rtelextdg2 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rtelextdg2		Structured version Visualization version GIF version

Theorem rtelextdg2 33871

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 20718	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		rtelextdg2.10	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	1	sdrgss 20770	. . . . . . . 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 4730	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
9	6, 8	unssd 4132	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ 𝑉)
10	1, 3, 9	fldgenssid 33374	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝑋})))
11		ssun2 4119	. . . . . 6 ⊢ {𝑋} ⊆ (𝐹 ∪ {𝑋})
12		snidg 4604	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
13	7, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
14	11, 13	sselid 3919	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐹 ∪ {𝑋}))
15	10, 14	sseldd 3922	. . . 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 33812	. . . . . 6 ⊢ (𝜑 → 𝐿/_FldExt𝐾)
20		extdg1id 33810	. . . . . 6 ⊢ ((𝐿/_FldExt𝐾 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
21	19, 20	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
22	21	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (Base‘𝐿) = (Base‘𝐾))
23	1, 3, 9	fldgenssv 33376	. . . . . 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 2781	. . 3 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (𝐸 fldGen (𝐹 ∪ {𝑋})) = 𝐹)
31	16, 30	eleqtrd 2838	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝑋 ∈ 𝐹)
32		simpr 484	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 2) → (𝐿[:]𝐾) = 2)
33		1zzd 12558	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
34		2z 12559	. . . . . 6 ⊢ 2 ∈ ℤ
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
36		extdgcl 33800	. . . . . . . 8 ⊢ (𝐿/_FldExt𝐾 → (𝐿[:]𝐾) ∈ ℕ₀^*)
37	19, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀^*)
38		2nn0 12454	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
49		eqid 2736	. . . . . . . 8 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
50		eqid 2736	. . . . . . . 8 ⊢ (._r‘𝑃) = (._r‘𝑃)
51		eqid 2736	. . . . . . . 8 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
52		eqid 2736	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53		eqid 2736	. . . . . . . 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 33870	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2)
55		xnn0lenn0nn0 13197	. . . . . . 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 12549	. . . . 5 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℤ)
58		extdggt0 33801	. . . . . . 7 ⊢ (𝐿/_FldExt𝐾 → 0 < (𝐿[:]𝐾))
59	19, 58	syl 17	. . . . . 6 ⊢ (𝜑 → 0 < (𝐿[:]𝐾))
60		zgt0ge1 12583	. . . . . . 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 13469	. . . 4 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ (1...2))
64		fz12pr 13535	. . . 4 ⊢ (1...2) = {1, 2}
65	63, 64	eleqtrdi 2846	. . 3 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ {1, 2})
66		elpri 4591	. . 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 32527	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 3887 ⊆ wss 3889 {csn 4567 {cpr 4569 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 0cc0 11038 1c1 11039 < clt 11179 ≤ cle 11180 2c2 12236 ℕ₀cn0 12437 ℕ₀^*cxnn0 12510 ℤcz 12524 ...cfz 13461 Basecbs 17179 ↾_s cress 17200 +_gcplusg 17220 ._rcmulr 17221 0_gc0g 17402 ._gcmg 19043 mulGrpcmgp 20121 Fieldcfield 20707 SubDRingcsdrg 20763 algSccascl 21832 var₁cv1 22139 Poly₁cpl1 22140 fldGen cfldgen 33371 /_FldExtcfldext 33782 [:]cextdg 33784

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-reg 9507 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-rpss 7677 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-inf 9356 df-oi 9425 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-ico 13304 df-fz 13462 df-fzo 13609 df-seq 13964 df-hash 14293 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 17472 df-qus 17473 df-mre 17548 df-mrc 17549 df-mri 17550 df-acs 17551 df-proset 18260 df-drs 18261 df-poset 18279 df-ipo 18494 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-gim 19234 df-cntz 19292 df-oppg 19321 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-srg 20168 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-irred 20339 df-invr 20368 df-dvr 20381 df-rhm 20452 df-nzr 20490 df-subrng 20523 df-subrg 20547 df-rlreg 20671 df-domn 20672 df-idom 20673 df-drng 20708 df-field 20709 df-sdrg 20764 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lmhm 21017 df-lmim 21018 df-lmic 21019 df-lbs 21070 df-lvec 21098 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-lpidl 21320 df-lpir 21321 df-pid 21335 df-cnfld 21353 df-dsmm 21712 df-frlm 21727 df-uvc 21763 df-lindf 21786 df-linds 21787 df-assa 21833 df-asp 21834 df-ascl 21835 df-psr 21889 df-mvr 21890 df-mpl 21891 df-opsr 21893 df-evls 22052 df-evl 22053 df-psr1 22143 df-vr1 22144 df-ply1 22145 df-coe1 22146 df-evls1 22280 df-evl1 22281 df-mdeg 26020 df-deg1 26021 df-mon1 26096 df-uc1p 26097 df-q1p 26098 df-r1p 26099 df-ig1p 26100 df-fldgen 33372 df-mxidl 33520 df-dim 33744 df-fldext 33785 df-extdg 33786 df-irng 33828 df-minply 33844

This theorem is referenced by: constrelextdg2 33891

W3C validator