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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 21424 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑅 = (Scalar‘𝑃)) |
| 3 | 2 | adantr 481 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃)) |
| 4 | 3 | fveq2d 6778 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
| 5 | 4 | oveq1d 7290 | . . . 4 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → ((1r‘𝑅)( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋))) |
| 6 | nzrring 20532 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
| 7 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑅 ∈ Ring) |
| 8 | 1 | ply1lmod 21423 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
| 9 | 7, 8 | syl 17 | . . . . 5 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑃 ∈ LMod) |
| 10 | 1 | ply1ring 21419 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
| 11 | deg1pw.n | . . . . . . . 8 ⊢ 𝑁 = (mulGrp‘𝑃) | |
| 12 | 11 | ringmgp 19789 | . . . . . . 7 ⊢ (𝑃 ∈ Ring → 𝑁 ∈ Mnd) |
| 13 | 7, 10, 12 | 3syl 18 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑁 ∈ Mnd) |
| 14 | simpr 485 | . . . . . 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 21388 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
| 18 | 7, 17 | syl 17 | . . . . . 6 ⊢ ((𝑅 ∈ NzRing ∧ 𝐹 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑃)) |
| 19 | 11, 16 | mgpbas 19726 | . . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑁) |
| 20 | deg1pw.e | . . . . . . 7 ⊢ ↑ = (.g‘𝑁) | |
| 21 | 19, 20 | mulgnn0cl 18720 | . . . . . 6 ⊢ ((𝑁 ∈ Mnd ∧ 𝐹 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑃)) → (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) |
| 22 | 13, 14, 18, 21 | syl3anc 1370 | . . . . 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 20151 | . . . . 5 ⊢ ((𝑃 ∈ LMod ∧ (𝐹 ↑ 𝑋) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝐹 ↑ 𝑋)) = (𝐹 ↑ 𝑋)) |
| 27 | 9, 22, 26 | syl2anc 584 | . . . 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 6778 | . 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 19807 | . . . 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 20531 | . . . 4 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
| 36 | 35 | adantr 481 | . . 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 25283 | . . 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 1374 | . 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 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6433 (class class class)co 7275 ℕ0cn0 12233 Basecbs 16912 Scalarcsca 16965 ·𝑠 cvsca 16966 0gc0g 17150 Mndcmnd 18385 .gcmg 18700 mulGrpcmgp 19720 1rcur 19737 Ringcrg 19783 LModclmod 20123 NzRingcnzr 20528 var1cv1 21347 Poly1cpl1 21348 deg1 cdg1 25216 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-gsum 17153 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-ghm 18832 df-cntz 18923 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-subrg 20022 df-lmod 20125 df-lss 20194 df-nzr 20529 df-cnfld 20598 df-psr 21112 df-mvr 21113 df-mpl 21114 df-opsr 21116 df-psr1 21351 df-vr1 21352 df-ply1 21353 df-coe1 21354 df-mdeg 25217 df-deg1 25218 |
| This theorem is referenced by: ply1remlem 25327 lgsqrlem4 26497 idomrootle 41020 |
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