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Mirrors > Home > MPE Home > Th. List > evls1pw | Structured version Visualization version GIF version |
Description: Univariate polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.) |
Ref | Expression |
---|---|
evls1pw.q | ⊢ 𝑄 = (𝑆 evalSub1 𝑅) |
evls1pw.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evls1pw.w | ⊢ 𝑊 = (Poly1‘𝑈) |
evls1pw.g | ⊢ 𝐺 = (mulGrp‘𝑊) |
evls1pw.k | ⊢ 𝐾 = (Base‘𝑆) |
evls1pw.b | ⊢ 𝐵 = (Base‘𝑊) |
evls1pw.e | ⊢ ↑ = (.g‘𝐺) |
evls1pw.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evls1pw.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evls1pw.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
evls1pw.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
evls1pw | ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s 𝐾)))(𝑄‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evls1pw.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
2 | evls1pw.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
3 | evls1pw.q | . . . . 5 ⊢ 𝑄 = (𝑆 evalSub1 𝑅) | |
4 | evls1pw.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
5 | eqid 2820 | . . . . 5 ⊢ (𝑆 ↑s 𝐾) = (𝑆 ↑s 𝐾) | |
6 | evls1pw.u | . . . . 5 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
7 | evls1pw.w | . . . . 5 ⊢ 𝑊 = (Poly1‘𝑈) | |
8 | 3, 4, 5, 6, 7 | evls1rhm 20478 | . . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾))) |
9 | 1, 2, 8 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾))) |
10 | evls1pw.g | . . . 4 ⊢ 𝐺 = (mulGrp‘𝑊) | |
11 | eqid 2820 | . . . 4 ⊢ (mulGrp‘(𝑆 ↑s 𝐾)) = (mulGrp‘(𝑆 ↑s 𝐾)) | |
12 | 10, 11 | rhmmhm 19467 | . . 3 ⊢ (𝑄 ∈ (𝑊 RingHom (𝑆 ↑s 𝐾)) → 𝑄 ∈ (𝐺 MndHom (mulGrp‘(𝑆 ↑s 𝐾)))) |
13 | 9, 12 | syl 17 | . 2 ⊢ (𝜑 → 𝑄 ∈ (𝐺 MndHom (mulGrp‘(𝑆 ↑s 𝐾)))) |
14 | evls1pw.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
15 | evls1pw.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
16 | evls1pw.b | . . . 4 ⊢ 𝐵 = (Base‘𝑊) | |
17 | 10, 16 | mgpbas 19238 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
18 | evls1pw.e | . . 3 ⊢ ↑ = (.g‘𝐺) | |
19 | eqid 2820 | . . 3 ⊢ (.g‘(mulGrp‘(𝑆 ↑s 𝐾))) = (.g‘(mulGrp‘(𝑆 ↑s 𝐾))) | |
20 | 17, 18, 19 | mhmmulg 18261 | . 2 ⊢ ((𝑄 ∈ (𝐺 MndHom (mulGrp‘(𝑆 ↑s 𝐾))) ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s 𝐾)))(𝑄‘𝑋))) |
21 | 13, 14, 15, 20 | syl3anc 1366 | 1 ⊢ (𝜑 → (𝑄‘(𝑁 ↑ 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s 𝐾)))(𝑄‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ℕ0cn0 11891 Basecbs 16476 ↾s cress 16477 ↑s cpws 16713 MndHom cmhm 17947 .gcmg 18217 mulGrpcmgp 19232 CRingccrg 19291 RingHom crh 19457 SubRingcsubrg 19524 Poly1cpl1 20338 evalSub1 ces1 20469 |
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 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
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 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-ofr 7403 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-oadd 8099 df-er 8282 df-map 8401 df-pm 8402 df-ixp 8455 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-fzo 13031 df-seq 13367 df-hash 13688 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-sca 16574 df-vsca 16575 df-ip 16576 df-tset 16577 df-ple 16578 df-ds 16580 df-hom 16582 df-cco 16583 df-0g 16708 df-gsum 16709 df-prds 16714 df-pws 16716 df-mre 16850 df-mrc 16851 df-acs 16853 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-mhm 17949 df-submnd 17950 df-grp 18099 df-minusg 18100 df-sbg 18101 df-mulg 18218 df-subg 18269 df-ghm 18349 df-cntz 18440 df-cmn 18901 df-abl 18902 df-mgp 19233 df-ur 19245 df-srg 19249 df-ring 19292 df-cring 19293 df-rnghom 19460 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-assa 20078 df-asp 20079 df-ascl 20080 df-psr 20129 df-mvr 20130 df-mpl 20131 df-opsr 20133 df-evls 20279 df-psr1 20341 df-ply1 20343 df-evls1 20471 |
This theorem is referenced by: evls1varpw 20483 |
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