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Mirrors > Home > MPE Home > Th. List > mon1pid | Structured version Visualization version GIF version |
Description: Monicity and degree of the unit polynomial. (Contributed by Stefan O'Rear, 12-Sep-2015.) |
Ref | Expression |
---|---|
mon1pid.p | ⊢ 𝑃 = (Poly1‘𝑅) |
mon1pid.o | ⊢ 1 = (1r‘𝑃) |
mon1pid.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
mon1pid.d | ⊢ 𝐷 = (deg1‘𝑅) |
Ref | Expression |
---|---|
mon1pid | ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mon1pid.p | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | 1 | ply1nz 26176 | . . . 4 ⊢ (𝑅 ∈ NzRing → 𝑃 ∈ NzRing) |
3 | nzrring 20533 | . . . 4 ⊢ (𝑃 ∈ NzRing → 𝑃 ∈ Ring) | |
4 | eqid 2735 | . . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | mon1pid.o | . . . . 5 ⊢ 1 = (1r‘𝑃) | |
6 | 4, 5 | ringidcl 20280 | . . . 4 ⊢ (𝑃 ∈ Ring → 1 ∈ (Base‘𝑃)) |
7 | 2, 3, 6 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ∈ (Base‘𝑃)) |
8 | eqid 2735 | . . . . 5 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
9 | 5, 8 | nzrnz 20532 | . . . 4 ⊢ (𝑃 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
10 | 2, 9 | syl 17 | . . 3 ⊢ (𝑅 ∈ NzRing → 1 ≠ (0g‘𝑃)) |
11 | nzrring 20533 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
12 | eqid 2735 | . . . . . . . . 9 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
13 | eqid 2735 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
14 | 1, 12, 13, 5 | ply1scl1 22312 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
15 | 11, 14 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → ((algSc‘𝑃)‘(1r‘𝑅)) = 1 ) |
16 | 15 | fveq2d 6911 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (coe1‘ 1 )) |
17 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
18 | 17, 13 | ringidcl 20280 | . . . . . . 7 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅)) |
19 | eqid 2735 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
20 | 1, 12, 17, 19 | coe1scl 22306 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅)) → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
21 | 11, 18, 20 | syl2anc2 585 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (coe1‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
22 | 16, 21 | eqtr3d 2777 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (coe1‘ 1 ) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))) |
23 | 15 | fveq2d 6911 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = (𝐷‘ 1 )) |
24 | 11, 18 | syl 17 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ∈ (Base‘𝑅)) |
25 | 13, 19 | nzrnz 20532 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
26 | mon1pid.d | . . . . . . . 8 ⊢ 𝐷 = (deg1‘𝑅) | |
27 | 26, 1, 17, 12, 19 | deg1scl 26167 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (1r‘𝑅) ∈ (Base‘𝑅) ∧ (1r‘𝑅) ≠ (0g‘𝑅)) → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
28 | 11, 24, 25, 27 | syl3anc 1370 | . . . . . 6 ⊢ (𝑅 ∈ NzRing → (𝐷‘((algSc‘𝑃)‘(1r‘𝑅))) = 0) |
29 | 23, 28 | eqtr3d 2777 | . . . . 5 ⊢ (𝑅 ∈ NzRing → (𝐷‘ 1 ) = 0) |
30 | 22, 29 | fveq12d 6914 | . . . 4 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0)) |
31 | 0nn0 12539 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
32 | iftrue 4537 | . . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)) = (1r‘𝑅)) | |
33 | eqid 2735 | . . . . . 6 ⊢ (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅))) | |
34 | fvex 6920 | . . . . . 6 ⊢ (1r‘𝑅) ∈ V | |
35 | 32, 33, 34 | fvmpt 7016 | . . . . 5 ⊢ (0 ∈ ℕ0 → ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0) = (1r‘𝑅)) |
36 | 31, 35 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, (1r‘𝑅), (0g‘𝑅)))‘0) = (1r‘𝑅) |
37 | 30, 36 | eqtrdi 2791 | . . 3 ⊢ (𝑅 ∈ NzRing → ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅)) |
38 | mon1pid.m | . . . 4 ⊢ 𝑀 = (Monic1p‘𝑅) | |
39 | 1, 4, 8, 26, 38, 13 | ismon1p 26197 | . . 3 ⊢ ( 1 ∈ 𝑀 ↔ ( 1 ∈ (Base‘𝑃) ∧ 1 ≠ (0g‘𝑃) ∧ ((coe1‘ 1 )‘(𝐷‘ 1 )) = (1r‘𝑅))) |
40 | 7, 10, 37, 39 | syl3anbrc 1342 | . 2 ⊢ (𝑅 ∈ NzRing → 1 ∈ 𝑀) |
41 | 40, 29 | jca 511 | 1 ⊢ (𝑅 ∈ NzRing → ( 1 ∈ 𝑀 ∧ (𝐷‘ 1 ) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ifcif 4531 ↦ cmpt 5231 ‘cfv 6563 0cc0 11153 ℕ0cn0 12524 Basecbs 17245 0gc0g 17486 1rcur 20199 Ringcrg 20251 NzRingcnzr 20529 algSccascl 21890 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-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-nzr 20530 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-cnfld 21383 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-mdeg 26109 df-deg1 26110 df-mon1 26185 |
This theorem is referenced by: ply1unit 33580 mon1psubm 43188 deg1mhm 43189 |
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