rtelextdg2 - Mathbox for Thierry Arnoux

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Theorem rtelextdg2 33717

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 20650	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		rtelextdg2.10	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	1	sdrgss 20702	. . . . . . . 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 4773	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
9	6, 8	unssd 4155	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ 𝑉)
10	1, 3, 9	fldgenssid 33263	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝑋})))
11		ssun2 4142	. . . . . 6 ⊢ {𝑋} ⊆ (𝐹 ∪ {𝑋})
12		snidg 4624	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
13	7, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
14	11, 13	sselid 3944	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐹 ∪ {𝑋}))
15	10, 14	sseldd 3947	. . . 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 33663	. . . . . 6 ⊢ (𝜑 → 𝐿/_FldExt𝐾)
20		extdg1id 33661	. . . . . 6 ⊢ ((𝐿/_FldExt𝐾 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
21	19, 20	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
22	21	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (Base‘𝐿) = (Base‘𝐾))
23	1, 3, 9	fldgenssv 33265	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝑋})) ⊆ 𝑉)
24	18, 1	ressbas2 17208	. . . . . 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 17208	. . . . . 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 2774	. . 3 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (𝐸 fldGen (𝐹 ∪ {𝑋})) = 𝐹)
31	16, 30	eleqtrd 2830	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝑋 ∈ 𝐹)
32		simpr 484	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 2) → (𝐿[:]𝐾) = 2)
33		1zzd 12564	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
34		2z 12565	. . . . . 6 ⊢ 2 ∈ ℤ
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
36		extdgcl 33652	. . . . . . . 8 ⊢ (𝐿/_FldExt𝐾 → (𝐿[:]𝐾) ∈ ℕ₀^*)
37	19, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀^*)
38		2nn0 12459	. . . . . . . 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 2729	. . . . . . . 8 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
49		eqid 2729	. . . . . . . 8 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
50		eqid 2729	. . . . . . . 8 ⊢ (._r‘𝑃) = (._r‘𝑃)
51		eqid 2729	. . . . . . . 8 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
52		eqid 2729	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53		eqid 2729	. . . . . . . 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 33716	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2)
55		xnn0lenn0nn0 13205	. . . . . . 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 12555	. . . . 5 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℤ)
58		extdggt0 33653	. . . . . . 7 ⊢ (𝐿/_FldExt𝐾 → 0 < (𝐿[:]𝐾))
59	19, 58	syl 17	. . . . . 6 ⊢ (𝜑 → 0 < (𝐿[:]𝐾))
60		zgt0ge1 12588	. . . . . . 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 13476	. . . 4 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ (1...2))
64		fz12pr 13542	. . . 4 ⊢ (1...2) = {1, 2}
65	63, 64	eleqtrdi 2838	. . 3 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ {1, 2})
66		elpri 4613	. . 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 32387	1 ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∪ cun 3912 ⊆ wss 3914 {csn 4589 {cpr 4591 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 < clt 11208 ≤ cle 11209 2c2 12241 ℕ₀cn0 12442 ℕ₀^*cxnn0 12515 ℤcz 12529 ...cfz 13468 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 ._rcmulr 17221 0_gc0g 17402 ._gcmg 18999 mulGrpcmgp 20049 Fieldcfield 20639 SubDRingcsdrg 20695 algSccascl 21761 var₁cv1 22060 Poly₁cpl1 22061 fldGen cfldgen 33260 /_FldExtcfldext 33634 [:]cextdg 33636

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-reg 9545 ax-inf2 9594 ax-ac2 10416 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-ofr 7654 df-rpss 7699 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-sup 9393 df-inf 9394 df-oi 9463 df-r1 9717 df-rank 9718 df-dju 9854 df-card 9892 df-acn 9895 df-ac 10069 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-ico 13312 df-fz 13469 df-fzo 13616 df-seq 13967 df-hash 14296 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ocomp 17241 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-imas 17471 df-qus 17472 df-mre 17547 df-mrc 17548 df-mri 17549 df-acs 17550 df-proset 18255 df-drs 18256 df-poset 18274 df-ipo 18487 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145 df-gim 19191 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-srg 20096 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-irred 20268 df-invr 20297 df-dvr 20310 df-rhm 20381 df-nzr 20422 df-subrng 20455 df-subrg 20479 df-rlreg 20603 df-domn 20604 df-idom 20605 df-drng 20640 df-field 20641 df-sdrg 20696 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lmhm 20929 df-lmim 20930 df-lmic 20931 df-lbs 20982 df-lvec 21010 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-2idl 21160 df-lpidl 21232 df-lpir 21233 df-pid 21247 df-cnfld 21265 df-dsmm 21641 df-frlm 21656 df-uvc 21692 df-lindf 21715 df-linds 21716 df-assa 21762 df-asp 21763 df-ascl 21764 df-psr 21818 df-mvr 21819 df-mpl 21820 df-opsr 21822 df-evls 21981 df-evl 21982 df-psr1 22064 df-vr1 22065 df-ply1 22066 df-coe1 22067 df-evls1 22202 df-evl1 22203 df-mdeg 25960 df-deg1 25961 df-mon1 26036 df-uc1p 26037 df-q1p 26038 df-r1p 26039 df-ig1p 26040 df-fldgen 33261 df-mxidl 33431 df-dim 33595 df-fldext 33637 df-extdg 33638 df-irng 33679 df-minply 33690

This theorem is referenced by: constrelextdg2 33737

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