fldext2rspun - Mathbox for Thierry Arnoux

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Theorem fldext2rspun 33878

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 2741	. . . . . 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 20769	. . . . . . 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 33864	. . . . 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 33858	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
13		extdgmul 33859	. . . . 5 ⊢ ((𝐸/_FldExt𝐼 ∧ 𝐼/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
14	9, 12, 13	syl2anc 591	. . . 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 12249	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
20		fldext2rspun.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	19, 20	nnexpcld 14202	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
22	17, 21	eqeltrd 2841	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ)
23	22	nnnn0d 12493	. . . . . . . . 9 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
24		fldext2rspun.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼[:]𝐾) = 2)
25	24, 18	eqeltrdi 2849	. . . . . . . . 9 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ)
26	11, 2, 15, 4, 10, 16, 5, 6, 23, 3, 25	fldextrspundgdvdslem 33876	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ₀)
27		elnn0 12434	. . . . . . . 8 ⊢ ((𝐸[:]𝐼) ∈ ℕ₀ ↔ ((𝐸[:]𝐼) ∈ ℕ ∨ (𝐸[:]𝐼) = 0))
28	26, 27	sylib 220	. . . . . . 7 ⊢ (𝜑 → ((𝐸[:]𝐼) ∈ ℕ ∨ (𝐸[:]𝐼) = 0))
29		extdggt0 33853	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐼 → 0 < (𝐸[:]𝐼))
30	9, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐸[:]𝐼))
31	30	gt0ne0d 11709	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ≠ 0)
32	31	neneqd 2941	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐸[:]𝐼) = 0)
33	28, 32	olcnd 884	. . . . . 6 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ)
34	33	nnred 12184	. . . . 5 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℝ)
35	25	nnred 12184	. . . . 5 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ)
36		rexmul 13218	. . . . 5 ⊢ (((𝐸[:]𝐼) ∈ ℝ ∧ (𝐼[:]𝐾) ∈ ℝ) → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
37	34, 35, 36	syl2anc 591	. . . 4 ⊢ (𝜑 → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
38	14, 37	eqtrd 2776	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
39	33, 25	nnmulcld 12225	. . 3 ⊢ (𝜑 → ((𝐸[:]𝐼) · (𝐼[:]𝐾)) ∈ ℕ)
40	38, 39	eqeltrd 2841	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ∈ ℕ)
41		2nn0 12449	. . . . 5 ⊢ 2 ∈ ℕ₀
42	24, 41	eqeltrdi 2849	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ₀)
43		uncom 4091	. . . . . . 7 ⊢ (𝐺 ∪ 𝐻) = (𝐻 ∪ 𝐺)
44	43	oveq2i 7371	. . . . . 6 ⊢ (𝐿 fldGen (𝐺 ∪ 𝐻)) = (𝐿 fldGen (𝐻 ∪ 𝐺))
45	44	oveq2i 7371	. . . . 5 ⊢ (𝐿 ↾_s (𝐿 fldGen (𝐺 ∪ 𝐻))) = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
46	3, 45	eqtri 2764	. . . 4 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
47	11, 15, 2, 4, 16, 10, 6, 5, 42, 46, 22	fldextrspundgdvds 33877	. . 3 ⊢ (𝜑 → (𝐽[:]𝐾) ∥ (𝐸[:]𝐾))
48	17, 47	eqbrtrrd 5099	. 2 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐸[:]𝐾))
49	11, 2, 15, 4, 10, 16, 5, 6, 23, 3	fldextrspundglemul 33875	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))
50	22	nnred 12184	. . . . 5 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℝ)
51		rexmul 13218	. . . . 5 ⊢ (((𝐼[:]𝐾) ∈ ℝ ∧ (𝐽[:]𝐾) ∈ ℝ) → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐼[:]𝐾) · (𝐽[:]𝐾)))
52	35, 50, 51	syl2anc 591	. . . 4 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐼[:]𝐾) · (𝐽[:]𝐾)))
53	24, 17	oveq12d 7378	. . . 4 ⊢ (𝜑 → ((𝐼[:]𝐾) · (𝐽[:]𝐾)) = (2 · (2↑𝑁)))
54		2cnd 12254	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
55	54, 20	expcld 14103	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
56	54, 55	mulcomd 11161	. . . . 5 ⊢ (𝜑 → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
57	54, 20	expp1d 14104	. . . . 5 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
58	56, 57	eqtr4d 2779	. . . 4 ⊢ (𝜑 → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
59	52, 53, 58	3eqtrd 2780	. . 3 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = (2↑(𝑁 + 1)))
60	49, 59	breqtrd 5101	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ (2↑(𝑁 + 1)))
61	40, 20, 48, 60	2exple2exp 32941	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (𝐸[:]𝐾) = (2↑𝑛))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 854 = wceq 1548 ∈ wcel 2121 ∃wrex 3065 ∪ cun 3883 ⊆ wss 3885 class class class wbr 5075 ‘cfv 6489 (class class class)co 7360 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 < clt 11174 ≤ cle 11175 ℕcn 12169 2c2 12231 ℕ₀cn0 12432 ·_e cxmu 13057 ↑cexp 14018 ∥ cdvds 16216 Basecbs 17174 ↾_s cress 17195 Fieldcfield 20706 SubDRingcsdrg 20762 fldGen cfldgen 33398 /_FldExtcfldext 33834 [:]cextdg 33836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-reg 9501 ax-inf2 9557 ax-ac2 10380 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-rpss 7670 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-inf 9350 df-oi 9419 df-r1 9683 df-rank 9684 df-dju 9820 df-card 9858 df-acn 9861 df-ac 10033 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-ind 12155 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-icc 13300 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-word 14471 df-lsw 14520 df-concat 14528 df-s1 14554 df-substr 14599 df-pfx 14629 df-s2 14805 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644 df-dvds 16217 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ocomp 17236 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mre 17543 df-mrc 17544 df-mri 17545 df-acs 17546 df-proset 18255 df-drs 18256 df-poset 18274 df-ipo 18489 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-ghm 19183 df-cntz 19287 df-cntr 19288 df-lsm 19606 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-nzr 20489 df-subrng 20522 df-subrg 20546 df-rgspn 20587 df-rlreg 20670 df-domn 20671 df-idom 20672 df-drng 20707 df-field 20708 df-sdrg 20763 df-lmod 20856 df-lss 20926 df-lsp 20966 df-lmhm 21016 df-lmim 21017 df-lbs 21069 df-lvec 21097 df-sra 21167 df-rgmod 21168 df-cnfld 21352 df-zring 21426 df-dsmm 21711 df-frlm 21726 df-uvc 21762 df-lindf 21785 df-linds 21786 df-assa 21832 df-fldgen 33399 df-dim 33796 df-fldext 33837 df-extdg 33838

This theorem is referenced by: (None)

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