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Mirrors > Home > MPE Home > Th. List > evlsvarpw | Structured version Visualization version GIF version |
Description: Polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by SN, 21-Feb-2024.) |
Ref | Expression |
---|---|
evlsvarpw.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsvarpw.w | ⊢ 𝑊 = (𝐼 mPoly 𝑈) |
evlsvarpw.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evlsvarpw.e | ⊢ ↑ = (.g‘𝐺) |
evlsvarpw.x | ⊢ 𝑋 = ((𝐼 mVar 𝑈)‘𝑌) |
evlsvarpw.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsvarpw.p | ⊢ 𝑃 = (𝑆 ↑s (𝐵 ↑m 𝐼)) |
evlsvarpw.h | ⊢ 𝐻 = (mulGrp‘𝑃) |
evlsvarpw.b | ⊢ 𝐵 = (Base‘𝑆) |
evlsvarpw.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsvarpw.y | ⊢ (𝜑 → 𝑌 ∈ 𝐼) |
evlsvarpw.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsvarpw.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsvarpw.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
evlsvarpw | ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsvarpw.q | . 2 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
2 | evlsvarpw.w | . 2 ⊢ 𝑊 = (𝐼 mPoly 𝑈) | |
3 | evlsvarpw.g | . 2 ⊢ 𝐺 = (mulGrp‘𝑊) | |
4 | evlsvarpw.e | . 2 ⊢ ↑ = (.g‘𝐺) | |
5 | evlsvarpw.u | . 2 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
6 | evlsvarpw.p | . 2 ⊢ 𝑃 = (𝑆 ↑s (𝐵 ↑m 𝐼)) | |
7 | evlsvarpw.h | . 2 ⊢ 𝐻 = (mulGrp‘𝑃) | |
8 | evlsvarpw.b | . 2 ⊢ 𝐵 = (Base‘𝑆) | |
9 | eqid 2820 | . 2 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
10 | evlsvarpw.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
11 | evlsvarpw.s | . 2 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
12 | evlsvarpw.r | . 2 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
13 | evlsvarpw.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
14 | evlsvarpw.x | . . 3 ⊢ 𝑋 = ((𝐼 mVar 𝑈)‘𝑌) | |
15 | eqid 2820 | . . . 4 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar 𝑈) | |
16 | 5 | subrgring 19534 | . . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring) |
17 | 12, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ Ring) |
18 | evlsvarpw.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐼) | |
19 | 2, 15, 9, 10, 17, 18 | mvrcl 20225 | . . 3 ⊢ (𝜑 → ((𝐼 mVar 𝑈)‘𝑌) ∈ (Base‘𝑊)) |
20 | 14, 19 | eqeltrid 2916 | . 2 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑊)) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20 | evlspw 20302 | 1 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘𝐻)(𝑄‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6352 (class class class)co 7153 ↑m cmap 8403 ℕ0cn0 11895 Basecbs 16479 ↾s cress 16480 ↑s cpws 16716 .gcmg 18220 mulGrpcmgp 19235 Ringcrg 19293 CRingccrg 19294 SubRingcsubrg 19527 mVar cmvr 20128 mPoly cmpl 20129 evalSub ces 20280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-iin 4919 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-se 5512 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-isom 6361 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-of 7406 df-ofr 7407 df-om 7578 df-1st 7686 df-2nd 7687 df-supp 7828 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-2o 8100 df-oadd 8103 df-er 8286 df-map 8405 df-pm 8406 df-ixp 8459 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-fsupp 8831 df-sup 8903 df-oi 8971 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-3 11699 df-4 11700 df-5 11701 df-6 11702 df-7 11703 df-8 11704 df-9 11705 df-n0 11896 df-z 11980 df-dec 12097 df-uz 12242 df-fz 12891 df-fzo 13032 df-seq 13368 df-hash 13689 df-struct 16481 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-mulr 16575 df-sca 16577 df-vsca 16578 df-ip 16579 df-tset 16580 df-ple 16581 df-ds 16583 df-hom 16585 df-cco 16586 df-0g 16711 df-gsum 16712 df-prds 16717 df-pws 16719 df-mre 16853 df-mrc 16854 df-acs 16856 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-mhm 17952 df-submnd 17953 df-grp 18102 df-minusg 18103 df-sbg 18104 df-mulg 18221 df-subg 18272 df-ghm 18352 df-cntz 18443 df-cmn 18904 df-abl 18905 df-mgp 19236 df-ur 19248 df-srg 19252 df-ring 19295 df-cring 19296 df-rnghom 19463 df-subrg 19529 df-lmod 19632 df-lss 19700 df-lsp 19740 df-assa 20081 df-asp 20082 df-ascl 20083 df-psr 20132 df-mvr 20133 df-mpl 20134 df-evls 20282 |
This theorem is referenced by: (None) |
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