fldext2rspun - Mathbox for Thierry Arnoux

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Theorem fldext2rspun 34225

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 2760	. . . . . 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 20989	. . . . . . 7 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
8	6, 7	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
9	1, 2, 3, 4, 5, 8	fldgenfldext 34211	. . . . 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 34205	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
13		extdgmul 34206	. . . . 5 ⊢ ((𝐸/_FldExt𝐼 ∧ 𝐼/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
14	9, 12, 13	syl2anc 596	. . . 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 12363	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
20		fldext2rspun.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	19, 20	nnexpcld 14334	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
22	17, 21	eqeltrd 2860	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ)
23	22	nnnn0d 12614	. . . . . . . . 9 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
24		fldext2rspun.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼[:]𝐾) = 2)
25	24, 18	eqeltrdi 2868	. . . . . . . . 9 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ)
26	11, 2, 15, 4, 10, 16, 5, 6, 23, 3, 25	fldextrspundgdvdslem 34223	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ₀)
27		elnn0 12555	. . . . . . . 8 ⊢ ((𝐸[:]𝐼) ∈ ℕ₀ ↔ ((𝐸[:]𝐼) ∈ ℕ ∨ (𝐸[:]𝐼) = 0))
28	26, 27	sylib 221	. . . . . . 7 ⊢ (𝜑 → ((𝐸[:]𝐼) ∈ ℕ ∨ (𝐸[:]𝐼) = 0))
29		extdggt0 34200	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐼 → 0 < (𝐸[:]𝐼))
30	9, 29	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐸[:]𝐼))
31	30	gt0ne0d 11827	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ≠ 0)
32	31	neneqd 2960	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐸[:]𝐼) = 0)
33	28, 32	olcnd 891	. . . . . 6 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ)
34	33	nnred 12297	. . . . 5 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℝ)
35	25	nnred 12297	. . . . 5 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ)
36		rexmul 13348	. . . . 5 ⊢ (((𝐸[:]𝐼) ∈ ℝ ∧ (𝐼[:]𝐾) ∈ ℝ) → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
37	34, 35, 36	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
38	14, 37	eqtrd 2795	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
39	33, 25	nnmulcld 12338	. . 3 ⊢ (𝜑 → ((𝐸[:]𝐼) · (𝐼[:]𝐾)) ∈ ℕ)
40	38, 39	eqeltrd 2860	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ∈ ℕ)
41		2nn0 12570	. . . . 5 ⊢ 2 ∈ ℕ₀
42	24, 41	eqeltrdi 2868	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ₀)
43		uncom 4105	. . . . . . 7 ⊢ (𝐺 ∪ 𝐻) = (𝐻 ∪ 𝐺)
44	43	oveq2i 7427	. . . . . 6 ⊢ (𝐿 fldGen (𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐻 ∪ 𝐺))
45	44	oveq2i 7427	. . . . 5 ⊢ (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻))) = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
46	3, 45	eqtri 2783	. . . 4 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
47	11, 15, 2, 4, 16, 10, 6, 5, 42, 46, 22	fldextrspundgdvds 34224	. . 3 ⊢ (𝜑 → (𝐽[:]𝐾) ∥ (𝐸[:]𝐾))
48	17, 47	eqbrtrrd 5129	. 2 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐸[:]𝐾))
49	11, 2, 15, 4, 10, 16, 5, 6, 23, 3	fldextrspundglemul 34222	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))
50	22	nnred 12297	. . . . 5 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℝ)
51		rexmul 13348	. . . . 5 ⊢ (((𝐼[:]𝐾) ∈ ℝ ∧ (𝐽[:]𝐾) ∈ ℝ) → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐼[:]𝐾) · (𝐽[:]𝐾)))
52	35, 50, 51	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐼[:]𝐾) · (𝐽[:]𝐾)))
53	24, 17	oveq12d 7434	. . . 4 ⊢ (𝜑 → ((𝐼[:]𝐾) · (𝐽[:]𝐾)) = (2 · (2↑𝑁)))
54		2cnd 12368	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
55	54, 20	expcld 14235	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
56	54, 55	mulcomd 11279	. . . . 5 ⊢ (𝜑 → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
57	54, 20	expp1d 14236	. . . . 5 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
58	56, 57	eqtr4d 2798	. . . 4 ⊢ (𝜑 → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
59	52, 53, 58	3eqtrd 2799	. . 3 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = (2↑(𝑁 + 1)))
60	49, 59	breqtrd 5131	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ (2↑(𝑁 + 1)))
61	40, 20, 48, 60	2exple2exp 33336	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (𝐸[:]𝐾) = (2↑𝑛))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ∃wrex 3086 ∪ cun 3897 ⊆ wss 3899 class class class wbr 5103 ‘cfv 6535 (class class class)co 7416 ℝcr 11148 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 < clt 11292 ≤ cle 11293 ℕcn 12282 2c2 12344 ℕ₀cn0 12553 ·_e cxmu 13187 ↑cexp 14150 ∥ cdvds 16367 Basecbs 17326 ↾_s cress 17347 Fieldcfield 20920 SubDRingcsdrg 20982 fldGen cfldgen 33783 /_FldExtcfldext 34181 [:]cextdg 34183

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-reg 9571 ax-inf2 9627 ax-ac2 10490 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-rpss 7730 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-tpos 8229 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-map 8835 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-sup 9419 df-inf 9420 df-oi 9489 df-r1 9753 df-rank 9754 df-scott 9893 df-dju 9931 df-card 9969 df-acn 9972 df-ac 10144 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-ind 12268 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-xnn0 12627 df-z 12641 df-dec 12762 df-uz 12913 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-icc 13430 df-fz 13587 df-fzo 13735 df-seq 14091 df-exp 14151 df-hash 14420 df-word 14604 df-lsw 14653 df-concat 14661 df-s1 14688 df-substr 14734 df-pfx 14766 df-s2 14944 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-clim 15600 df-sum 15799 df-dvds 16368 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ocomp 17388 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-0g 17551 df-gsum 17552 df-prds 17557 df-pws 17559 df-mre 17695 df-mrc 17696 df-mri 17697 df-acs 17698 df-proset 18407 df-drs 18408 df-poset 18426 df-ipo 18641 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-mhm 18917 df-submnd 18918 df-grp 19086 df-minusg 19087 df-sbg 19088 df-mulg 19217 df-subg 19272 df-ghm 19367 df-cntz 19470 df-cntr 19471 df-lsm 19789 df-cmn 19935 df-abl 19936 df-mgp 20300 df-rng 20314 df-ur 20347 df-ring 20400 df-cring 20401 df-oppr 20506 df-dvdsr 20526 df-unit 20527 df-invr 20557 df-dvr 20570 df-nzr 20702 df-subrng 20737 df-subrg 20761 df-rgspn 20802 df-rlreg 20885 df-domn 20886 df-idom 20887 df-drng 20921 df-field 20922 df-sdrg 20983 df-lmod 21076 df-lss 21146 df-lsp 21186 df-lmhm 21236 df-lmim 21237 df-lbs 21289 df-lvec 21317 df-sra 21387 df-rgmod 21388 df-cnfld 21618 df-zring 21692 df-dsmm 21977 df-frlm 21992 df-uvc 22028 df-lindf 22051 df-linds 22052 df-assa 22100 df-fldgen 33784 df-dim 34143 df-fldext 34184 df-extdg 34185

This theorem is used by: (None)

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