rtelextdg2 - Mathbox for Thierry Arnoux

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Theorem rtelextdg2 33905

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 20686	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		rtelextdg2.10	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	1	sdrgss 20738	. . . . . . . 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 4767	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
9	6, 8	unssd 4146	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ 𝑉)
10	1, 3, 9	fldgenssid 33407	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝑋})))
11		ssun2 4133	. . . . . 6 ⊢ {𝑋} ⊆ (𝐹 ∪ {𝑋})
12		snidg 4619	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
13	7, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
14	11, 13	sselid 3933	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐹 ∪ {𝑋}))
15	10, 14	sseldd 3936	. . . 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 33846	. . . . . 6 ⊢ (𝜑 → 𝐿/_FldExt𝐾)
20		extdg1id 33844	. . . . . 6 ⊢ ((𝐿/_FldExt𝐾 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
21	19, 20	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
22	21	fveq2d 6846	. . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (Base‘𝐿) = (Base‘𝐾))
23	1, 3, 9	fldgenssv 33409	. . . . . 6 ⊢ (𝜑 → (𝐸 fldGen (𝐹 ∪ {𝑋})) ⊆ 𝑉)
24	18, 1	ressbas2 17177	. . . . . 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 17177	. . . . . 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 2782	. . 3 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (𝐸 fldGen (𝐹 ∪ {𝑋})) = 𝐹)
31	16, 30	eleqtrd 2839	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝑋 ∈ 𝐹)
32		simpr 484	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 2) → (𝐿[:]𝐾) = 2)
33		1zzd 12534	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
34		2z 12535	. . . . . 6 ⊢ 2 ∈ ℤ
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
36		extdgcl 33834	. . . . . . . 8 ⊢ (𝐿/_FldExt𝐾 → (𝐿[:]𝐾) ∈ ℕ₀^*)
37	19, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀^*)
38		2nn0 12430	. . . . . . . 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 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 33904	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2)
55		xnn0lenn0nn0 13172	. . . . . . 7 ⊢ (((𝐿[:]𝐾) ∈ ℕ₀^* ∧ 2 ∈ ℕ₀ ∧ (𝐿[:]𝐾) ≤ 2) → (𝐿[:]𝐾) ∈ ℕ₀)
56	37, 39, 54, 55	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀)
57	56	nn0zd 12525	. . . . 5 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℤ)
58		extdggt0 33835	. . . . . . 7 ⊢ (𝐿/_FldExt𝐾 → 0 < (𝐿[:]𝐾))
59	19, 58	syl 17	. . . . . 6 ⊢ (𝜑 → 0 < (𝐿[:]𝐾))
60		zgt0ge1 12558	. . . . . . 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 585	. . . . 5 ⊢ (𝜑 → 1 ≤ (𝐿[:]𝐾))
63	33, 35, 57, 62, 54	elfzd 13443	. . . 4 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ (1...2))
64		fz12pr 13509	. . . 4 ⊢ (1...2) = {1, 2}
65	63, 64	eleqtrdi 2847	. . 3 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ {1, 2})
66		elpri 4606	. . 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 32544	1 ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∪ cun 3901 ⊆ wss 3903 {csn 4582 {cpr 4584 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 0cc0 11038 1c1 11039 < clt 11178 ≤ cle 11179 2c2 12212 ℕ₀cn0 12413 ℕ₀^*cxnn0 12486 ℤcz 12500 ...cfz 13435 Basecbs 17148 ↾_s cress 17169 +_gcplusg 17189 ._rcmulr 17190 0_gc0g 17371 ._gcmg 19009 mulGrpcmgp 20087 Fieldcfield 20675 SubDRingcsdrg 20731 algSccascl 21819 var₁cv1 22128 Poly₁cpl1 22129 fldGen cfldgen 33404 /_FldExtcfldext 33816 [:]cextdg 33818

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-reg 9509 ax-inf2 9562 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-rpss 7678 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-ec 8647 df-qs 8651 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-inf 9358 df-oi 9427 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-ico 13279 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ocomp 17210 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-imas 17441 df-qus 17442 df-mre 17517 df-mrc 17518 df-mri 17519 df-acs 17520 df-proset 18229 df-drs 18230 df-poset 18248 df-ipo 18463 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-nsg 19066 df-eqg 19067 df-ghm 19154 df-gim 19200 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-irred 20307 df-invr 20336 df-dvr 20349 df-rhm 20420 df-nzr 20458 df-subrng 20491 df-subrg 20515 df-rlreg 20639 df-domn 20640 df-idom 20641 df-drng 20676 df-field 20677 df-sdrg 20732 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lmhm 20986 df-lmim 20987 df-lmic 20988 df-lbs 21039 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-lidl 21175 df-rsp 21176 df-2idl 21217 df-lpidl 21289 df-lpir 21290 df-pid 21304 df-cnfld 21322 df-dsmm 21699 df-frlm 21714 df-uvc 21750 df-lindf 21773 df-linds 21774 df-assa 21820 df-asp 21821 df-ascl 21822 df-psr 21877 df-mvr 21878 df-mpl 21879 df-opsr 21881 df-evls 22041 df-evl 22042 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-evls1 22271 df-evl1 22272 df-mdeg 26028 df-deg1 26029 df-mon1 26104 df-uc1p 26105 df-q1p 26106 df-r1p 26107 df-ig1p 26108 df-fldgen 33405 df-mxidl 33553 df-dim 33777 df-fldext 33819 df-extdg 33820 df-irng 33862 df-minply 33878

This theorem is referenced by: constrelextdg2 33925

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