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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 2738 | . . . . . 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 20902 | . . . . 5 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
10 | 1, 2, 3, 9 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼)))) |
11 | rhmrcl1 19593 | . . . 4 ⊢ (𝑄 ∈ (𝑃 RingHom (𝑆 ↑s (𝐾 ↑m 𝐼))) → 𝑃 ∈ Ring) | |
12 | eqid 2738 | . . . . 5 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃) | |
13 | 12 | ringmgp 19422 | . . . 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 498 | . . 3 ⊢ (𝜑 → 𝑀 ∈ 𝐵) |
18 | evlsaddval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑃) | |
19 | 12, 18 | mgpbas 19364 | . . . 4 ⊢ 𝐵 = (Base‘(mulGrp‘𝑃)) |
20 | evlsexpval.g | . . . 4 ⊢ ∙ = (.g‘(mulGrp‘𝑃)) | |
21 | 19, 20 | mulgnn0cl 18362 | . . 3 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ 𝐵) → (𝑁 ∙ 𝑀) ∈ 𝐵) |
22 | 14, 15, 17, 21 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝑁 ∙ 𝑀) ∈ 𝐵) |
23 | eqid 2738 | . . . . 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 20907 | . . . 4 ⊢ (𝜑 → (𝑄‘(𝑁 ∙ 𝑀)) = (𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))) |
25 | 24 | fveq1d 6676 | . . 3 ⊢ (𝜑 → ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = ((𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))‘𝐴)) |
26 | eqid 2738 | . . . 4 ⊢ (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) = (Base‘(𝑆 ↑s (𝐾 ↑m 𝐼))) | |
27 | eqid 2738 | . . . 4 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆) | |
28 | eqid 2738 | . . . 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 19429 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Ring) |
31 | ovexd 7205 | . . . 4 ⊢ (𝜑 → (𝐾 ↑m 𝐼) ∈ V) | |
32 | 18, 26 | rhmf 19600 | . . . . . 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 | ffvelrnd 6862 | . . . 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 39869 | . . 3 ⊢ (𝜑 → ((𝑁(.g‘(mulGrp‘(𝑆 ↑s (𝐾 ↑m 𝐼))))(𝑄‘𝑀))‘𝐴) = (𝑁 ↑ ((𝑄‘𝑀)‘𝐴))) |
37 | 16 | simprd 499 | . . . 4 ⊢ (𝜑 → ((𝑄‘𝑀)‘𝐴) = 𝑉) |
38 | 37 | oveq2d 7186 | . . 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 515 | 1 ⊢ (𝜑 → ((𝑁 ∙ 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 ∙ 𝑀))‘𝐴) = (𝑁 ↑ 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ↑m cmap 8437 ℕ0cn0 11976 Basecbs 16586 ↾s cress 16587 ↑s cpws 16823 Mndcmnd 18027 .gcmg 18342 mulGrpcmgp 19358 Ringcrg 19416 CRingccrg 19417 RingHom crh 19586 SubRingcsubrg 19650 mPoly cmpl 20719 evalSub ces 20884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-ofr 7426 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-fzo 13125 df-seq 13461 df-hash 13783 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-hom 16692 df-cco 16693 df-0g 16818 df-gsum 16819 df-prds 16824 df-pws 16826 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-submnd 18073 df-grp 18222 df-minusg 18223 df-sbg 18224 df-mulg 18343 df-subg 18394 df-ghm 18474 df-cntz 18565 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-srg 19375 df-ring 19418 df-cring 19419 df-rnghom 19589 df-subrg 19652 df-lmod 19755 df-lss 19823 df-lsp 19863 df-assa 20669 df-asp 20670 df-ascl 20671 df-psr 20722 df-mvr 20723 df-mpl 20724 df-evls 20886 |
This theorem is referenced by: (None) |
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