Nearby theorems
|
|
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > evl1scad | Structured version Visualization version GIF version | ||
| Description: Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.) |
| Ref | Expression |
|---|---|
| evl1sca.o | β’ π = (eval1βπ ) |
| evl1sca.p | β’ π = (Poly1βπ ) |
| evl1sca.b | β’ π΅ = (Baseβπ ) |
| evl1sca.a | β’ π΄ = (algScβπ) |
| evl1scad.u | β’ π = (Baseβπ) |
| evl1scad.1 | β’ (π β π β CRing) |
| evl1scad.2 | β’ (π β π β π΅) |
| evl1scad.3 | β’ (π β π β π΅) |
| Ref | Expression |
|---|---|
| evl1scad | β’ (π β ((π΄βπ) β π β§ ((πβ(π΄βπ))βπ) = π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | evl1scad.1 | . . . 4 β’ (π β π β CRing) | |
| 2 | crngring 20150 | . . . 4 β’ (π β CRing β π β Ring) | |
| 3 | evl1sca.p | . . . . 5 β’ π = (Poly1βπ ) | |
| 4 | evl1sca.a | . . . . 5 β’ π΄ = (algScβπ) | |
| 5 | evl1sca.b | . . . . 5 β’ π΅ = (Baseβπ ) | |
| 6 | evl1scad.u | . . . . 5 β’ π = (Baseβπ) | |
| 7 | 3, 4, 5, 6 | ply1sclf 22159 | . . . 4 β’ (π β Ring β π΄:π΅βΆπ) |
| 8 | 1, 2, 7 | 3syl 18 | . . 3 β’ (π β π΄:π΅βΆπ) |
| 9 | evl1scad.2 | . . 3 β’ (π β π β π΅) | |
| 10 | 8, 9 | ffvelcdmd 7081 | . 2 β’ (π β (π΄βπ) β π) |
| 11 | evl1sca.o | . . . . . 6 β’ π = (eval1βπ ) | |
| 12 | 11, 3, 5, 4 | evl1sca 22208 | . . . . 5 β’ ((π β CRing β§ π β π΅) β (πβ(π΄βπ)) = (π΅ Γ {π})) |
| 13 | 1, 9, 12 | syl2anc 583 | . . . 4 β’ (π β (πβ(π΄βπ)) = (π΅ Γ {π})) |
| 14 | 13 | fveq1d 6887 | . . 3 β’ (π β ((πβ(π΄βπ))βπ) = ((π΅ Γ {π})βπ)) |
| 15 | evl1scad.3 | . . . 4 β’ (π β π β π΅) | |
| 16 | fvconst2g 7199 | . . . 4 β’ ((π β π΅ β§ π β π΅) β ((π΅ Γ {π})βπ) = π) | |
| 17 | 9, 15, 16 | syl2anc 583 | . . 3 β’ (π β ((π΅ Γ {π})βπ) = π) |
| 18 | 14, 17 | eqtrd 2766 | . 2 β’ (π β ((πβ(π΄βπ))βπ) = π) |
| 19 | 10, 18 | jca 511 | 1 β’ (π β ((π΄βπ) β π β§ ((πβ(π΄βπ))βπ) = π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {csn 4623 Γ cxp 5667 βΆwf 6533 βcfv 6537 Basecbs 17153 Ringcrg 20138 CRingccrg 20139 algSccascl 21747 Poly1cpl1 22051 eval1ce1 22188 |
| 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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| 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-rmo 3370 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-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 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-se 5625 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-srg 20092 df-ring 20140 df-cring 20141 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-lsp 20819 df-assa 21748 df-asp 21749 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-evls 21977 df-evl 21978 df-psr1 22054 df-ply1 22056 df-evl1 22190 |
| This theorem is referenced by: evl1vsd 22218 evl1gsumd 22231 ply1remlem 26054 idomrootle 26062 lgsqrlem1 27234 aks6d1c1p2 41486 aks6d1c1p3 41487 aks6d1c1p6 41491 evl1gprodd 41494 aks6d1c5lem1 41512 aks6d1c5lem2 41514 evl1at0 47344 evl1at1 47345 lineval 47347 |
| Copyright terms: Public domain | W3C validator |