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Mirrors > Home > MPE Home > Th. List > deg1pw | Structured version Visualization version GIF version |
Description: Exact degree of a variable power over a nontrivial ring. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
Ref | Expression |
---|---|
deg1pw.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1pw.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1pw.x | ⊢ 𝑋 = (var1‘𝑅) |
deg1pw.n | ⊢ 𝑁 = (mulGrp‘𝑃) |
deg1pw.e | ⊢ ↑ = (.g‘𝑁) |
Ref | Expression |
---|---|
deg1pw | ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1pw.p | . . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1sca 20424 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑅 = (Scalar‘𝑃)) |
3 | 2 | adantr 483 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃)) |
4 | 3 | fveq2d 6677 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
5 | 4 | oveq1d 7174 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) |
6 | nzrring 20037 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
7 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑅 ∈ Ring) |
8 | 1 | ply1lmod 20423 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑃 ∈ LMod) |
10 | 1 | ply1ring 20419 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
11 | deg1pw.n | . . . . . . . 8 ⊢ 𝑁 = (mulGrp‘𝑃) | |
12 | 11 | ringmgp 19306 | . . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑁 ∈ Mnd) |
13 | 7, 10, 12 | 3syl 18 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑁 ∈ Mnd) |
14 | simpr 487 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℕ0) | |
15 | deg1pw.x | . . . . . . . 8 ⊢ 𝑋 = (var1‘𝑅) | |
16 | eqid 2824 | . . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
17 | 15, 1, 16 | vr1cl 20388 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
18 | 7, 17 | syl 17 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃)) |
19 | 11, 16 | mgpbas 19248 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑁) |
20 | deg1pw.e | . . . . . . 7 ⊢ ↑ = (.g‘𝑁) | |
21 | 19, 20 | mulgnn0cl 18247 | . . . . . 6 ⊢ ((𝑁 ∈ Mnd ∧ 𝐹 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑃)) → (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) |
22 | 13, 14, 18, 21 | syl3anc 1367 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) |
23 | eqid 2824 | . . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
24 | eqid 2824 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
25 | eqid 2824 | . . . . . 6 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
26 | 16, 23, 24, 25 | lmodvs1 19665 | . . . . 5 ⊢ ((𝑃 ∈ LMod ∧ (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
27 | 9, 22, 26 | syl2anc 586 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
28 | 5, 27 | eqtrd 2859 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
29 | 28 | fveq2d 6677 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) = (𝐷‘(𝐹 ↑ 𝑋))) |
30 | eqid 2824 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
31 | eqid 2824 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
32 | 30, 31 | ringidcl 19321 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
33 | 7, 32 | syl 17 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (1r‘𝑅) ∈ (Base‘𝑅)) |
34 | eqid 2824 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
35 | 31, 34 | nzrnz 20036 | . . . 4 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
36 | 35 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (1r‘𝑅) ≠ (0g‘𝑅)) |
37 | deg1pw.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
38 | 37, 30, 1, 15, 24, 11, 20, 34 | deg1tm 24715 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ ((1r‘𝑅) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) = 𝐹) |
39 | 7, 33, 36, 14, 38 | syl121anc 1371 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) = 𝐹) |
40 | 29, 39 | eqtr3d 2861 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 (class class class)co 7159 ℕ0cn0 11900 Basecbs 16486 Scalarcsca 16571 ·𝑠 cvsca 16572 0gc0g 16716 Mndcmnd 17914 .gcmg 18227 mulGrpcmgp 19242 1rcur 19254 Ringcrg 19300 LModclmod 19637 NzRingcnzr 20033 var1cv1 20347 Poly1cpl1 20348 deg1 cdg1 24651 |
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 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-ofr 7413 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-gsum 16719 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-ghm 18359 df-cntz 18450 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-subrg 19536 df-lmod 19639 df-lss 19707 df-nzr 20034 df-psr 20139 df-mvr 20140 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-vr1 20352 df-ply1 20353 df-coe1 20354 df-cnfld 20549 df-mdeg 24652 df-deg1 24653 |
This theorem is referenced by: ply1remlem 24759 lgsqrlem4 25928 idomrootle 39801 |
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