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Mirrors > Home > MPE Home > Th. List > deg1submon1p | Structured version Visualization version GIF version |
Description: The difference of two monic polynomials of the same degree is a polynomial of lesser degree. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
Ref | Expression |
---|---|
deg1submon1p.d | ⊢ 𝐷 = (deg1‘𝑅) |
deg1submon1p.o | ⊢ 𝑂 = (Monic1p‘𝑅) |
deg1submon1p.p | ⊢ 𝑃 = (Poly1‘𝑅) |
deg1submon1p.m | ⊢ − = (-g‘𝑃) |
deg1submon1p.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
deg1submon1p.f1 | ⊢ (𝜑 → 𝐹 ∈ 𝑂) |
deg1submon1p.f2 | ⊢ (𝜑 → (𝐷‘𝐹) = 𝑋) |
deg1submon1p.g1 | ⊢ (𝜑 → 𝐺 ∈ 𝑂) |
deg1submon1p.g2 | ⊢ (𝜑 → (𝐷‘𝐺) = 𝑋) |
Ref | Expression |
---|---|
deg1submon1p | ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | deg1submon1p.d | . 2 ⊢ 𝐷 = (deg1‘𝑅) | |
2 | deg1submon1p.p | . 2 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | eqid 2735 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | deg1submon1p.m | . 2 ⊢ − = (-g‘𝑃) | |
5 | deg1submon1p.f2 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) = 𝑋) | |
6 | deg1submon1p.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
7 | deg1submon1p.f1 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑂) | |
8 | deg1submon1p.o | . . . . . 6 ⊢ 𝑂 = (Monic1p‘𝑅) | |
9 | 2, 3, 8 | mon1pcl 26199 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → 𝐹 ∈ (Base‘𝑃)) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘𝑃)) |
11 | eqid 2735 | . . . . . 6 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
12 | 2, 11, 8 | mon1pn0 26201 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → 𝐹 ≠ (0g‘𝑃)) |
13 | 7, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ≠ (0g‘𝑃)) |
14 | 1, 2, 11, 3 | deg1nn0cl 26142 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ (0g‘𝑃)) → (𝐷‘𝐹) ∈ ℕ0) |
15 | 6, 10, 13, 14 | syl3anc 1370 | . . 3 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0) |
16 | 5, 15 | eqeltrrd 2840 | . 2 ⊢ (𝜑 → 𝑋 ∈ ℕ0) |
17 | 16 | nn0red 12586 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ) |
18 | 17 | leidd 11827 | . . 3 ⊢ (𝜑 → 𝑋 ≤ 𝑋) |
19 | 5, 18 | eqbrtrd 5170 | . 2 ⊢ (𝜑 → (𝐷‘𝐹) ≤ 𝑋) |
20 | deg1submon1p.g1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑂) | |
21 | 2, 3, 8 | mon1pcl 26199 | . . 3 ⊢ (𝐺 ∈ 𝑂 → 𝐺 ∈ (Base‘𝑃)) |
22 | 20, 21 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝑃)) |
23 | deg1submon1p.g2 | . . 3 ⊢ (𝜑 → (𝐷‘𝐺) = 𝑋) | |
24 | 23, 18 | eqbrtrd 5170 | . 2 ⊢ (𝜑 → (𝐷‘𝐺) ≤ 𝑋) |
25 | eqid 2735 | . 2 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
26 | eqid 2735 | . 2 ⊢ (coe1‘𝐺) = (coe1‘𝐺) | |
27 | 5 | fveq2d 6911 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = ((coe1‘𝐹)‘𝑋)) |
28 | eqid 2735 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
29 | 1, 28, 8 | mon1pldg 26204 | . . . . 5 ⊢ (𝐹 ∈ 𝑂 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = (1r‘𝑅)) |
30 | 7, 29 | syl 17 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐹)‘(𝐷‘𝐹)) = (1r‘𝑅)) |
31 | 27, 30 | eqtr3d 2777 | . . 3 ⊢ (𝜑 → ((coe1‘𝐹)‘𝑋) = (1r‘𝑅)) |
32 | 1, 28, 8 | mon1pldg 26204 | . . . 4 ⊢ (𝐺 ∈ 𝑂 → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (1r‘𝑅)) |
33 | 20, 32 | syl 17 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) = (1r‘𝑅)) |
34 | 23 | fveq2d 6911 | . . 3 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) = ((coe1‘𝐺)‘𝑋)) |
35 | 31, 33, 34 | 3eqtr2d 2781 | . 2 ⊢ (𝜑 → ((coe1‘𝐹)‘𝑋) = ((coe1‘𝐺)‘𝑋)) |
36 | 1, 2, 3, 4, 16, 6, 10, 19, 22, 24, 25, 26, 35 | deg1sublt 26164 | 1 ⊢ (𝜑 → (𝐷‘(𝐹 − 𝐺)) < 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 < clt 11293 ≤ cle 11294 ℕ0cn0 12524 Basecbs 17245 0gc0g 17486 -gcsg 18966 1rcur 20199 Ringcrg 20251 Poly1cpl1 22194 coe1cco1 22195 deg1cdg1 26108 Monic1pcmn1 26180 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-lmod 20877 df-lss 20948 df-cnfld 21383 df-psr 21947 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-ply1 22199 df-coe1 22200 df-mdeg 26109 df-deg1 26110 df-mon1 26185 |
This theorem is referenced by: ig1peu 26229 |
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