Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > drnguc1p | Structured version Visualization version GIF version |
Description: Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
drnguc1p.p | ⊢ 𝑃 = (Poly1‘𝑅) |
drnguc1p.b | ⊢ 𝐵 = (Base‘𝑃) |
drnguc1p.z | ⊢ 0 = (0g‘𝑃) |
drnguc1p.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
Ref | Expression |
---|---|
drnguc1p | ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1132 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐵) | |
2 | simp3 1133 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐹 ≠ 0 ) | |
3 | eqid 2820 | . . . . . 6 ⊢ (coe1‘𝐹) = (coe1‘𝐹) | |
4 | drnguc1p.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑃) | |
5 | drnguc1p.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | 3, 4, 5, 6 | coe1f 20374 | . . . . 5 ⊢ (𝐹 ∈ 𝐵 → (coe1‘𝐹):ℕ0⟶(Base‘𝑅)) |
8 | 7 | 3ad2ant2 1129 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (coe1‘𝐹):ℕ0⟶(Base‘𝑅)) |
9 | drngring 19504 | . . . . 5 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring) | |
10 | eqid 2820 | . . . . . 6 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
11 | drnguc1p.z | . . . . . 6 ⊢ 0 = (0g‘𝑃) | |
12 | 10, 5, 11, 4 | deg1nn0cl 24680 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (( deg1 ‘𝑅)‘𝐹) ∈ ℕ0) |
13 | 9, 12 | syl3an1 1158 | . . . 4 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (( deg1 ‘𝑅)‘𝐹) ∈ ℕ0) |
14 | 8, 13 | ffvelrnd 6845 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Base‘𝑅)) |
15 | eqid 2820 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
16 | 10, 5, 11, 4, 15, 3 | deg1ldg 24684 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ≠ (0g‘𝑅)) |
17 | 9, 16 | syl3an1 1158 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ≠ (0g‘𝑅)) |
18 | eqid 2820 | . . . . 5 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
19 | 6, 18, 15 | drngunit 19502 | . . . 4 ⊢ (𝑅 ∈ DivRing → (((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅) ↔ (((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Base‘𝑅) ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ≠ (0g‘𝑅)))) |
20 | 19 | 3ad2ant1 1128 | . . 3 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅) ↔ (((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Base‘𝑅) ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ≠ (0g‘𝑅)))) |
21 | 14, 17, 20 | mpbir2and 711 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅)) |
22 | drnguc1p.c | . . 3 ⊢ 𝐶 = (Unic1p‘𝑅) | |
23 | 5, 4, 11, 10, 22, 18 | isuc1p 24732 | . 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅))) |
24 | 1, 2, 21, 23 | syl3anbrc 1338 | 1 ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → 𝐹 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ⟶wf 6344 ‘cfv 6348 ℕ0cn0 11891 Basecbs 16478 0gc0g 16708 Ringcrg 19292 Unitcui 19384 DivRingcdr 19497 Poly1cpl1 20340 coe1cco1 20341 deg1 cdg1 24646 Unic1pcuc1p 24718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-sup 8899 df-oi 8967 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 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 12890 df-fzo 13031 df-seq 13367 df-hash 13688 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-sca 16576 df-vsca 16577 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-0g 16710 df-gsum 16711 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-grp 18101 df-minusg 18102 df-mulg 18220 df-subg 18271 df-cntz 18442 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-cring 19295 df-drng 19499 df-psr 20131 df-mpl 20133 df-opsr 20135 df-psr1 20343 df-ply1 20345 df-coe1 20346 df-cnfld 20541 df-mdeg 24647 df-deg1 24648 df-uc1p 24723 |
This theorem is referenced by: ig1peu 24763 |
Copyright terms: Public domain | W3C validator |