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Mirrors > Home > MPE Home > Th. List > evls1fvcl | Structured version Visualization version GIF version |
Description: Variant of fveval1fvcl 22271 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 2728 | . . . . . 6 β’ (π βΎs π) = (π βΎs π) | |
5 | evls1maprhm.u | . . . . . 6 β’ π = (Baseβπ) | |
6 | eqid 2728 | . . . . . 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 22308 | . . . . 5 β’ (π β π = ((eval1βπ ) βΎ π)) |
10 | 9 | fveq1d 6904 | . . . 4 β’ (π β (πβπ) = (((eval1βπ ) βΎ π)βπ)) |
11 | evls1fvcl.2 | . . . . 5 β’ (π β π β π) | |
12 | 11 | fvresd 6922 | . . . 4 β’ (π β (((eval1βπ ) βΎ π)βπ) = ((eval1βπ )βπ)) |
13 | 10, 12 | eqtrd 2768 | . . 3 β’ (π β (πβπ) = ((eval1βπ )βπ)) |
14 | 13 | fveq1d 6904 | . 2 β’ (π β ((πβπ)βπ) = (((eval1βπ )βπ)βπ)) |
15 | eqid 2728 | . . 3 β’ (Poly1βπ ) = (Poly1βπ ) | |
16 | eqid 2728 | . . 3 β’ (Baseβ(Poly1βπ )) = (Baseβ(Poly1βπ )) | |
17 | evls1fvcl.1 | . . 3 β’ (π β π β π΅) | |
18 | eqid 2728 | . . . . . 6 β’ ((Poly1βπ ) βΎs π) = ((Poly1βπ ) βΎs π) | |
19 | 15, 4, 3, 5, 8, 18 | ressply1bas 22166 | . . . . 5 β’ (π β π = (Baseβ((Poly1βπ ) βΎs π))) |
20 | 18, 16 | ressbasss 17228 | . . . . 5 β’ (Baseβ((Poly1βπ ) βΎs π)) β (Baseβ(Poly1βπ )) |
21 | 19, 20 | eqsstrdi 4036 | . . . 4 β’ (π β π β (Baseβ(Poly1βπ ))) |
22 | 21, 11 | sseldd 3983 | . . 3 β’ (π β π β (Baseβ(Poly1βπ ))) |
23 | 6, 15, 2, 16, 7, 17, 22 | fveval1fvcl 22271 | . 2 β’ (π β (((eval1βπ )βπ)βπ) β π΅) |
24 | 14, 23 | eqeltrd 2829 | 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 5684 βcfv 6553 (class class class)co 7426 Basecbs 17189 βΎs cress 17218 CRingccrg 20188 SubRingcsubrg 20520 Poly1cpl1 22114 evalSub1 ces1 22251 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-vr1 22118 df-ply1 22119 df-coe1 22120 df-evls1 22253 df-evl1 22254 |
This theorem is referenced by: minplyirred 33422 irredminply 33425 |
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