rtelextdg2 - Mathbox for Thierry Arnoux

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Theorem rtelextdg2 33718

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.)

**Hypotheses**
Ref	Expression
rtelextdg2.1	⊢ 𝐾 = (𝐸 ↾_s 𝐹)
rtelextdg2.2	⊢ 𝐿 = (𝐸 ↾_s (𝐸 fldGen (𝐹 ∪ {𝑋})))
rtelextdg2.3	⊢ 0 = (0_g‘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 )

**Assertion**
Ref	Expression
rtelextdg2	⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))

**Proof of Theorem rtelextdg2**
Step	Hyp	Ref	Expression
1		rtelextdg2.5	. . . . . 6 ⊢ 𝑉 = (Base‘𝐸)
2		rtelextdg2.9	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ Field)
3	2	flddrngd 20763	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		rtelextdg2.10	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	1	sdrgss 20816	. . . . . . . 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 4834	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
9	6, 8	unssd 4215	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ 𝑉)
10	1, 3, 9	fldgenssid 33280	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝑋})))
11		ssun2 4202	. . . . . 6 ⊢ {𝑋} ⊆ (𝐹 ∪ {𝑋})
12		snidg 4682	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
13	7, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
14	11, 13	sselid 4006	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐹 ∪ {𝑋}))
15	10, 14	sseldd 4009	. . . 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 33678	. . . . . 6 ⊢ (𝜑 → 𝐿/_FldExt𝐾)
20		extdg1id 33676	. . . . . 6 ⊢ ((𝐿/_FldExt𝐾 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
21	19, 20	sylan 579	. . . . 5 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
22	21	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (Base‘𝐿) = (Base‘𝐾))
23	1, 3, 9	fldgenssv 33282	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝑋})) ⊆ 𝑉)
24	18, 1	ressbas2 17296	. . . . . 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 17296	. . . . . 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 2790	. . 3 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (𝐸 fldGen (𝐹 ∪ {𝑋})) = 𝐹)
31	16, 30	eleqtrd 2846	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝑋 ∈ 𝐹)
32		simpr 484	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 2) → (𝐿[:]𝐾) = 2)
33		1zzd 12674	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
34		2z 12675	. . . . . 6 ⊢ 2 ∈ ℤ
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
36		extdgcl 33669	. . . . . . . 8 ⊢ (𝐿/_FldExt𝐾 → (𝐿[:]𝐾) ∈ ℕ₀^*)
37	19, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀^*)
38		2nn0 12570	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
39	38	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℕ₀)
40		rtelextdg2.3	. . . . . . . 8 ⊢ 0 = (0_g‘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 2740	. . . . . . . 8 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
49		eqid 2740	. . . . . . . 8 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
50		eqid 2740	. . . . . . . 8 ⊢ (._r‘𝑃) = (._r‘𝑃)
51		eqid 2740	. . . . . . . 8 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
52		eqid 2740	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53		eqid 2740	. . . . . . . 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 33717	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2)
55		xnn0lenn0nn0 13307	. . . . . . 7 ⊢ (((𝐿[:]𝐾) ∈ ℕ₀^* ∧ 2 ∈ ℕ₀ ∧ (𝐿[:]𝐾) ≤ 2) → (𝐿[:]𝐾) ∈ ℕ₀)
56	37, 39, 54, 55	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀)
57	56	nn0zd 12665	. . . . 5 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℤ)
58		extdggt0 33670	. . . . . . 7 ⊢ (𝐿/_FldExt𝐾 → 0 < (𝐿[:]𝐾))
59	19, 58	syl 17	. . . . . 6 ⊢ (𝜑 → 0 < (𝐿[:]𝐾))
60		zgt0ge1 12697	. . . . . . 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 583	. . . . 5 ⊢ (𝜑 → 1 ≤ (𝐿[:]𝐾))
63	33, 35, 57, 62, 54	elfzd 13575	. . . 4 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ (1...2))
64		fz12pr 13641	. . . 4 ⊢ (1...2) = {1, 2}
65	63, 64	eleqtrdi 2854	. . 3 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ {1, 2})
66		elpri 4671	. . 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 32487	1 ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1537 ∈ wcel 2108 ∪ cun 3974 ⊆ wss 3976 {csn 4648 {cpr 4650 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 0cc0 11184 1c1 11185 < clt 11324 ≤ cle 11325 2c2 12348 ℕ₀cn0 12553 ℕ₀^*cxnn0 12625 ℤcz 12639 ...cfz 13567 Basecbs 17258 ↾_s cress 17287 +_gcplusg 17311 ._rcmulr 17312 0_gc0g 17499 ._gcmg 19107 mulGrpcmgp 20161 Fieldcfield 20752 SubDRingcsdrg 20809 algSccascl 21895 var₁cv1 22198 Poly₁cpl1 22199 fldGen cfldgen 33277 /_FldExtcfldext 33651 [:]cextdg 33654

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-reg 9661 ax-inf2 9710 ax-ac2 10532 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-rpss 7758 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-r1 9833 df-rank 9834 df-dju 9970 df-card 10008 df-acn 10011 df-ac 10185 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-ico 13413 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ocomp 17332 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-imas 17568 df-qus 17569 df-mre 17644 df-mrc 17645 df-mri 17646 df-acs 17647 df-proset 18365 df-drs 18366 df-poset 18383 df-ipo 18598 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-gim 19299 df-cntz 19357 df-oppg 19386 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-irred 20385 df-invr 20414 df-dvr 20427 df-rhm 20498 df-nzr 20539 df-subrng 20572 df-subrg 20597 df-rlreg 20716 df-domn 20717 df-idom 20718 df-drng 20753 df-field 20754 df-sdrg 20810 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lmhm 21044 df-lmim 21045 df-lmic 21046 df-lbs 21097 df-lvec 21125 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-lpidl 21355 df-lpir 21356 df-pid 21370 df-cnfld 21388 df-dsmm 21775 df-frlm 21790 df-uvc 21826 df-lindf 21849 df-linds 21850 df-assa 21896 df-asp 21897 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-evls 22121 df-evl 22122 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205 df-evls1 22340 df-evl1 22341 df-mdeg 26114 df-deg1 26115 df-mon1 26190 df-uc1p 26191 df-q1p 26192 df-r1p 26193 df-ig1p 26194 df-fldgen 33278 df-mxidl 33453 df-dim 33612 df-fldext 33655 df-extdg 33656 df-irng 33684 df-minply 33693

This theorem is referenced by: constrelextdg2 33737

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