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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > evls1fvcl | Structured version Visualization version GIF version |
Description: Variant of fveval1fvcl 22207 for the subring evaluation function evalSub1 (Contributed by Thierry Arnoux, 22-Mar-2025.) |
Ref | Expression |
---|---|
evls1maprhm.q | β’ π = (π evalSub1 π) |
evls1maprhm.p | β’ π = (Poly1β(π βΎs π)) |
evls1maprhm.b | β’ π΅ = (Baseβπ ) |
evls1maprhm.u | β’ π = (Baseβπ) |
evls1maprhm.r | β’ (π β π β CRing) |
evls1maprhm.s | β’ (π β π β (SubRingβπ )) |
evls1fvcl.1 | β’ (π β π β π΅) |
evls1fvcl.2 | β’ (π β π β π) |
Ref | Expression |
---|---|
evls1fvcl | β’ (π β ((πβπ)βπ) β π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1maprhm.q | . . . . . 6 β’ π = (π evalSub1 π) | |
2 | evls1maprhm.b | . . . . . 6 β’ π΅ = (Baseβπ ) | |
3 | evls1maprhm.p | . . . . . 6 β’ π = (Poly1β(π βΎs π)) | |
4 | eqid 2726 | . . . . . 6 β’ (π βΎs π) = (π βΎs π) | |
5 | evls1maprhm.u | . . . . . 6 β’ π = (Baseβπ) | |
6 | eqid 2726 | . . . . . 6 β’ (eval1βπ ) = (eval1βπ ) | |
7 | evls1maprhm.r | . . . . . 6 β’ (π β π β CRing) | |
8 | evls1maprhm.s | . . . . . 6 β’ (π β π β (SubRingβπ )) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ressply1evl 33156 | . . . . 5 β’ (π β π = ((eval1βπ ) βΎ π)) |
10 | 9 | fveq1d 6887 | . . . 4 β’ (π β (πβπ) = (((eval1βπ ) βΎ π)βπ)) |
11 | evls1fvcl.2 | . . . . 5 β’ (π β π β π) | |
12 | 11 | fvresd 6905 | . . . 4 β’ (π β (((eval1βπ ) βΎ π)βπ) = ((eval1βπ )βπ)) |
13 | 10, 12 | eqtrd 2766 | . . 3 β’ (π β (πβπ) = ((eval1βπ )βπ)) |
14 | 13 | fveq1d 6887 | . 2 β’ (π β ((πβπ)βπ) = (((eval1βπ )βπ)βπ)) |
15 | eqid 2726 | . . 3 β’ (Poly1βπ ) = (Poly1βπ ) | |
16 | eqid 2726 | . . 3 β’ (Baseβ(Poly1βπ )) = (Baseβ(Poly1βπ )) | |
17 | evls1fvcl.1 | . . 3 β’ (π β π β π΅) | |
18 | eqid 2726 | . . . . . 6 β’ ((Poly1βπ ) βΎs π) = ((Poly1βπ ) βΎs π) | |
19 | 15, 4, 3, 5, 8, 18 | ressply1bas 22102 | . . . . 5 β’ (π β π = (Baseβ((Poly1βπ ) βΎs π))) |
20 | 18, 16 | ressbasss 17192 | . . . . 5 β’ (Baseβ((Poly1βπ ) βΎs π)) β (Baseβ(Poly1βπ )) |
21 | 19, 20 | eqsstrdi 4031 | . . . 4 β’ (π β π β (Baseβ(Poly1βπ ))) |
22 | 21, 11 | sseldd 3978 | . . 3 β’ (π β π β (Baseβ(Poly1βπ ))) |
23 | 6, 15, 2, 16, 7, 17, 22 | fveval1fvcl 22207 | . 2 β’ (π β (((eval1βπ )βπ)βπ) β π΅) |
24 | 14, 23 | eqeltrd 2827 | 1 β’ (π β ((πβπ)βπ) β π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βΎ cres 5671 βcfv 6537 (class class class)co 7405 Basecbs 17153 βΎs cress 17182 CRingccrg 20139 SubRingcsubrg 20469 Poly1cpl1 22051 evalSub1 ces1 22187 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-vr1 22055 df-ply1 22056 df-coe1 22057 df-evls1 22189 df-evl1 22190 |
This theorem is referenced by: minplyirred 33290 irredminply 33293 |
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