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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 21334 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑅 = (Scalar‘𝑃)) |
| 3 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃)) |
| 4 | 3 | fveq2d 6760 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
| 5 | 4 | oveq1d 7270 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) |
| 6 | nzrring 20445 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
| 7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑅 ∈ Ring) |
| 8 | 1 | ply1lmod 21333 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
| 9 | 7, 8 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑃 ∈ LMod) |
| 10 | 1 | ply1ring 21329 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
| 11 | deg1pw.n | . . . . . . . 8 ⊢ 𝑁 = (mulGrp‘𝑃) | |
| 12 | 11 | ringmgp 19704 | . . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑁 ∈ Mnd) |
| 13 | 7, 10, 12 | 3syl 18 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑁 ∈ Mnd) |
| 14 | simpr 484 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝐹 ∈ ℕ0) | |
| 15 | deg1pw.x | . . . . . . . 8 ⊢ 𝑋 = (var1‘𝑅) | |
| 16 | eqid 2738 | . . . . . . . 8 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
| 17 | 15, 1, 16 | vr1cl 21298 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
| 18 | 7, 17 | syl 17 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃)) |
| 19 | 11, 16 | mgpbas 19641 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑁) |
| 20 | deg1pw.e | . . . . . . 7 ⊢ ↑ = (.g‘𝑁) | |
| 21 | 19, 20 | mulgnn0cl 18635 | . . . . . 6 ⊢ ((𝑁 ∈ Mnd ∧ 𝐹 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑃)) → (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) |
| 22 | 13, 14, 18, 21 | syl3anc 1369 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) |
| 23 | eqid 2738 | . . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
| 24 | eqid 2738 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
| 25 | eqid 2738 | . . . . . 6 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
| 26 | 16, 23, 24, 25 | lmodvs1 20066 | . . . . 5 ⊢ ((𝑃 ∈ LMod ∧ (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
| 27 | 9, 22, 26 | syl2anc 583 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
| 28 | 5, 27 | eqtrd 2778 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
| 29 | 28 | fveq2d 6760 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) = (𝐷‘(𝐹 ↑ 𝑋))) |
| 30 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 31 | eqid 2738 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 32 | 30, 31 | ringidcl 19722 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 33 | 7, 32 | syl 17 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (1r‘𝑅) ∈ (Base‘𝑅)) |
| 34 | eqid 2738 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 35 | 31, 34 | nzrnz 20444 | . . . 4 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
| 36 | 35 | adantr 480 | . . 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 25188 | . . 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 1373 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) = 𝐹) |
| 40 | 29, 39 | eqtr3d 2780 | 1 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (𝐷‘(𝐹 ↑ 𝑋)) = 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ‘cfv 6418 (class class class)co 7255 ℕ0cn0 12163 Basecbs 16840 Scalarcsca 16891 ·𝑠 cvsca 16892 0gc0g 17067 Mndcmnd 18300 .gcmg 18615 mulGrpcmgp 19635 1rcur 19652 Ringcrg 19698 LModclmod 20038 NzRingcnzr 20441 var1cv1 21257 Poly1cpl1 21258 deg1 cdg1 25121 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-gsum 17070 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-subrg 19937 df-lmod 20040 df-lss 20109 df-nzr 20442 df-cnfld 20511 df-psr 21022 df-mvr 21023 df-mpl 21024 df-opsr 21026 df-psr1 21261 df-vr1 21262 df-ply1 21263 df-coe1 21264 df-mdeg 25122 df-deg1 25123 |
| This theorem is referenced by: ply1remlem 25232 lgsqrlem4 26402 idomrootle 40936 |
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