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Mirrors > Home > MPE Home > Th. List > deg1mul3 | Structured version Visualization version GIF version |
Description: Degree of multiplication of a polynomial on the left by a nonzero-dividing scalar. (Contributed by Stefan O'Rear, 29-Mar-2015.) (Proof shortened by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
deg1mul3.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
deg1mul3.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1mul3.e | ⊢ 𝐸 = (RLReg‘𝑅) |
deg1mul3.b | ⊢ 𝐵 = (Base‘𝑃) |
deg1mul3.t | ⊢ · = (.r‘𝑃) |
deg1mul3.a | ⊢ 𝐴 = (algSc‘𝑃) |
Ref | Expression |
---|---|
deg1mul3 | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1mul3.e | . . . . . . . 8 ⊢ 𝐸 = (RLReg‘𝑅) | |
2 | eqid 2622 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | rrgss 19292 | . . . . . . 7 ⊢ 𝐸 ⊆ (Base‘𝑅) |
4 | 3 | sseli 3599 | . . . . . 6 ⊢ (𝐹 ∈ 𝐸 → 𝐹 ∈ (Base‘𝑅)) |
5 | deg1mul3.p | . . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | deg1mul3.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑃) | |
7 | deg1mul3.a | . . . . . . 7 ⊢ 𝐴 = (algSc‘𝑃) | |
8 | deg1mul3.t | . . . . . . 7 ⊢ · = (.r‘𝑃) | |
9 | eqid 2622 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | 5, 6, 2, 7, 8, 9 | coe1sclmul 19652 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘𝑅) ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘𝑓 (.r‘𝑅)(coe1‘𝐺))) |
11 | 4, 10 | syl3an2 1360 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘𝑓 (.r‘𝑅)(coe1‘𝐺))) |
12 | 11 | oveq1d 6665 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)) = (((ℕ0 × {𝐹}) ∘𝑓 (.r‘𝑅)(coe1‘𝐺)) supp (0g‘𝑅))) |
13 | eqid 2622 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | nn0ex 11298 | . . . . . 6 ⊢ ℕ0 ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ℕ0 ∈ V) |
16 | simp1 1061 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ Ring) | |
17 | simp2 1062 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐸) | |
18 | eqid 2622 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
19 | 18, 6, 5, 2 | coe1f 19581 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
20 | 19 | 3ad2ant3 1084 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
21 | 1, 2, 9, 13, 15, 16, 17, 20 | rrgsupp 19291 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (((ℕ0 × {𝐹}) ∘𝑓 (.r‘𝑅)(coe1‘𝐺)) supp (0g‘𝑅)) = ((coe1‘𝐺) supp (0g‘𝑅))) |
22 | 12, 21 | eqtrd 2656 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)) = ((coe1‘𝐺) supp (0g‘𝑅))) |
23 | 22 | supeq1d 8352 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → sup(((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)), ℝ*, < ) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
24 | 5 | ply1ring 19618 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
25 | 24 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑃 ∈ Ring) |
26 | 5, 7, 2, 6 | ply1sclf 19655 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘𝑅)⟶𝐵) |
27 | 26 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐴:(Base‘𝑅)⟶𝐵) |
28 | 4 | 3ad2ant2 1083 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑅)) |
29 | 27, 28 | ffvelrnd 6360 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐴‘𝐹) ∈ 𝐵) |
30 | simp3 1063 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
31 | 6, 8 | ringcl 18561 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐴‘𝐹) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
32 | 25, 29, 30, 31 | syl3anc 1326 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
33 | deg1mul3.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
34 | eqid 2622 | . . . 4 ⊢ (coe1‘((𝐴‘𝐹) · 𝐺)) = (coe1‘((𝐴‘𝐹) · 𝐺)) | |
35 | 33, 5, 6, 13, 34 | deg1val 23856 | . . 3 ⊢ (((𝐴‘𝐹) · 𝐺) ∈ 𝐵 → (𝐷‘((𝐴‘𝐹) · 𝐺)) = sup(((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)), ℝ*, < )) |
36 | 32, 35 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = sup(((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)), ℝ*, < )) |
37 | 33, 5, 6, 13, 18 | deg1val 23856 | . . 3 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
38 | 37 | 3ad2ant3 1084 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
39 | 23, 36, 38 | 3eqtr4d 2666 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 supp csupp 7295 supcsup 8346 ℝ*cxr 10073 < clt 10074 ℕ0cn0 11292 Basecbs 15857 .rcmulr 15942 0gc0g 16100 Ringcrg 18547 RLRegcrlreg 19279 algSccascl 19311 Poly1cpl1 19547 coe1cco1 19548 deg1 cdg1 23814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-gsum 16103 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-lmod 18865 df-lss 18933 df-rlreg 19283 df-ascl 19314 df-psr 19356 df-mvr 19357 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-vr1 19551 df-ply1 19552 df-coe1 19553 df-cnfld 19747 df-mdeg 23815 df-deg1 23816 |
This theorem is referenced by: uc1pmon1p 23911 ig1peu 23931 |
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