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Mirrors > Home > MPE Home > Th. List > evls1scasrng | Structured version Visualization version GIF version |
Description: The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 13-Sep-2019.) |
Ref | Expression |
---|---|
evls1scasrng.q | β’ π = (π evalSub1 π ) |
evls1scasrng.o | β’ π = (eval1βπ) |
evls1scasrng.w | β’ π = (Poly1βπ) |
evls1scasrng.u | β’ π = (π βΎs π ) |
evls1scasrng.p | β’ π = (Poly1βπ) |
evls1scasrng.b | β’ π΅ = (Baseβπ) |
evls1scasrng.a | β’ π΄ = (algScβπ) |
evls1scasrng.c | β’ πΆ = (algScβπ) |
evls1scasrng.s | β’ (π β π β CRing) |
evls1scasrng.r | β’ (π β π β (SubRingβπ)) |
evls1scasrng.x | β’ (π β π β π ) |
Ref | Expression |
---|---|
evls1scasrng | β’ (π β (πβ(π΄βπ)) = (πβ(πΆβπ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1scasrng.c | . . . . . 6 β’ πΆ = (algScβπ) | |
2 | evls1scasrng.p | . . . . . . . 8 β’ π = (Poly1βπ) | |
3 | evls1scasrng.s | . . . . . . . . . 10 β’ (π β π β CRing) | |
4 | evls1scasrng.b | . . . . . . . . . . . 12 β’ π΅ = (Baseβπ) | |
5 | 4 | ressid 17210 | . . . . . . . . . . 11 β’ (π β CRing β (π βΎs π΅) = π) |
6 | 5 | eqcomd 2733 | . . . . . . . . . 10 β’ (π β CRing β π = (π βΎs π΅)) |
7 | 3, 6 | syl 17 | . . . . . . . . 9 β’ (π β π = (π βΎs π΅)) |
8 | 7 | fveq2d 6895 | . . . . . . . 8 β’ (π β (Poly1βπ) = (Poly1β(π βΎs π΅))) |
9 | 2, 8 | eqtrid 2779 | . . . . . . 7 β’ (π β π = (Poly1β(π βΎs π΅))) |
10 | 9 | fveq2d 6895 | . . . . . 6 β’ (π β (algScβπ) = (algScβ(Poly1β(π βΎs π΅)))) |
11 | 1, 10 | eqtrid 2779 | . . . . 5 β’ (π β πΆ = (algScβ(Poly1β(π βΎs π΅)))) |
12 | 11 | fveq1d 6893 | . . . 4 β’ (π β (πΆβπ) = ((algScβ(Poly1β(π βΎs π΅)))βπ)) |
13 | 12 | fveq2d 6895 | . . 3 β’ (π β ((π evalSub1 π΅)β(πΆβπ)) = ((π evalSub1 π΅)β((algScβ(Poly1β(π βΎs π΅)))βπ))) |
14 | eqid 2727 | . . . 4 β’ (π evalSub1 π΅) = (π evalSub1 π΅) | |
15 | eqid 2727 | . . . 4 β’ (Poly1β(π βΎs π΅)) = (Poly1β(π βΎs π΅)) | |
16 | eqid 2727 | . . . 4 β’ (π βΎs π΅) = (π βΎs π΅) | |
17 | eqid 2727 | . . . 4 β’ (algScβ(Poly1β(π βΎs π΅))) = (algScβ(Poly1β(π βΎs π΅))) | |
18 | crngring 20169 | . . . . 5 β’ (π β CRing β π β Ring) | |
19 | 4 | subrgid 20494 | . . . . 5 β’ (π β Ring β π΅ β (SubRingβπ)) |
20 | 3, 18, 19 | 3syl 18 | . . . 4 β’ (π β π΅ β (SubRingβπ)) |
21 | evls1scasrng.r | . . . . . 6 β’ (π β π β (SubRingβπ)) | |
22 | 4 | subrgss 20493 | . . . . . 6 β’ (π β (SubRingβπ) β π β π΅) |
23 | 21, 22 | syl 17 | . . . . 5 β’ (π β π β π΅) |
24 | evls1scasrng.x | . . . . 5 β’ (π β π β π ) | |
25 | 23, 24 | sseldd 3979 | . . . 4 β’ (π β π β π΅) |
26 | 14, 15, 16, 4, 17, 3, 20, 25 | evls1sca 22216 | . . 3 β’ (π β ((π evalSub1 π΅)β((algScβ(Poly1β(π βΎs π΅)))βπ)) = (π΅ Γ {π})) |
27 | 13, 26 | eqtrd 2767 | . 2 β’ (π β ((π evalSub1 π΅)β(πΆβπ)) = (π΅ Γ {π})) |
28 | evls1scasrng.o | . . . . 5 β’ π = (eval1βπ) | |
29 | 28, 4 | evl1fval1 22224 | . . . 4 β’ π = (π evalSub1 π΅) |
30 | 29 | a1i 11 | . . 3 β’ (π β π = (π evalSub1 π΅)) |
31 | 30 | fveq1d 6893 | . 2 β’ (π β (πβ(πΆβπ)) = ((π evalSub1 π΅)β(πΆβπ))) |
32 | evls1scasrng.q | . . 3 β’ π = (π evalSub1 π ) | |
33 | evls1scasrng.w | . . 3 β’ π = (Poly1βπ) | |
34 | evls1scasrng.u | . . 3 β’ π = (π βΎs π ) | |
35 | evls1scasrng.a | . . 3 β’ π΄ = (algScβπ) | |
36 | 32, 33, 34, 4, 35, 3, 21, 24 | evls1sca 22216 | . 2 β’ (π β (πβ(π΄βπ)) = (π΅ Γ {π})) |
37 | 27, 31, 36 | 3eqtr4rd 2778 | 1 β’ (π β (πβ(π΄βπ)) = (πβ(πΆβπ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wss 3944 {csn 4624 Γ cxp 5670 βcfv 6542 (class class class)co 7414 Basecbs 17165 βΎs cress 17194 Ringcrg 20157 CRingccrg 20158 SubRingcsubrg 20488 algSccascl 21766 Poly1cpl1 22070 evalSub1 ces1 22206 eval1ce1 22207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-sup 9451 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-hom 17242 df-cco 17243 df-0g 17408 df-gsum 17409 df-prds 17414 df-pws 17416 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-ghm 19152 df-cntz 19252 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-srg 20111 df-ring 20159 df-cring 20160 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-lmod 20727 df-lss 20798 df-lsp 20838 df-assa 21767 df-asp 21768 df-ascl 21769 df-psr 21822 df-mvr 21823 df-mpl 21824 df-opsr 21826 df-evls 21996 df-evl 21997 df-psr1 22073 df-ply1 22075 df-evls1 22208 df-evl1 22209 |
This theorem is referenced by: (None) |
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