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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rtelextdg2 | Structured version Visualization version GIF version | ||
| Description: If an element 𝑋 is a solution of a quadratic equation, then it is either in the base field, or the degree of its field extension is exactly 2. (Contributed by Thierry Arnoux, 22-Jun-2025.) |
| Ref | Expression |
|---|---|
| rtelextdg2.1 | ⊢ 𝐾 = (𝐸 ↾s 𝐹) |
| rtelextdg2.2 | ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋}))) |
| rtelextdg2.3 | ⊢ 0 = (0g‘𝐸) |
| rtelextdg2.4 | ⊢ 𝑃 = (Poly1‘𝐾) |
| rtelextdg2.5 | ⊢ 𝑉 = (Base‘𝐸) |
| rtelextdg2.6 | ⊢ · = (.r‘𝐸) |
| rtelextdg2.7 | ⊢ + = (+g‘𝐸) |
| rtelextdg2.8 | ⊢ ↑ = (.g‘(mulGrp‘𝐸)) |
| rtelextdg2.9 | ⊢ (𝜑 → 𝐸 ∈ Field) |
| rtelextdg2.10 | ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) |
| rtelextdg2.11 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
| rtelextdg2.12 | ⊢ (𝜑 → 𝐴 ∈ 𝐹) |
| rtelextdg2.13 | ⊢ (𝜑 → 𝐵 ∈ 𝐹) |
| rtelextdg2.14 | ⊢ (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 ) |
| Ref | Expression |
|---|---|
| rtelextdg2 | ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rtelextdg2.5 | . . . . . 6 ⊢ 𝑉 = (Base‘𝐸) | |
| 2 | rtelextdg2.9 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field) | |
| 3 | 2 | flddrngd 20741 | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing) |
| 4 | rtelextdg2.10 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸)) | |
| 5 | 1 | sdrgss 20794 | . . . . . . . 8 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝑉) |
| 6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ⊆ 𝑉) |
| 7 | rtelextdg2.11 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 8 | 7 | snssd 4809 | . . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
| 9 | 6, 8 | unssd 4192 | . . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ 𝑉) |
| 10 | 1, 3, 9 | fldgenssid 33315 | . . . . 5 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝑋}))) |
| 11 | ssun2 4179 | . . . . . 6 ⊢ {𝑋} ⊆ (𝐹 ∪ {𝑋}) | |
| 12 | snidg 4660 | . . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋}) | |
| 13 | 7, 12 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑋}) |
| 14 | 11, 13 | sselid 3981 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐹 ∪ {𝑋})) |
| 15 | 10, 14 | sseldd 3984 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐸 fldGen (𝐹 ∪ {𝑋}))) |
| 16 | 15 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝑋 ∈ (𝐸 fldGen (𝐹 ∪ {𝑋}))) |
| 17 | rtelextdg2.1 | . . . . . . 7 ⊢ 𝐾 = (𝐸 ↾s 𝐹) | |
| 18 | rtelextdg2.2 | . . . . . . 7 ⊢ 𝐿 = (𝐸 ↾s (𝐸 fldGen (𝐹 ∪ {𝑋}))) | |
| 19 | 1, 17, 18, 2, 4, 8 | fldgenfldext 33718 | . . . . . 6 ⊢ (𝜑 → 𝐿/FldExt𝐾) |
| 20 | extdg1id 33716 | . . . . . 6 ⊢ ((𝐿/FldExt𝐾 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾) | |
| 21 | 19, 20 | sylan 580 | . . . . 5 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾) |
| 22 | 21 | fveq2d 6910 | . . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (Base‘𝐿) = (Base‘𝐾)) |
| 23 | 1, 3, 9 | fldgenssv 33317 | . . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝑋})) ⊆ 𝑉) |
| 24 | 18, 1 | ressbas2 17283 | . . . . . 6 ⊢ ((𝐸 fldGen (𝐹 ∪ {𝑋})) ⊆ 𝑉 → (𝐸 fldGen (𝐹 ∪ {𝑋})) = (Base‘𝐿)) |
| 25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝑋})) = (Base‘𝐿)) |
| 26 | 25 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (𝐸 fldGen (𝐹 ∪ {𝑋})) = (Base‘𝐿)) |
| 27 | 17, 1 | ressbas2 17283 | . . . . . 6 ⊢ (𝐹 ⊆ 𝑉 → 𝐹 = (Base‘𝐾)) |
| 28 | 6, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘𝐾)) |
| 29 | 28 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝐹 = (Base‘𝐾)) |
| 30 | 22, 26, 29 | 3eqtr4d 2787 | . . 3 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (𝐸 fldGen (𝐹 ∪ {𝑋})) = 𝐹) |
| 31 | 16, 30 | eleqtrd 2843 | . 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝑋 ∈ 𝐹) |
| 32 | simpr 484 | . 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 2) → (𝐿[:]𝐾) = 2) | |
| 33 | 1zzd 12648 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 34 | 2z 12649 | . . . . . 6 ⊢ 2 ∈ ℤ | |
| 35 | 34 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℤ) |
| 36 | extdgcl 33707 | . . . . . . . 8 ⊢ (𝐿/FldExt𝐾 → (𝐿[:]𝐾) ∈ ℕ0*) | |
| 37 | 19, 36 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ0*) |
| 38 | 2nn0 12543 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
| 39 | 38 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ0) |
| 40 | rtelextdg2.3 | . . . . . . . 8 ⊢ 0 = (0g‘𝐸) | |
| 41 | rtelextdg2.4 | . . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝐾) | |
| 42 | rtelextdg2.6 | . . . . . . . 8 ⊢ · = (.r‘𝐸) | |
| 43 | rtelextdg2.7 | . . . . . . . 8 ⊢ + = (+g‘𝐸) | |
| 44 | rtelextdg2.8 | . . . . . . . 8 ⊢ ↑ = (.g‘(mulGrp‘𝐸)) | |
| 45 | rtelextdg2.12 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝐹) | |
| 46 | rtelextdg2.13 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝐹) | |
| 47 | rtelextdg2.14 | . . . . . . . 8 ⊢ (𝜑 → ((2 ↑ 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 ) | |
| 48 | eqid 2737 | . . . . . . . 8 ⊢ (var1‘𝐾) = (var1‘𝐾) | |
| 49 | eqid 2737 | . . . . . . . 8 ⊢ (+g‘𝑃) = (+g‘𝑃) | |
| 50 | eqid 2737 | . . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃) | |
| 51 | eqid 2737 | . . . . . . . 8 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃)) | |
| 52 | eqid 2737 | . . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃) | |
| 53 | eqid 2737 | . . . . . . . 8 ⊢ ((2(.g‘(mulGrp‘𝑃))(var1‘𝐾))(+g‘𝑃)((((algSc‘𝑃)‘𝐴)(.r‘𝑃)(var1‘𝐾))(+g‘𝑃)((algSc‘𝑃)‘𝐵))) = ((2(.g‘(mulGrp‘𝑃))(var1‘𝐾))(+g‘𝑃)((((algSc‘𝑃)‘𝐴)(.r‘𝑃)(var1‘𝐾))(+g‘𝑃)((algSc‘𝑃)‘𝐵))) | |
| 54 | 17, 18, 40, 41, 1, 42, 43, 44, 2, 4, 7, 45, 46, 47, 48, 49, 50, 51, 52, 53 | rtelextdg2lem 33767 | . . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2) |
| 55 | xnn0lenn0nn0 13287 | . . . . . . 7 ⊢ (((𝐿[:]𝐾) ∈ ℕ0* ∧ 2 ∈ ℕ0 ∧ (𝐿[:]𝐾) ≤ 2) → (𝐿[:]𝐾) ∈ ℕ0) | |
| 56 | 37, 39, 54, 55 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ0) |
| 57 | 56 | nn0zd 12639 | . . . . 5 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℤ) |
| 58 | extdggt0 33708 | . . . . . . 7 ⊢ (𝐿/FldExt𝐾 → 0 < (𝐿[:]𝐾)) | |
| 59 | 19, 58 | syl 17 | . . . . . 6 ⊢ (𝜑 → 0 < (𝐿[:]𝐾)) |
| 60 | zgt0ge1 12672 | . . . . . . 7 ⊢ ((𝐿[:]𝐾) ∈ ℤ → (0 < (𝐿[:]𝐾) ↔ 1 ≤ (𝐿[:]𝐾))) | |
| 61 | 60 | biimpa 476 | . . . . . 6 ⊢ (((𝐿[:]𝐾) ∈ ℤ ∧ 0 < (𝐿[:]𝐾)) → 1 ≤ (𝐿[:]𝐾)) |
| 62 | 57, 59, 61 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 1 ≤ (𝐿[:]𝐾)) |
| 63 | 33, 35, 57, 62, 54 | elfzd 13555 | . . . 4 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ (1...2)) |
| 64 | fz12pr 13621 | . . . 4 ⊢ (1...2) = {1, 2} | |
| 65 | 63, 64 | eleqtrdi 2851 | . . 3 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ {1, 2}) |
| 66 | elpri 4649 | . . 3 ⊢ ((𝐿[:]𝐾) ∈ {1, 2} → ((𝐿[:]𝐾) = 1 ∨ (𝐿[:]𝐾) = 2)) | |
| 67 | 65, 66 | syl 17 | . 2 ⊢ (𝜑 → ((𝐿[:]𝐾) = 1 ∨ (𝐿[:]𝐾) = 2)) |
| 68 | 31, 32, 67 | orim12da 32477 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 < clt 11295 ≤ cle 11296 2c2 12321 ℕ0cn0 12526 ℕ0*cxnn0 12599 ℤcz 12613 ...cfz 13547 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 .rcmulr 17298 0gc0g 17484 .gcmg 19085 mulGrpcmgp 20137 Fieldcfield 20730 SubDRingcsdrg 20787 algSccascl 21872 var1cv1 22177 Poly1cpl1 22178 fldGen cfldgen 33312 /FldExtcfldext 33689 [:]cextdg 33692 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-rpss 7743 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-r1 9804 df-rank 9805 df-dju 9941 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ocomp 17318 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-imas 17553 df-qus 17554 df-mre 17629 df-mrc 17630 df-mri 17631 df-acs 17632 df-proset 18340 df-drs 18341 df-poset 18359 df-ipo 18573 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19231 df-gim 19277 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-irred 20359 df-invr 20388 df-dvr 20401 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-idom 20696 df-drng 20731 df-field 20732 df-sdrg 20788 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lmhm 21021 df-lmim 21022 df-lmic 21023 df-lbs 21074 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-2idl 21260 df-lpidl 21332 df-lpir 21333 df-pid 21347 df-cnfld 21365 df-dsmm 21752 df-frlm 21767 df-uvc 21803 df-lindf 21826 df-linds 21827 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-evls1 22319 df-evl1 22320 df-mdeg 26094 df-deg1 26095 df-mon1 26170 df-uc1p 26171 df-q1p 26172 df-r1p 26173 df-ig1p 26174 df-fldgen 33313 df-mxidl 33488 df-dim 33650 df-fldext 33693 df-extdg 33694 df-irng 33734 df-minply 33743 |
| This theorem is referenced by: constrelextdg2 33788 |
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