fldext2rspun - Mathbox for Thierry Arnoux

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Theorem fldext2rspun 33841

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 2737	. . . . . 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 20730	. . . . . . 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 33827	. . . . 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 33821	. . . . 5 ⊢ (𝜑 → 𝐼/_FldExt𝐾)
13		extdgmul 33822	. . . . 5 ⊢ ((𝐸/_FldExt𝐼 ∧ 𝐼/_FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)))
14	9, 12, 13	syl2anc 585	. . . 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 12222	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℕ
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℕ)
20		fldext2rspun.n	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	19, 20	nnexpcld 14172	. . . . . . . . . . 11 ⊢ (𝜑 → (2↑𝑁) ∈ ℕ)
22	17, 21	eqeltrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ)
23	22	nnnn0d 12466	. . . . . . . . 9 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
24		fldext2rspun.1	. . . . . . . . . 10 ⊢ (𝜑 → (𝐼[:]𝐾) = 2)
25	24, 18	eqeltrdi 2845	. . . . . . . . 9 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ)
26	11, 2, 15, 4, 10, 16, 5, 6, 23, 3, 25	fldextrspundgdvdslem 33839	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ₀)
27		elnn0 12407	. . . . . . . 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 33816	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐼 → 0 < (𝐸[:]𝐼))
30	9, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 < (𝐸[:]𝐼))
31	30	gt0ne0d 11705	. . . . . . . 8 ⊢ (𝜑 → (𝐸[:]𝐼) ≠ 0)
32	31	neneqd 2938	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐸[:]𝐼) = 0)
33	28, 32	olcnd 878	. . . . . 6 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℕ)
34	33	nnred 12164	. . . . 5 ⊢ (𝜑 → (𝐸[:]𝐼) ∈ ℝ)
35	25	nnred 12164	. . . . 5 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℝ)
36		rexmul 13190	. . . . 5 ⊢ (((𝐸[:]𝐼) ∈ ℝ ∧ (𝐼[:]𝐾) ∈ ℝ) → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
37	34, 35, 36	syl2anc 585	. . . 4 ⊢ (𝜑 → ((𝐸[:]𝐼) ·_e (𝐼[:]𝐾)) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
38	14, 37	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) = ((𝐸[:]𝐼) · (𝐼[:]𝐾)))
39	33, 25	nnmulcld 12202	. . 3 ⊢ (𝜑 → ((𝐸[:]𝐼) · (𝐼[:]𝐾)) ∈ ℕ)
40	38, 39	eqeltrd 2837	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ∈ ℕ)
41		2nn0 12422	. . . . 5 ⊢ 2 ∈ ℕ₀
42	24, 41	eqeltrdi 2845	. . . 4 ⊢ (𝜑 → (𝐼[:]𝐾) ∈ ℕ₀)
43		uncom 4111	. . . . . . 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 2760	. . . 4 ⊢ 𝐸 = (𝐿 ↾_s (𝐿 fldGen (𝐻 ∪ 𝐺)))
47	11, 15, 2, 4, 16, 10, 6, 5, 42, 46, 22	fldextrspundgdvds 33840	. . 3 ⊢ (𝜑 → (𝐽[:]𝐾) ∥ (𝐸[:]𝐾))
48	17, 47	eqbrtrrd 5123	. 2 ⊢ (𝜑 → (2↑𝑁) ∥ (𝐸[:]𝐾))
49	11, 2, 15, 4, 10, 16, 5, 6, 23, 3	fldextrspundglemul 33838	. . 3 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)))
50	22	nnred 12164	. . . . 5 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℝ)
51		rexmul 13190	. . . . 5 ⊢ (((𝐼[:]𝐾) ∈ ℝ ∧ (𝐽[:]𝐾) ∈ ℝ) → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = ((𝐼[:]𝐾) · (𝐽[:]𝐾)))
52	35, 50, 51	syl2anc 585	. . . 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 12227	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℂ)
55	54, 20	expcld 14073	. . . . . 6 ⊢ (𝜑 → (2↑𝑁) ∈ ℂ)
56	54, 55	mulcomd 11157	. . . . 5 ⊢ (𝜑 → (2 · (2↑𝑁)) = ((2↑𝑁) · 2))
57	54, 20	expp1d 14074	. . . . 5 ⊢ (𝜑 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2))
58	56, 57	eqtr4d 2775	. . . 4 ⊢ (𝜑 → (2 · (2↑𝑁)) = (2↑(𝑁 + 1)))
59	52, 53, 58	3eqtrd 2776	. . 3 ⊢ (𝜑 → ((𝐼[:]𝐾) ·_e (𝐽[:]𝐾)) = (2↑(𝑁 + 1)))
60	49, 59	breqtrd 5125	. 2 ⊢ (𝜑 → (𝐸[:]𝐾) ≤ (2↑(𝑁 + 1)))
61	40, 20, 48, 60	2exple2exp 32928	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ₀ (𝐸[:]𝐾) = (2↑𝑛))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 ∪ cun 3900 ⊆ wss 3902 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 < clt 11170 ≤ cle 11171 ℕcn 12149 2c2 12204 ℕ₀cn0 12405 ·_e cxmu 13029 ↑cexp 13988 ∥ cdvds 16183 Basecbs 17140 ↾_s cress 17161 Fieldcfield 20667 SubDRingcsdrg 20723 fldGen cfldgen 33394 /_FldExtcfldext 33797 [:]cextdg 33799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-reg 9501 ax-inf2 9554 ax-ac2 10377 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 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 9680 df-rank 9681 df-dju 9817 df-card 9855 df-acn 9858 df-ac 10030 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-xnn0 12479 df-z 12493 df-dec 12612 df-uz 12756 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-icc 13272 df-fz 13428 df-fzo 13575 df-seq 13929 df-exp 13989 df-hash 14258 df-word 14441 df-lsw 14490 df-concat 14498 df-s1 14524 df-substr 14569 df-pfx 14599 df-s2 14775 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-sum 15614 df-dvds 16184 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ocomp 17202 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-0g 17365 df-gsum 17366 df-prds 17371 df-pws 17373 df-mre 17509 df-mrc 17510 df-mri 17511 df-acs 17512 df-proset 18221 df-drs 18222 df-poset 18240 df-ipo 18455 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18712 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-cntz 19250 df-cntr 19251 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-nzr 20450 df-subrng 20483 df-subrg 20507 df-rgspn 20548 df-rlreg 20631 df-domn 20632 df-idom 20633 df-drng 20668 df-field 20669 df-sdrg 20724 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 21691 df-frlm 21706 df-uvc 21742 df-lindf 21765 df-linds 21766 df-assa 21812 df-ind 32932 df-fldgen 33395 df-dim 33758 df-fldext 33800 df-extdg 33801

This theorem is referenced by: (None)

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