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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 | evlsaddval.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑍) | |
2 | evlsaddval.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
3 | evlsaddval.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
4 | evlsaddval.q | . . . . . 6 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
5 | evlsaddval.p | . . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑈) | |
6 | evlsaddval.u | . . . . . 6 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
7 | eqid 2733 | . . . . . 6 ⊢ (𝑆 ↑s (𝐾 ↑m 𝐼)) = (𝑆 ↑s (𝐾 ↑m 𝐼)) | |
8 | evlsaddval.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝑆) | |
9 | 4, 5, 6, 7, 8 | evlsrhm 21326 | . . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
10 | 1, 2, 3, 9 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
11 | rhmrcl1 19991 | . . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑃 ∈ Ring) | |
12 | eqid 2733 | . . . . 5 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
13 | 12 | ringmgp 19817 | . . . 4 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd) |
14 | 10, 11, 13 | 3syl 18 | . . 3 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd) |
15 | evlsexpval.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
16 | evlsaddval.m | . . . 4 ⊢ (𝜑 → (𝑀 ∈ 𝐵 ∧ ((𝑄‘𝑀)‘𝐴) = 𝑉)) | |
17 | 16 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
18 | evlsaddval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
19 | 12, 18 | mgpbas 19754 | . . . 4 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃)) |
20 | evlsexpval.g | . . . 4 ⊢ ∙ = (.g‘(mulGrp‘𝑃)) | |
21 | 19, 20 | mulgnn0cl 18748 | . . 3 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ 𝐵) → (𝑁 ∙ 𝑀) ∈ 𝐵) |
22 | 14, 15, 17, 21 | syl3anc 1369 | . 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵) |
23 | eqid 2733 | . . . . 5 ⊢ (mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
24 | 4, 5, 12, 20, 6, 7, 23, 8, 18, 1, 2, 3, 15, 17 | evlspw 21331 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))) |
25 | 24 | fveq1d 6794 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))‘𝐴)) |
26 | eqid 2733 | . . . 4 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
27 | eqid 2733 | . . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
28 | eqid 2733 | . . . 4 ⊢ (.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) = (.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) | |
29 | evlsexpval.f | . . . 4 ⊢ ↑ = (.g‘(mulGrp‘𝑆)) | |
30 | 2 | crngringd 19824 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) |
31 | ovexd 7330 | . . . 4 ⊢ (𝜑 → (𝐾 ↑m 𝐼) ∈ V) | |
32 | 18, 26 | rhmf 19998 | . . . . . 6 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
33 | 10, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑄:𝐵⟶(Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
34 | 33, 17 | ffvelcdmd 6982 | . . . 4 ⊢ (𝜑 → (𝑄‘𝑀) ∈ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼)))) |
35 | evlsaddval.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
36 | 7, 26, 23, 27, 28, 29, 30, 31, 15, 34, 35 | pwsexpg 40291 | . . 3 ⊢ (𝜑 → ((𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))‘𝐴) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐴))) |
37 | 16 | simprd 495 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉) |
38 | 37 | oveq2d 7311 | . . 3 ⊢ (𝜑 → (𝑁 ↑ ((𝑄‘𝑀)‘𝐴)) = (𝑁 ↑ 𝑉)) |
39 | 25, 36, 38 | 3eqtrd 2777 | . 2 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉)) |
40 | 22, 39 | jca 511 | 1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2101 Vcvv 3434 ⟶wf 6443 ‘cfv 6447 (class class class)co 7295 ↑m cmap 8635 ℕ0cn0 12261 Basecbs 16940 ↾s cress 16969 ↑s cpws 17185 Mndcmnd 18413 .gcmg 18728 mulGrpcmgp 19748 Ringcrg 19811 CRingccrg 19812 RingHom crh 19984 SubRingcsubrg 20048 mPoly cmpl 21137 evalSub ces 21308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-ofr 7554 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-er 8518 df-map 8637 df-pm 8638 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-sup 9229 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-fz 13268 df-fzo 13411 df-seq 13750 df-hash 14073 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-hom 17014 df-cco 17015 df-0g 17180 df-gsum 17181 df-prds 17186 df-pws 17188 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-mhm 18458 df-submnd 18459 df-grp 18608 df-minusg 18609 df-sbg 18610 df-mulg 18729 df-subg 18780 df-ghm 18860 df-cntz 18951 df-cmn 19416 df-abl 19417 df-mgp 19749 df-ur 19766 df-srg 19770 df-ring 19813 df-cring 19814 df-rnghom 19987 df-subrg 20050 df-lmod 20153 df-lss 20222 df-lsp 20262 df-assa 21088 df-asp 21089 df-ascl 21090 df-psr 21140 df-mvr 21141 df-mpl 21142 df-evls 21310 |
This theorem is referenced by: (None) |
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