fldext2rspun - Mathbox for Thierry Arnoux

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Theorem fldext2rspun 33695

Description: Given two field extensions 𝐼 / 𝐾 and 𝐽 / 𝐾, 𝐼 / 𝐾 being a quadratic extension, and the degree of 𝐽 / 𝐾 being a power of 2, the degree of the extension 𝐸 / 𝐾 is a power of 2 , 𝐸 being the composite field 𝐼𝐽. (Contributed by Thierry Arnoux, 19-Oct-2025.)

**Hypotheses**
Ref	Expression
fldextrspun.k	⊢ 𝐾 = (𝐿 ↾_s 𝐹)
fldextrspun.i	⊢ 𝐼 = (𝐿 ↾_s 𝐺)
fldextrspun.j	⊢ 𝐽 = (𝐿 ↾_s 𝐻)
fldextrspun.2	⊢ (𝜑 → 𝐿 ∈ Field)
fldextrspun.3	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
fldextrspun.4	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
fldextrspun.5	⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
fldextrspun.6	⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
fldext2rspun.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
fldext2rspun.1	⊢ (𝜑 → (𝐼[:]𝐾) = 2)
fldext2rspun.2	⊢ (𝜑 → (𝐽[:]𝐾) = (2↑𝑁))
fldext2rspun.e	⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻)))

**Assertion**
Ref	Expression
fldext2rspun	⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (𝐸[:]𝐾) = (2↑𝑛))

Distinct variable groups: 𝑛,𝐸 𝑛,𝐾 𝑛,𝑁 𝜑,𝑛 Allowed substitution hints: 𝐹(𝑛) 𝐺(𝑛) 𝐻(𝑛) 𝐼(𝑛) 𝐽(𝑛) 𝐿(𝑛)

**Proof of Theorem fldext2rspun**
Step	Hyp	Ref	Expression
1		eqid 2731	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
2		fldextrspun.i	. . . . . 6 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
3		fldext2rspun.e	. . . . . 6 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻)))
4		fldextrspun.2	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ Field)
5		fldextrspun.5	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
6		fldextrspun.6	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
7	1	sdrgss 20708	. . . . . . 7 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
8	6, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
9	1, 2, 3, 4, 5, 8	fldgenfldext 33681	. . . . 5 ⊢ (𝜑 → 𝐸/_FldExt𝐼)
10		fldextrspun.3	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
11		fldextrspun.k	. . . . . 6 ⊢ 𝐾 = (𝐿 ↾_s 𝐹)
12	2, 4, 5, 10, 11	fldsdrgfldext2 33675	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
13		extdgmul 33676	. . . . 5 ⊢ ((𝐸/_FldExt𝐼 ∧ 𝐼/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
14	9, 12, 13	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
15		fldextrspun.j	. . . . . . . . 9 ⊢ 𝐽 = (𝐿 ↾_s 𝐻)
16		fldextrspun.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
17		fldext2rspun.2	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽[:]𝐾) = (2↑𝑁))
18		2nn 12198	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
20		fldext2rspun.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	19, 20	nnexpcld 14152	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
22	17, 21	eqeltrd 2831	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ)
23	22	nnnn0d 12442	. . . . . . . . 9 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
24		fldext2rspun.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼[:]𝐾) = 2)
25	24, 18	eqeltrdi 2839	. . . . . . . . 9 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ)
26	11, 2, 15, 4, 10, 16, 5, 6, 23, 3, 25	fldextrspundgdvdslem 33693	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ₀)
27		elnn0 12383	. . . . . . . 8 ⊢ ((𝐸[:]𝐼) ∈ ℕ₀ ↔ ((𝐸[:]𝐼) ∈ ℕ ∨ (𝐸[:]𝐼) = 0))
28	26, 27	sylib 218	. . . . . . 7 ⊢ (𝜑 → ((𝐸[:]𝐼) ∈ ℕ ∨ (𝐸[:]𝐼) = 0))
29		extdggt0 33670	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐼 → 0 < (𝐸[:]𝐼))
30	9, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐸[:]𝐼))
31	30	gt0ne0d 11681	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ≠ 0)
32	31	neneqd 2933	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐸[:]𝐼) = 0)
33	28, 32	olcnd 877	. . . . . 6 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ)
34	33	nnred 12140	. . . . 5 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℝ)
35	25	nnred 12140	. . . . 5 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ)
36		rexmul 13170	. . . . 5 ⊢ (((𝐸[:]𝐼) ∈ ℝ ∧ (𝐼[:]𝐾) ∈ ℝ) → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
37	34, 35, 36	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
38	14, 37	eqtrd 2766	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
39	33, 25	nnmulcld 12178	. . 3 ⊢ (𝜑 → ((𝐸[:]𝐼) · (𝐼[:]𝐾)) ∈ ℕ)
40	38, 39	eqeltrd 2831	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ∈ ℕ)
41		2nn0 12398	. . . . 5 ⊢ 2 ∈ ℕ₀
42	24, 41	eqeltrdi 2839	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ₀)
43		uncom 4105	. . . . . . 7 ⊢ (𝐺 ∪ 𝐻) = (𝐻 ∪ 𝐺)
44	43	oveq2i 7357	. . . . . 6 ⊢ (𝐿 fldGen (𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐻 ∪ 𝐺))
45	44	oveq2i 7357	. . . . 5 ⊢ (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻))) = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
46	3, 45	eqtri 2754	. . . 4 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
47	11, 15, 2, 4, 16, 10, 6, 5, 42, 46, 22	fldextrspundgdvds 33694	. . 3 ⊢ (𝜑 → (𝐽[:]𝐾) ∥ (𝐸[:]𝐾))
48	17, 47	eqbrtrrd 5113	. 2 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐸[:]𝐾))
49	11, 2, 15, 4, 10, 16, 5, 6, 23, 3	fldextrspundglemul 33692	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))
50	22	nnred 12140	. . . . 5 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℝ)
51		rexmul 13170	. . . . 5 ⊢ (((𝐼[:]𝐾) ∈ ℝ ∧ (𝐽[:]𝐾) ∈ ℝ) → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐼[:]𝐾) · (𝐽[:]𝐾)))
52	35, 50, 51	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐼[:]𝐾) · (𝐽[:]𝐾)))
53	24, 17	oveq12d 7364	. . . 4 ⊢ (𝜑 → ((𝐼[:]𝐾) · (𝐽[:]𝐾)) = (2 · (2↑𝑁)))
54		2cnd 12203	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
55	54, 20	expcld 14053	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
56	54, 55	mulcomd 11133	. . . . 5 ⊢ (𝜑 → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
57	54, 20	expp1d 14054	. . . . 5 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
58	56, 57	eqtr4d 2769	. . . 4 ⊢ (𝜑 → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
59	52, 53, 58	3eqtrd 2770	. . 3 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = (2↑(𝑁 + 1)))
60	49, 59	breqtrd 5115	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ (2↑(𝑁 + 1)))
61	40, 20, 48, 60	2exple2exp 32828	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (𝐸[:]𝐾) = (2↑𝑛))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 ∪ cun 3895 ⊆ wss 3897 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 < clt 11146 ≤ cle 11147 ℕcn 12125 2c2 12180 ℕ₀cn0 12381 ·_e cxmu 13010 ↑cexp 13968 ∥ cdvds 16163 Basecbs 17120 ↾_s cress 17141 Fieldcfield 20645 SubDRingcsdrg 20701 fldGen cfldgen 33276 /_FldExtcfldext 33651 [:]cextdg 33653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-reg 9478 ax-inf2 9531 ax-ac2 10354 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-rpss 7656 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-inf 9327 df-oi 9396 df-r1 9657 df-rank 9658 df-dju 9794 df-card 9832 df-acn 9835 df-ac 10007 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-word 14421 df-lsw 14470 df-concat 14478 df-s1 14504 df-substr 14549 df-pfx 14579 df-s2 14755 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-dvds 16164 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ocomp 17182 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-mri 17490 df-acs 17491 df-proset 18200 df-drs 18201 df-poset 18219 df-ipo 18434 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cntr 19230 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rgspn 20526 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-sdrg 20702 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lmhm 20956 df-lmim 20957 df-lbs 21009 df-lvec 21037 df-sra 21107 df-rgmod 21108 df-cnfld 21292 df-zring 21384 df-dsmm 21669 df-frlm 21684 df-uvc 21720 df-lindf 21743 df-linds 21744 df-assa 21790 df-ind 32832 df-fldgen 33277 df-dim 33612 df-fldext 33654 df-extdg 33655

This theorem is referenced by: (None)

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