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| Mirrors > Home > MPE Home > Th. List > q1peqb | Structured version Visualization version GIF version | ||
| Description: Characterizing property of the polynomial quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.) |
| Ref | Expression |
|---|---|
| q1pval.q | ⊢ 𝑄 = (quot1p‘𝑅) |
| q1pval.p | ⊢ 𝑃 = (Poly1‘𝑅) |
| q1pval.b | ⊢ 𝐵 = (Base‘𝑃) |
| q1pval.d | ⊢ 𝐷 = (deg1‘𝑅) |
| q1pval.m | ⊢ − = (-g‘𝑃) |
| q1pval.t | ⊢ · = (.r‘𝑃) |
| q1peqb.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
| Ref | Expression |
|---|---|
| q1peqb | ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝐹𝑄𝐺) = 𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3451 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ V) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) → 𝑋 ∈ V) |
| 3 | 2 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) → 𝑋 ∈ V)) |
| 4 | ovex 7391 | . . . 4 ⊢ (𝐹𝑄𝐺) ∈ V | |
| 5 | eleq1 2825 | . . . 4 ⊢ ((𝐹𝑄𝐺) = 𝑋 → ((𝐹𝑄𝐺) ∈ V ↔ 𝑋 ∈ V)) | |
| 6 | 4, 5 | mpbii 233 | . . 3 ⊢ ((𝐹𝑄𝐺) = 𝑋 → 𝑋 ∈ V) |
| 7 | 6 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝐹𝑄𝐺) = 𝑋 → 𝑋 ∈ V)) |
| 8 | simpr 484 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) ∧ 𝑋 ∈ V) → 𝑋 ∈ V) | |
| 9 | q1pval.p | . . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 10 | q1pval.d | . . . . . . . 8 ⊢ 𝐷 = (deg1‘𝑅) | |
| 11 | q1pval.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝑃) | |
| 12 | q1pval.m | . . . . . . . 8 ⊢ − = (-g‘𝑃) | |
| 13 | eqid 2737 | . . . . . . . 8 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
| 14 | q1pval.t | . . . . . . . 8 ⊢ · = (.r‘𝑃) | |
| 15 | simp1 1137 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝑅 ∈ Ring) | |
| 16 | simp2 1138 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐹 ∈ 𝐵) | |
| 17 | q1peqb.c | . . . . . . . . . 10 ⊢ 𝐶 = (Unic1p‘𝑅) | |
| 18 | 9, 11, 17 | uc1pcl 26121 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
| 19 | 18 | 3ad2ant3 1136 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ∈ 𝐵) |
| 20 | 9, 13, 17 | uc1pn0 26123 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ≠ (0g‘𝑃)) |
| 21 | 20 | 3ad2ant3 1136 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → 𝐺 ≠ (0g‘𝑃)) |
| 22 | eqid 2737 | . . . . . . . . . 10 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
| 23 | 10, 22, 17 | uc1pldg 26126 | . . . . . . . . 9 ⊢ (𝐺 ∈ 𝐶 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Unit‘𝑅)) |
| 24 | 23 | 3ad2ant3 1136 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Unit‘𝑅)) |
| 25 | 9, 10, 11, 12, 13, 14, 15, 16, 19, 21, 24, 22 | ply1divalg2 26116 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) |
| 26 | df-reu 3344 | . . . . . . 7 ⊢ (∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺) ↔ ∃!𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) | |
| 27 | 25, 26 | sylib 218 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ∃!𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) |
| 28 | 27 | adantr 480 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) ∧ 𝑋 ∈ V) → ∃!𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) |
| 29 | eleq1 2825 | . . . . . . 7 ⊢ (𝑞 = 𝑋 → (𝑞 ∈ 𝐵 ↔ 𝑋 ∈ 𝐵)) | |
| 30 | oveq1 7365 | . . . . . . . . . 10 ⊢ (𝑞 = 𝑋 → (𝑞 · 𝐺) = (𝑋 · 𝐺)) | |
| 31 | 30 | oveq2d 7374 | . . . . . . . . 9 ⊢ (𝑞 = 𝑋 → (𝐹 − (𝑞 · 𝐺)) = (𝐹 − (𝑋 · 𝐺))) |
| 32 | 31 | fveq2d 6836 | . . . . . . . 8 ⊢ (𝑞 = 𝑋 → (𝐷‘(𝐹 − (𝑞 · 𝐺))) = (𝐷‘(𝐹 − (𝑋 · 𝐺)))) |
| 33 | 32 | breq1d 5096 | . . . . . . 7 ⊢ (𝑞 = 𝑋 → ((𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺))) |
| 34 | 29, 33 | anbi12d 633 | . . . . . 6 ⊢ (𝑞 = 𝑋 → ((𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)))) |
| 35 | 34 | adantl 481 | . . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) ∧ 𝑋 ∈ V) ∧ 𝑞 = 𝑋) → ((𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)))) |
| 36 | 8, 28, 35 | iota2d 6478 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) ∧ 𝑋 ∈ V) → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) ↔ (℩𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) = 𝑋)) |
| 37 | q1pval.q | . . . . . . . . 9 ⊢ 𝑄 = (quot1p‘𝑅) | |
| 38 | 37, 9, 11, 10, 12, 14 | q1pval 26132 | . . . . . . . 8 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) |
| 39 | 16, 19, 38 | syl2anc 585 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) = (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) |
| 40 | df-riota 7315 | . . . . . . 7 ⊢ (℩𝑞 ∈ 𝐵 (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)) = (℩𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) | |
| 41 | 39, 40 | eqtrdi 2788 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝐹𝑄𝐺) = (℩𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))) |
| 42 | 41 | adantr 480 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) ∧ 𝑋 ∈ V) → (𝐹𝑄𝐺) = (℩𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺)))) |
| 43 | 42 | eqeq1d 2739 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) ∧ 𝑋 ∈ V) → ((𝐹𝑄𝐺) = 𝑋 ↔ (℩𝑞(𝑞 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑞 · 𝐺))) < (𝐷‘𝐺))) = 𝑋)) |
| 44 | 36, 43 | bitr4d 282 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) ∧ 𝑋 ∈ V) → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝐹𝑄𝐺) = 𝑋)) |
| 45 | 44 | ex 412 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → (𝑋 ∈ V → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝐹𝑄𝐺) = 𝑋))) |
| 46 | 3, 7, 45 | pm5.21ndd 379 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶) → ((𝑋 ∈ 𝐵 ∧ (𝐷‘(𝐹 − (𝑋 · 𝐺))) < (𝐷‘𝐺)) ↔ (𝐹𝑄𝐺) = 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∃!weu 2569 ≠ wne 2933 ∃!wreu 3341 Vcvv 3430 class class class wbr 5086 ℩cio 6444 ‘cfv 6490 ℩crio 7314 (class class class)co 7358 < clt 11168 Basecbs 17168 .rcmulr 17210 0gc0g 17391 -gcsg 18900 Ringcrg 20203 Unitcui 20324 Poly1cpl1 22149 coe1cco1 22150 deg1cdg1 26031 Unic1pcuc1p 26104 quot1pcq1p 26105 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-ghm 19177 df-cntz 19281 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-oppr 20306 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-subrng 20512 df-subrg 20536 df-rlreg 20660 df-lmod 20846 df-lss 20916 df-cnfld 21343 df-psr 21897 df-mvr 21898 df-mpl 21899 df-opsr 21901 df-psr1 22152 df-vr1 22153 df-ply1 22154 df-coe1 22155 df-mdeg 26032 df-deg1 26033 df-uc1p 26109 df-q1p 26110 |
| This theorem is referenced by: q1pcl 26134 r1pdeglt 26137 r1pid2 26139 dvdsq1p 26140 q1pdir 33683 q1pvsca 33684 r1pid2OLD 33689 irredminply 33881 aks6d1c5lem3 42587 |
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