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Mirrors > Home > MPE Home > Th. List > coe1mul4 | Structured version Visualization version GIF version |
Description: Value of the "leading" coefficient of a product of two nonzero polynomials. This will fail to actually be the leading coefficient only if it is zero (requiring the basic ring to contain zero divisors). (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
coe1mul3.s | ⊢ 𝑌 = (Poly1‘𝑅) |
coe1mul3.t | ⊢ ∙ = (.r‘𝑌) |
coe1mul3.u | ⊢ · = (.r‘𝑅) |
coe1mul3.b | ⊢ 𝐵 = (Base‘𝑌) |
coe1mul3.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
coe1mul4.z | ⊢ 0 = (0g‘𝑌) |
coe1mul4.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
coe1mul4.f1 | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
coe1mul4.f2 | ⊢ (𝜑 → 𝐹 ≠ 0 ) |
coe1mul4.g1 | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
coe1mul4.g2 | ⊢ (𝜑 → 𝐺 ≠ 0 ) |
Ref | Expression |
---|---|
coe1mul4 | ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹)) · ((coe1‘𝐺)‘(𝐷‘𝐺)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coe1mul3.s | . 2 ⊢ 𝑌 = (Poly1‘𝑅) | |
2 | coe1mul3.t | . 2 ⊢ ∙ = (.r‘𝑌) | |
3 | coe1mul3.u | . 2 ⊢ · = (.r‘𝑅) | |
4 | coe1mul3.b | . 2 ⊢ 𝐵 = (Base‘𝑌) | |
5 | coe1mul3.d | . 2 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
6 | coe1mul4.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | coe1mul4.f1 | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
8 | coe1mul4.f2 | . . 3 ⊢ (𝜑 → 𝐹 ≠ 0 ) | |
9 | coe1mul4.z | . . . 4 ⊢ 0 = (0g‘𝑌) | |
10 | 5, 1, 9, 4 | deg1nn0cl 24285 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ0) |
11 | 6, 7, 8, 10 | syl3anc 1439 | . 2 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
12 | 11 | nn0red 11703 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℝ) |
13 | 12 | leidd 10941 | . 2 ⊢ (𝜑 → (𝐷‘𝐹) ≤ (𝐷‘𝐹)) |
14 | coe1mul4.g1 | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
15 | coe1mul4.g2 | . . 3 ⊢ (𝜑 → 𝐺 ≠ 0 ) | |
16 | 5, 1, 9, 4 | deg1nn0cl 24285 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵 ∧ 𝐺 ≠ 0 ) → (𝐷‘𝐺) ∈ ℕ0) |
17 | 6, 14, 15, 16 | syl3anc 1439 | . 2 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℕ0) |
18 | 17 | nn0red 11703 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) ∈ ℝ) |
19 | 18 | leidd 10941 | . 2 ⊢ (𝜑 → (𝐷‘𝐺) ≤ (𝐷‘𝐺)) |
20 | 1, 2, 3, 4, 5, 6, 7, 11, 13, 14, 17, 19 | coe1mul3 24296 | 1 ⊢ (𝜑 → ((coe1‘(𝐹 ∙ 𝐺))‘((𝐷‘𝐹) + (𝐷‘𝐺))) = (((coe1‘𝐹)‘(𝐷‘𝐹)) · ((coe1‘𝐺)‘(𝐷‘𝐺)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ‘cfv 6135 (class class class)co 6922 + caddc 10275 ℕ0cn0 11642 Basecbs 16255 .rcmulr 16339 0gc0g 16486 Ringcrg 18934 Poly1cpl1 19943 coe1cco1 19944 deg1 cdg1 24251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-gsum 16489 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-submnd 17722 df-grp 17812 df-minusg 17813 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-psr 19753 df-mpl 19755 df-opsr 19757 df-psr1 19946 df-ply1 19948 df-coe1 19949 df-cnfld 20143 df-mdeg 24252 df-deg1 24253 |
This theorem is referenced by: deg1mul2 24311 mon1psubm 38743 |
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