rtelextdg2 - Mathbox for Thierry Arnoux

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Theorem rtelextdg2 33733

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 20758	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		rtelextdg2.10	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	1	sdrgss 20811	. . . . . . . 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 4814	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
9	6, 8	unssd 4202	. . . . . 6 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ 𝑉)
10	1, 3, 9	fldgenssid 33295	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ {𝑋}) ⊆ (𝐸 fldGen (𝐹 ∪ {𝑋})))
11		ssun2 4189	. . . . . 6 ⊢ {𝑋} ⊆ (𝐹 ∪ {𝑋})
12		snidg 4665	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
13	7, 12	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
14	11, 13	sselid 3993	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐹 ∪ {𝑋}))
15	10, 14	sseldd 3996	. . . 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 33693	. . . . . 6 ⊢ (𝜑 → 𝐿/_FldExt𝐾)
20		extdg1id 33691	. . . . . 6 ⊢ ((𝐿/_FldExt𝐾 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
21	19, 20	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝐿 = 𝐾)
22	21	fveq2d 6911	. . . 4 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (Base‘𝐿) = (Base‘𝐾))
23	1, 3, 9	fldgenssv 33297	. . . . . 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 2785	. . 3 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → (𝐸 fldGen (𝐹 ∪ {𝑋})) = 𝐹)
31	16, 30	eleqtrd 2841	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 1) → 𝑋 ∈ 𝐹)
32		simpr 484	. 2 ⊢ ((𝜑 ∧ (𝐿[:]𝐾) = 2) → (𝐿[:]𝐾) = 2)
33		1zzd 12646	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
34		2z 12647	. . . . . 6 ⊢ 2 ∈ ℤ
35	34	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
36		extdgcl 33684	. . . . . . . 8 ⊢ (𝐿/_FldExt𝐾 → (𝐿[:]𝐾) ∈ ℕ₀^*)
37	19, 36	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀^*)
38		2nn0 12541	. . . . . . . 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 2735	. . . . . . . 8 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
49		eqid 2735	. . . . . . . 8 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
50		eqid 2735	. . . . . . . 8 ⊢ (._r‘𝑃) = (._r‘𝑃)
51		eqid 2735	. . . . . . . 8 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
52		eqid 2735	. . . . . . . 8 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53		eqid 2735	. . . . . . . 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 33732	. . . . . . 7 ⊢ (𝜑 → (𝐿[:]𝐾) ≤ 2)
55		xnn0lenn0nn0 13284	. . . . . . 7 ⊢ (((𝐿[:]𝐾) ∈ ℕ₀^* ∧ 2 ∈ ℕ₀ ∧ (𝐿[:]𝐾) ≤ 2) → (𝐿[:]𝐾) ∈ ℕ₀)
56	37, 39, 54, 55	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℕ₀)
57	56	nn0zd 12637	. . . . 5 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ ℤ)
58		extdggt0 33685	. . . . . . 7 ⊢ (𝐿/_FldExt𝐾 → 0 < (𝐿[:]𝐾))
59	19, 58	syl 17	. . . . . 6 ⊢ (𝜑 → 0 < (𝐿[:]𝐾))
60		zgt0ge1 12670	. . . . . . 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 13552	. . . 4 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ (1...2))
64		fz12pr 13618	. . . 4 ⊢ (1...2) = {1, 2}
65	63, 64	eleqtrdi 2849	. . 3 ⊢ (𝜑 → (𝐿[:]𝐾) ∈ {1, 2})
66		elpri 4654	. . 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 847 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 {csn 4631 {cpr 4633 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 < clt 11293 ≤ cle 11294 2c2 12319 ℕ₀cn0 12524 ℕ₀^*cxnn0 12597 ℤcz 12611 ...cfz 13544 Basecbs 17245 ↾_s cress 17274 +_gcplusg 17298 ._rcmulr 17299 0_gc0g 17486 ._gcmg 19098 mulGrpcmgp 20152 Fieldcfield 20747 SubDRingcsdrg 20804 algSccascl 21890 var₁cv1 22193 Poly₁cpl1 22194 fldGen cfldgen 33292 /_FldExtcfldext 33666 [:]cextdg 33669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-reg 9630 ax-inf2 9679 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-rpss 7742 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-inf 9481 df-oi 9548 df-r1 9802 df-rank 9803 df-dju 9939 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-ico 13390 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ocomp 17319 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-imas 17555 df-qus 17556 df-mre 17631 df-mrc 17632 df-mri 17633 df-acs 17634 df-proset 18352 df-drs 18353 df-poset 18371 df-ipo 18586 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-gim 19290 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-irred 20376 df-invr 20405 df-dvr 20418 df-rhm 20489 df-nzr 20530 df-subrng 20563 df-subrg 20587 df-rlreg 20711 df-domn 20712 df-idom 20713 df-drng 20748 df-field 20749 df-sdrg 20805 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lmim 21040 df-lmic 21041 df-lbs 21092 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-lpidl 21350 df-lpir 21351 df-pid 21365 df-cnfld 21383 df-dsmm 21770 df-frlm 21785 df-uvc 21821 df-lindf 21844 df-linds 21845 df-assa 21891 df-asp 21892 df-ascl 21893 df-psr 21947 df-mvr 21948 df-mpl 21949 df-opsr 21951 df-evls 22116 df-evl 22117 df-psr1 22197 df-vr1 22198 df-ply1 22199 df-coe1 22200 df-evls1 22335 df-evl1 22336 df-mdeg 26109 df-deg1 26110 df-mon1 26185 df-uc1p 26186 df-q1p 26187 df-r1p 26188 df-ig1p 26189 df-fldgen 33293 df-mxidl 33468 df-dim 33627 df-fldext 33670 df-extdg 33671 df-irng 33699 df-minply 33708

This theorem is referenced by: constrelextdg2 33752

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