fldext2rspun - Mathbox for Thierry Arnoux

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Theorem fldext2rspun 33669

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 2735	. . . . . 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 20751	. . . . . . 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 33655	. . . . 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 33650	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
13		extdgmul 33651	. . . . 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 12311	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
20		fldext2rspun.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	19, 20	nnexpcld 14261	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
22	17, 21	eqeltrd 2834	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ)
23	22	nnnn0d 12560	. . . . . . . . 9 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
24		fldext2rspun.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼[:]𝐾) = 2)
25	24, 18	eqeltrdi 2842	. . . . . . . . 9 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ)
26	11, 2, 15, 4, 10, 16, 5, 6, 23, 3, 25	fldextrspundgdvdslem 33667	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ₀)
27		elnn0 12501	. . . . . . . 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 33645	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐼 → 0 < (𝐸[:]𝐼))
30	9, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐸[:]𝐼))
31	30	gt0ne0d 11799	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ≠ 0)
32	31	neneqd 2937	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐸[:]𝐼) = 0)
33	28, 32	olcnd 877	. . . . . 6 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ)
34	33	nnred 12253	. . . . 5 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℝ)
35	25	nnred 12253	. . . . 5 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ)
36		rexmul 13285	. . . . 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 2770	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
39	33, 25	nnmulcld 12291	. . 3 ⊢ (𝜑 → ((𝐸[:]𝐼) · (𝐼[:]𝐾)) ∈ ℕ)
40	38, 39	eqeltrd 2834	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ∈ ℕ)
41		2nn0 12516	. . . . 5 ⊢ 2 ∈ ℕ₀
42	24, 41	eqeltrdi 2842	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ₀)
43		uncom 4133	. . . . . . 7 ⊢ (𝐺 ∪ 𝐻) = (𝐻 ∪ 𝐺)
44	43	oveq2i 7414	. . . . . 6 ⊢ (𝐿 fldGen (𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐻 ∪ 𝐺))
45	44	oveq2i 7414	. . . . 5 ⊢ (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻))) = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
46	3, 45	eqtri 2758	. . . 4 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
47	11, 15, 2, 4, 16, 10, 6, 5, 42, 46, 22	fldextrspundgdvds 33668	. . 3 ⊢ (𝜑 → (𝐽[:]𝐾) ∥ (𝐸[:]𝐾))
48	17, 47	eqbrtrrd 5143	. 2 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐸[:]𝐾))
49	11, 2, 15, 4, 10, 16, 5, 6, 23, 3	fldextrspundglemul 33666	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))
50	22	nnred 12253	. . . . 5 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℝ)
51		rexmul 13285	. . . . 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 7421	. . . 4 ⊢ (𝜑 → ((𝐼[:]𝐾) · (𝐽[:]𝐾)) = (2 · (2↑𝑁)))
54		2cnd 12316	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
55	54, 20	expcld 14162	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
56	54, 55	mulcomd 11254	. . . . 5 ⊢ (𝜑 → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
57	54, 20	expp1d 14163	. . . . 5 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
58	56, 57	eqtr4d 2773	. . . 4 ⊢ (𝜑 → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
59	52, 53, 58	3eqtrd 2774	. . 3 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = (2↑(𝑁 + 1)))
60	49, 59	breqtrd 5145	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ (2↑(𝑁 + 1)))
61	40, 20, 48, 60	2exple2exp 32770	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (𝐸[:]𝐾) = (2↑𝑛))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 847 = wceq 1540 ∈ wcel 2108 ∃wrex 3060 ∪ cun 3924 ⊆ wss 3926 class class class wbr 5119 ‘cfv 6530 (class class class)co 7403 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 < clt 11267 ≤ cle 11268 ℕcn 12238 2c2 12293 ℕ₀cn0 12499 ·_e cxmu 13125 ↑cexp 14077 ∥ cdvds 16270 Basecbs 17226 ↾_s cress 17249 Fieldcfield 20688 SubDRingcsdrg 20744 fldGen cfldgen 33250 /_FldExtcfldext 33624 [:]cextdg 33627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-reg 9604 ax-inf2 9653 ax-ac2 10475 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-rpss 7715 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-sup 9452 df-inf 9453 df-oi 9522 df-r1 9776 df-rank 9777 df-dju 9913 df-card 9951 df-acn 9954 df-ac 10128 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12851 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-icc 13367 df-fz 13523 df-fzo 13670 df-seq 14018 df-exp 14078 df-hash 14347 df-word 14530 df-lsw 14579 df-concat 14587 df-s1 14612 df-substr 14657 df-pfx 14687 df-s2 14865 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-clim 15502 df-sum 15701 df-dvds 16271 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ocomp 17290 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-0g 17453 df-gsum 17454 df-prds 17459 df-pws 17461 df-mre 17596 df-mrc 17597 df-mri 17598 df-acs 17599 df-proset 18304 df-drs 18305 df-poset 18323 df-ipo 18536 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-mhm 18759 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-mulg 19049 df-subg 19104 df-ghm 19194 df-cntz 19298 df-cntr 19299 df-lsm 19615 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-cring 20194 df-oppr 20295 df-dvdsr 20315 df-unit 20316 df-invr 20346 df-dvr 20359 df-nzr 20471 df-subrng 20504 df-subrg 20528 df-rgspn 20569 df-rlreg 20652 df-domn 20653 df-idom 20654 df-drng 20689 df-field 20690 df-sdrg 20745 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lmhm 20978 df-lmim 20979 df-lbs 21031 df-lvec 21059 df-sra 21129 df-rgmod 21130 df-cnfld 21314 df-zring 21406 df-dsmm 21690 df-frlm 21705 df-uvc 21741 df-lindf 21764 df-linds 21765 df-assa 21811 df-ind 32774 df-fldgen 33251 df-dim 33585 df-fldext 33628 df-extdg 33629

This theorem is referenced by: (None)

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