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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 2821 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | rrgss 20059 | . . . . . . 7 ⊢ 𝐸 ⊆ (Base‘𝑅) |
4 | 3 | sseli 3962 | . . . . . 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 2821 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | 5, 6, 2, 7, 8, 9 | coe1sclmul 20444 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘𝑅) ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺))) |
11 | 4, 10 | syl3an2 1160 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺))) |
12 | 11 | oveq1d 7165 | . . . 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 2821 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | nn0ex 11897 | . . . . . 6 ⊢ ℕ0 ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ℕ0 ∈ V) |
16 | simp1 1132 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ Ring) | |
17 | simp2 1133 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐸) | |
18 | eqid 2821 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
19 | 18, 6, 5, 2 | coe1f 20373 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
20 | 19 | 3ad2ant3 1131 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
21 | 1, 2, 9, 13, 15, 16, 17, 20 | rrgsupp 20058 | . . . 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 2856 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)) = ((coe1‘𝐺) supp (0g‘𝑅))) |
23 | 22 | supeq1d 8904 | . 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 20410 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
25 | 24 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑃 ∈ Ring) |
26 | 5, 7, 2, 6 | ply1sclf 20447 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘𝑅)⟶𝐵) |
27 | 26 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐴:(Base‘𝑅)⟶𝐵) |
28 | 4 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑅)) |
29 | 27, 28 | ffvelrnd 6846 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐴‘𝐹) ∈ 𝐵) |
30 | simp3 1134 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
31 | 6, 8 | ringcl 19305 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐴‘𝐹) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
32 | 25, 29, 30, 31 | syl3anc 1367 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
33 | deg1mul3.d | . . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
34 | eqid 2821 | . . . 4 ⊢ (coe1‘((𝐴‘𝐹) · 𝐺)) = (coe1‘((𝐴‘𝐹) · 𝐺)) | |
35 | 33, 5, 6, 13, 34 | deg1val 24684 | . . 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 24684 | . . 3 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
38 | 37 | 3ad2ant3 1131 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
39 | 23, 36, 38 | 3eqtr4d 2866 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4560 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∘f cof 7401 supp csupp 7824 supcsup 8898 ℝ*cxr 10668 < clt 10669 ℕ0cn0 11891 Basecbs 16477 .rcmulr 16560 0gc0g 16707 Ringcrg 19291 RLRegcrlreg 20046 algSccascl 20078 Poly1cpl1 20339 coe1cco1 20340 deg1 cdg1 24642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-lmod 19630 df-lss 19698 df-rlreg 20050 df-ascl 20081 df-psr 20130 df-mvr 20131 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-vr1 20343 df-ply1 20344 df-coe1 20345 df-cnfld 20540 df-mdeg 24643 df-deg1 24644 |
This theorem is referenced by: uc1pmon1p 24739 ig1peu 24759 |
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