![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2740 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | rrgss 20724 | . . . . . . 7 ⊢ 𝐸 ⊆ (Base‘𝑅) |
4 | 3 | sseli 4004 | . . . . . 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 2740 | . . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
10 | 5, 6, 2, 7, 8, 9 | coe1sclmul 22306 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘𝑅) ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺))) |
11 | 4, 10 | syl3an2 1164 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘((𝐴‘𝐹) · 𝐺)) = ((ℕ0 × {𝐹}) ∘f (.r‘𝑅)(coe1‘𝐺))) |
12 | 11 | oveq1d 7463 | . . . 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 2740 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
14 | nn0ex 12559 | . . . . . 6 ⊢ ℕ0 ∈ V | |
15 | 14 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ℕ0 ∈ V) |
16 | simp1 1136 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ Ring) | |
17 | simp2 1137 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐸) | |
18 | eqid 2740 | . . . . . . 7 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
19 | 18, 6, 5, 2 | coe1f 22234 | . . . . . 6 ⊢ (𝐺 ∈ 𝐵 → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
20 | 19 | 3ad2ant3 1135 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (coe1‘𝐺):ℕ0⟶(Base‘𝑅)) |
21 | 1, 2, 9, 13, 15, 16, 17, 20 | rrgsupp 20723 | . . . 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 2780 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((coe1‘((𝐴‘𝐹) · 𝐺)) supp (0g‘𝑅)) = ((coe1‘𝐺) supp (0g‘𝑅))) |
23 | 22 | supeq1d 9515 | . 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 22270 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
25 | 24 | 3ad2ant1 1133 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝑃 ∈ Ring) |
26 | 5, 7, 2, 6 | ply1sclf 22309 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘𝑅)⟶𝐵) |
27 | 26 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐴:(Base‘𝑅)⟶𝐵) |
28 | 4 | 3ad2ant2 1134 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑅)) |
29 | 27, 28 | ffvelcdmd 7119 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐴‘𝐹) ∈ 𝐵) |
30 | simp3 1138 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵) | |
31 | 6, 8 | ringcl 20277 | . . . 4 ⊢ ((𝑃 ∈ Ring ∧ (𝐴‘𝐹) ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
32 | 25, 29, 30, 31 | syl3anc 1371 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → ((𝐴‘𝐹) · 𝐺) ∈ 𝐵) |
33 | deg1mul3.d | . . . 4 ⊢ 𝐷 = (deg1‘𝑅) | |
34 | eqid 2740 | . . . 4 ⊢ (coe1‘((𝐴‘𝐹) · 𝐺)) = (coe1‘((𝐴‘𝐹) · 𝐺)) | |
35 | 33, 5, 6, 13, 34 | deg1val 26155 | . . 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 26155 | . . 3 ⊢ (𝐺 ∈ 𝐵 → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
38 | 37 | 3ad2ant3 1135 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘𝐺) = sup(((coe1‘𝐺) supp (0g‘𝑅)), ℝ*, < )) |
39 | 23, 36, 38 | 3eqtr4d 2790 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐸 ∧ 𝐺 ∈ 𝐵) → (𝐷‘((𝐴‘𝐹) · 𝐺)) = (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 Vcvv 3488 {csn 4648 × cxp 5698 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∘f cof 7712 supp csupp 8201 supcsup 9509 ℝ*cxr 11323 < clt 11324 ℕ0cn0 12553 Basecbs 17258 .rcmulr 17312 0gc0g 17499 Ringcrg 20260 RLRegcrlreg 20713 algSccascl 21895 Poly1cpl1 22199 coe1cco1 22200 deg1cdg1 26113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-addf 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-subrng 20572 df-subrg 20597 df-rlreg 20716 df-lmod 20882 df-lss 20953 df-cnfld 21388 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205 df-mdeg 26114 df-deg1 26115 |
This theorem is referenced by: uc1pmon1p 26211 ig1peu 26234 |
Copyright terms: Public domain | W3C validator |