel 2098 Vcvv 3473 ran crn 5683 βΆwf 6549 βcfv 6553 (class class class)co 7426 βf cof 7690 Basecbs 17189 +gcplusg 17242 βs cpws 17437 CRingccrg 20188 SubRingcsubrg 20520 eval1ce1 22252 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-sup 9475 df-oi 9543 df-card 9972 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-fzo 13670 df-seq 14009 df-hash 14332 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-hom 17266 df-cco 17267 df-0g 17432 df-gsum 17433 df-prds 17438 df-pws 17440 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 df-submnd 18750 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19038 df-subg 19092 df-ghm 19182 df-cntz 19282 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-srg 20141 df-ring 20189 df-cring 20190 df-rhm 20425 df-subrng 20497 df-subrg 20522 df-lmod 20759 df-lss 20830 df-lsp 20870 df-assa 21801 df-asp 21802 df-ascl 21803 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-evls 22035 df-evl 22036 df-psr1 22117 df-ply1 22119 df-evl1 22254 |
This theorem is referenced by: (None) |
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