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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsexpval | Structured version Visualization version GIF version |
Description: Polynomial evaluation builder for exponentiation. (Contributed by SN, 27-Jul-2024.) |
Ref | Expression |
---|---|
evlsaddval.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
evlsaddval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑈) |
evlsaddval.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsaddval.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsaddval.b | ⊢ 𝐵 = (Base‘𝑃) |
evlsaddval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑍) |
evlsaddval.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsaddval.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsaddval.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
evlsaddval.m | ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) |
evlsexpval.g | ⊢ ∙ = (.g‘(mulGrp‘𝑃)) |
evlsexpval.f | ⊢ ↑ = (.g‘(mulGrp‘𝑆)) |
evlsexpval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
evlsexpval | ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . . 4 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
2 | evlsaddval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
3 | 1, 2 | mgpbas 19954 | . . 3 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃)) |
4 | evlsexpval.g | . . 3 ⊢ ∙ = (.g‘(mulGrp‘𝑃)) | |
5 | evlsaddval.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
6 | evlsaddval.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
7 | evlsaddval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
8 | evlsaddval.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
9 | evlsaddval.p | . . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑈) | |
10 | evlsaddval.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
11 | eqid 2732 | . . . . . 6 ⊢ (𝑆 ↑s (𝐾 ↑m 𝐼)) = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
12 | evlsaddval.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
13 | 8, 9, 10, 11, 12 | evlsrhm 21582 | . . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
14 | 5, 6, 7, 13 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
15 | rhmrcl1 20207 | . . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑃 ∈ Ring) | |
16 | 1 | ringmgp 20022 | . . . 4 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd) |
17 | 14, 15, 16 | 3syl 18 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd) |
18 | evlsexpval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
19 | evlsaddval.m | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) | |
20 | 19 | simpld 495 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
21 | 3, 4, 17, 18, 20 | mulgnn0cld 18949 | . 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵) |
22 | eqid 2732 | . . . . 5 ⊢ (mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
23 | 8, 9, 1, 4, 10, 11, 22, 12, 2, 5, 6, 7, 18, 20 | evlspw 21587 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))) |
24 | 23 | fveq1d 6881 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))‘𝐴)) |
25 | eqid 2732 | . . . 4 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
26 | eqid 2732 | . . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
27 | eqid 2732 | . . . 4 ⊢ (.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) = (.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) | |
28 | evlsexpval.f | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑆)) | |
29 | 6 | crngringd 20029 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) |
30 | ovexd 7429 | . . . 4 ⊢ (𝜑 → (𝐾 ↑m 𝐼) ∈ V) | |
31 | 2, 25 | rhmf 20215 | . . . . . 6 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
32 | 14, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
33 | 32, 20 | ffvelcdmd 7073 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
34 | evlsaddval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
35 | 11, 25, 22, 26, 27, 28, 29, 30, 18, 33, 34 | pwsexpg 20099 | . . 3 ⊢ (𝜑 → ((𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))‘𝐴) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐴))) |
36 | 19 | simprd 496 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉) |
37 | 36 | oveq2d 7410 | . . 3 ⊢ (𝜑 → (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)) = (𝑁 ↑ 𝑉)) |
38 | 24, 35, 37 | 3eqtrd 2776 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)) |
39 | 21, 38 | jca 512 | 1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟶wf 6529 ‘cfv 6533 (class class class)co 7394 ↑m cmap 8805 ℕ0cn0 12456 Basecbs 17128 ↾s cress 17157 ↑s cpws 17376 Mndcmnd 18604 .gcmg 18924 mulGrpcmgp 19948 Ringcrg 20016 CRingccrg 20017 RingHom crh 20200 SubRingcsubrg 20310 mPoly cmpl 21392 evalSub ces 21564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7654 df-ofr 7655 df-om 7840 df-1st 7959 df-2nd 7960 df-supp 8131 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-map 8807 df-pm 8808 df-ixp 8877 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-fsupp 9347 df-sup 9421 df-oi 9489 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-z 12543 df-dec 12662 df-uz 12807 df-fz 13469 df-fzo 13612 df-seq 13951 df-hash 14275 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-hom 17205 df-cco 17206 df-0g 17371 df-gsum 17372 df-prds 17377 df-pws 17379 df-mre 17514 df-mrc 17515 df-acs 17517 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-mhm 18649 df-submnd 18650 df-grp 18799 df-minusg 18800 df-sbg 18801 df-mulg 18925 df-subg 18977 df-ghm 19058 df-cntz 19149 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-srg 19970 df-ring 20018 df-cring 20019 df-rnghom 20203 df-subrg 20312 df-lmod 20424 df-lss 20494 df-lsp 20534 df-assa 21343 df-asp 21344 df-ascl 21345 df-psr 21395 df-mvr 21396 df-mpl 21397 df-evls 21566 |
This theorem is referenced by: (None) |
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