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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1divalg3 | Structured version Visualization version GIF version |
Description: Uniqueness of polynomial remainder: convert the subtraction in ply1divalg2 26193 to addition. (Contributed by SN, 20-Jun-2025.) |
Ref | Expression |
---|---|
ply1divalg3.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1divalg3.d | ⊢ 𝐷 = (deg1‘𝑅) |
ply1divalg3.b | ⊢ 𝐵 = (Base‘𝑃) |
ply1divalg3.m | ⊢ + = (+g‘𝑃) |
ply1divalg3.t | ⊢ ∙ = (.r‘𝑃) |
ply1divalg3.c | ⊢ 𝐶 = (Unic1p‘𝑅) |
ply1divalg3.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ply1divalg3.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
ply1divalg3.g | ⊢ (𝜑 → 𝐺 ∈ 𝐶) |
Ref | Expression |
---|---|
ply1divalg3 | ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 + (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1divalg3.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
2 | ply1divalg3.d | . . . 4 ⊢ 𝐷 = (deg1‘𝑅) | |
3 | ply1divalg3.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
4 | eqid 2735 | . . . 4 ⊢ (-g‘𝑃) = (-g‘𝑃) | |
5 | eqid 2735 | . . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃) | |
6 | ply1divalg3.t | . . . 4 ⊢ ∙ = (.r‘𝑃) | |
7 | ply1divalg3.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | ply1divalg3.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
9 | ply1divalg3.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐶) | |
10 | ply1divalg3.c | . . . . . 6 ⊢ 𝐶 = (Unic1p‘𝑅) | |
11 | 1, 3, 10 | uc1pcl 26198 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵) |
12 | 9, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
13 | 1, 5, 10 | uc1pn0 26200 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ≠ (0g‘𝑃)) |
14 | 9, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ≠ (0g‘𝑃)) |
15 | eqid 2735 | . . . . . 6 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
16 | 2, 15, 10 | uc1pldg 26203 | . . . . 5 ⊢ (𝐺 ∈ 𝐶 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Unit‘𝑅)) |
17 | 9, 16 | syl 17 | . . . 4 ⊢ (𝜑 → ((coe1‘𝐺)‘(𝐷‘𝐺)) ∈ (Unit‘𝑅)) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 12, 14, 17, 15 | ply1divalg2 26193 | . . 3 ⊢ (𝜑 → ∃!𝑝 ∈ 𝐵 (𝐷‘(𝐹(-g‘𝑃)(𝑝 ∙ 𝐺))) < (𝐷‘𝐺)) |
19 | eqid 2735 | . . . . 5 ⊢ (invg‘𝑃) = (invg‘𝑃) | |
20 | 1 | ply1ring 22265 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring) |
21 | 7, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Ring) |
22 | 21 | ringgrpd 20260 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Grp) |
23 | 22 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Grp) |
24 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑞 ∈ 𝐵) | |
25 | 3, 19, 23, 24 | grpinvcld 19019 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((invg‘𝑃)‘𝑞) ∈ 𝐵) |
26 | 3, 19, 22 | grpinvf1o 19040 | . . . . . 6 ⊢ (𝜑 → (invg‘𝑃):𝐵–1-1-onto→𝐵) |
27 | f1ofveu 7425 | . . . . . 6 ⊢ (((invg‘𝑃):𝐵–1-1-onto→𝐵 ∧ 𝑝 ∈ 𝐵) → ∃!𝑞 ∈ 𝐵 ((invg‘𝑃)‘𝑞) = 𝑝) | |
28 | 26, 27 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → ∃!𝑞 ∈ 𝐵 ((invg‘𝑃)‘𝑞) = 𝑝) |
29 | eqcom 2742 | . . . . . 6 ⊢ (𝑝 = ((invg‘𝑃)‘𝑞) ↔ ((invg‘𝑃)‘𝑞) = 𝑝) | |
30 | 29 | reubii 3387 | . . . . 5 ⊢ (∃!𝑞 ∈ 𝐵 𝑝 = ((invg‘𝑃)‘𝑞) ↔ ∃!𝑞 ∈ 𝐵 ((invg‘𝑃)‘𝑞) = 𝑝) |
31 | 28, 30 | sylibr 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → ∃!𝑞 ∈ 𝐵 𝑝 = ((invg‘𝑃)‘𝑞)) |
32 | oveq1 7438 | . . . . . . 7 ⊢ (𝑝 = ((invg‘𝑃)‘𝑞) → (𝑝 ∙ 𝐺) = (((invg‘𝑃)‘𝑞) ∙ 𝐺)) | |
33 | 32 | oveq2d 7447 | . . . . . 6 ⊢ (𝑝 = ((invg‘𝑃)‘𝑞) → (𝐹(-g‘𝑃)(𝑝 ∙ 𝐺)) = (𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺))) |
34 | 33 | fveq2d 6911 | . . . . 5 ⊢ (𝑝 = ((invg‘𝑃)‘𝑞) → (𝐷‘(𝐹(-g‘𝑃)(𝑝 ∙ 𝐺))) = (𝐷‘(𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺)))) |
35 | 34 | breq1d 5158 | . . . 4 ⊢ (𝑝 = ((invg‘𝑃)‘𝑞) → ((𝐷‘(𝐹(-g‘𝑃)(𝑝 ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺))) < (𝐷‘𝐺))) |
36 | 25, 31, 35 | reuxfr1ds 3760 | . . 3 ⊢ (𝜑 → (∃!𝑝 ∈ 𝐵 (𝐷‘(𝐹(-g‘𝑃)(𝑝 ∙ 𝐺))) < (𝐷‘𝐺) ↔ ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺))) < (𝐷‘𝐺))) |
37 | 18, 36 | mpbid 232 | . 2 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺))) < (𝐷‘𝐺)) |
38 | 21 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝑃 ∈ Ring) |
39 | 12 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → 𝐺 ∈ 𝐵) |
40 | 3, 6, 38, 25, 39 | ringcld 20277 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (((invg‘𝑃)‘𝑞) ∙ 𝐺) ∈ 𝐵) |
41 | ply1divalg3.m | . . . . . . . 8 ⊢ + = (+g‘𝑃) | |
42 | 3, 41, 19, 4 | grpsubval 19016 | . . . . . . 7 ⊢ ((𝐹 ∈ 𝐵 ∧ (((invg‘𝑃)‘𝑞) ∙ 𝐺) ∈ 𝐵) → (𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺)) = (𝐹 + ((invg‘𝑃)‘(((invg‘𝑃)‘𝑞) ∙ 𝐺)))) |
43 | 8, 40, 42 | syl2an2r 685 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺)) = (𝐹 + ((invg‘𝑃)‘(((invg‘𝑃)‘𝑞) ∙ 𝐺)))) |
44 | 3, 6, 19, 38, 24, 39 | ringmneg1 20318 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (((invg‘𝑃)‘𝑞) ∙ 𝐺) = ((invg‘𝑃)‘(𝑞 ∙ 𝐺))) |
45 | 44 | fveq2d 6911 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((invg‘𝑃)‘(((invg‘𝑃)‘𝑞) ∙ 𝐺)) = ((invg‘𝑃)‘((invg‘𝑃)‘(𝑞 ∙ 𝐺)))) |
46 | 3, 6, 38, 24, 39 | ringcld 20277 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝑞 ∙ 𝐺) ∈ 𝐵) |
47 | 3, 19 | grpinvinv 19036 | . . . . . . . . 9 ⊢ ((𝑃 ∈ Grp ∧ (𝑞 ∙ 𝐺) ∈ 𝐵) → ((invg‘𝑃)‘((invg‘𝑃)‘(𝑞 ∙ 𝐺))) = (𝑞 ∙ 𝐺)) |
48 | 22, 46, 47 | syl2an2r 685 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((invg‘𝑃)‘((invg‘𝑃)‘(𝑞 ∙ 𝐺))) = (𝑞 ∙ 𝐺)) |
49 | 45, 48 | eqtrd 2775 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((invg‘𝑃)‘(((invg‘𝑃)‘𝑞) ∙ 𝐺)) = (𝑞 ∙ 𝐺)) |
50 | 49 | oveq2d 7447 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹 + ((invg‘𝑃)‘(((invg‘𝑃)‘𝑞) ∙ 𝐺))) = (𝐹 + (𝑞 ∙ 𝐺))) |
51 | 43, 50 | eqtrd 2775 | . . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺)) = (𝐹 + (𝑞 ∙ 𝐺))) |
52 | 51 | fveq2d 6911 | . . . 4 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → (𝐷‘(𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺))) = (𝐷‘(𝐹 + (𝑞 ∙ 𝐺)))) |
53 | 52 | breq1d 5158 | . . 3 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐵) → ((𝐷‘(𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺))) < (𝐷‘𝐺) ↔ (𝐷‘(𝐹 + (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))) |
54 | 53 | reubidva 3394 | . 2 ⊢ (𝜑 → (∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹(-g‘𝑃)(((invg‘𝑃)‘𝑞) ∙ 𝐺))) < (𝐷‘𝐺) ↔ ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 + (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))) |
55 | 37, 54 | mpbid 232 | 1 ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 + (𝑞 ∙ 𝐺))) < (𝐷‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃!wreu 3376 class class class wbr 5148 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 < clt 11293 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 0gc0g 17486 Grpcgrp 18964 invgcminusg 18965 -gcsg 18966 Ringcrg 20251 Unitcui 20372 Poly1cpl1 22194 coe1cco1 22195 deg1cdg1 26108 Unic1pcuc1p 26181 |
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-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-uc1p 26186 |
This theorem is referenced by: r1peuqusdeg1 35628 |
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