fldextrspunfld - Mathbox for Thierry Arnoux

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Theorem fldextrspunfld 34007

Description: The ring generated by the union of two field extensions is a field. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)

**Hypotheses**
Ref	Expression
fldextrspunfld.k	⊢ 𝐾 = (𝐿 ↾_s 𝐹)
fldextrspunfld.i	⊢ 𝐼 = (𝐿 ↾_s 𝐺)
fldextrspunfld.j	⊢ 𝐽 = (𝐿 ↾_s 𝐻)
fldextrspunfld.2	⊢ (𝜑 → 𝐿 ∈ Field)
fldextrspunfld.3	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
fldextrspunfld.4	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
fldextrspunfld.5	⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
fldextrspunfld.6	⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
fldextrspunfld.7	⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
fldextrspunfld.n	⊢ 𝑁 = (RingSpan‘𝐿)
fldextrspunfld.c	⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
fldextrspunfld.e	⊢ 𝐸 = (𝐿 ↾_s 𝐶)

**Assertion**
Ref	Expression
fldextrspunfld	⊢ (𝜑 → 𝐸 ∈ Field)

Proof of Theorem fldextrspunfld Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2769	. . 3 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) = (Scalar‘((subringAlg ‘𝐸)‘𝐺))
2		fldextrspunfld.e	. . . . . 6 ⊢ 𝐸 = (𝐿 ↾_s 𝐶)
3		fldextrspunfld.2	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ Field)
4	3	flddrngd 20821	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
5	4	drngringd 20817	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
6		eqidd 2770	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
7		fldextrspunfld.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
8		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝐿) = (Base‘𝐿)
9	8	sdrgss 20870	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
10	7, 9	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
11		fldextrspunfld.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
12	8	sdrgss 20870	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
13	11, 12	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
14	10, 13	unssd 4153	. . . . . . . 8 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ (Base‘𝐿))
15		fldextrspunfld.n	. . . . . . . . 9 ⊢ 𝑁 = (RingSpan‘𝐿)
16	15	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (RingSpan‘𝐿))
17		fldextrspunfld.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)))
19	5, 6, 14, 16, 18	rgspncl 20694	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
20	3, 19	subrfld 33544	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ IDomn)
21	2, 20	eqeltrid 2873	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ IDomn)
22	21	idomcringd 20807	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 20868	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
24	7, 23	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
25	5, 6, 14, 16, 18	rgspnssid 20695	. . . . . 6 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
26	25	unssad 4154	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
27	2	subsubrg 20679	. . . . . 6 ⊢ (𝐶 ∈ (SubRing‘𝐿) → (𝐺 ∈ (SubRing‘𝐸) ↔ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)))
28	27	biimpar 482	. . . . 5 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)) → 𝐺 ∈ (SubRing‘𝐸))
29	19, 24, 26, 28	syl12anc 849	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐸))
30		eqid 2769	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
31	30	sraassa 21984	. . . 4 ⊢ ((𝐸 ∈ CRing ∧ 𝐺 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
32	22, 29, 31	syl2anc 595	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
33		eqid 2769	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
34	8	subrgss 20653	. . . . . . 7 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
35	19, 34	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
36	2, 8	ressbas2 17294	. . . . . 6 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
37	35, 36	syl 18	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
38	26, 37	sseqtrd 3981	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
39	30, 33, 21, 38	sraidom 33914	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ IDomn)
40		ressabs 17304	. . . . . . 7 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶) → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
41	19, 26, 40	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
42	2	oveq1i 7418	. . . . . 6 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
43		fldextrspunfld.i	. . . . . 6 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
44	41, 42, 43	3eqtr4g 2829	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
45		eqidd 2770	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
46	45, 38	srasca 21275	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
47	44, 46	eqtr3d 2806	. . . 4 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
48	43	sdrgdrng 20867	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
49	7, 48	syl 18	. . . 4 ⊢ (𝜑 → 𝐼 ∈ DivRing)
50	47, 49	eqeltrrd 2870	. . 3 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
51	30	sralmod 21282	. . . . . . 7 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
52	29, 51	syl 18	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
53	1	islvec 21199	. . . . . 6 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec ↔ (((subringAlg ‘𝐸)‘𝐺) ∈ LMod ∧ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing))
54	52, 50, 53	sylanbrc 594	. . . . 5 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LVec)
55		dimcl 33934	. . . . 5 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀^*)
56	54, 55	syl 18	. . . 4 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀^*)
57		fldextrspunfld.7	. . . 4 ⊢ (𝜑 → (𝐽[:]𝐾) ∈ ℕ₀)
58		fldextrspunfld.k	. . . . 5 ⊢ 𝐾 = (𝐿 ↾_s 𝐹)
59		fldextrspunfld.j	. . . . 5 ⊢ 𝐽 = (𝐿 ↾_s 𝐻)
60		fldextrspunfld.3	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
61		fldextrspunfld.4	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
62	58, 43, 59, 3, 60, 61, 7, 11, 57, 15, 17, 2	fldextrspunlem1 34006	. . . 4 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
63		xnn0lenn0nn0 13267	. . . 4 ⊢ (((dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀^* ∧ (𝐽[:]𝐾) ∈ ℕ₀ ∧ (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾)) → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀)
64	56, 57, 62, 63	syl3anc 1396	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀)
65	1, 32, 39, 50, 64	assafld 33968	. 2 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ Field)
66	45, 38	srabase 21272	. . . 4 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
67	37, 66	eqtrd 2804	. . 3 ⊢ (𝜑 → 𝐶 = (Base‘((subringAlg ‘𝐸)‘𝐺)))
68	45, 38	sraaddg 21273	. . . 4 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘((subringAlg ‘𝐸)‘𝐺)))
69	68	oveqdr 7436	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘𝐸)𝑦) = (𝑥(+_g‘((subringAlg ‘𝐸)‘𝐺))𝑦))
70	45, 38	sramulr 21274	. . . 4 ⊢ (𝜑 → (._r‘𝐸) = (._r‘((subringAlg ‘𝐸)‘𝐺)))
71	70	oveqdr 7436	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐸)𝑦) = (𝑥(._r‘((subringAlg ‘𝐸)‘𝐺))𝑦))
72	37, 67, 69, 71	fldpropd 20848	. 2 ⊢ (𝜑 → (𝐸 ∈ Field ↔ ((subringAlg ‘𝐸)‘𝐺) ∈ Field))
73	65, 72	mpbird 260	1 ⊢ (𝜑 → 𝐸 ∈ Field)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∪ cun 3911 ⊆ wss 3913 class class class wbr 5110 ‘cfv 6534 (class class class)co 7408 ≤ cle 11240 ℕ₀cn0 12500 ℕ₀^*cxnn0 12573 Basecbs 17265 ↾_s cress 17286 +_gcplusg 17306 ._rcmulr 17307 Scalarcsca 17309 CRingccrg 20312 SubRingcsubrg 20650 RingSpancrgspn 20691 IDomncidom 20774 DivRingcdr 20809 Fieldcfield 20810 SubDRingcsdrg 20863 LModclmod 20955 LVecclvec 21197 subringAlg csra 21266 AssAlgcasa 21965 dimcldim 33930 [:]cextdg 33971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-reg 9550 ax-inf2 9606 ax-ac2 10443 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174 ax-addf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-rpss 7718 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-sup 9398 df-inf 9399 df-oi 9468 df-r1 9732 df-rank 9733 df-dju 9883 df-card 9921 df-acn 9924 df-ac 10096 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-ind 12215 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-xnn0 12574 df-z 12588 df-dec 12708 df-uz 12859 df-rp 13013 df-xadd 13134 df-fz 13532 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-word 14547 df-lsw 14596 df-concat 14604 df-s1 14630 df-substr 14675 df-pfx 14705 df-s2 14881 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ocomp 17327 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-0g 17490 df-gsum 17491 df-prds 17496 df-pws 17498 df-mre 17634 df-mrc 17635 df-mri 17636 df-acs 17637 df-proset 18346 df-drs 18347 df-poset 18365 df-ipo 18580 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-mhm 18837 df-submnd 18838 df-grp 18999 df-minusg 19000 df-sbg 19001 df-mulg 19130 df-subg 19185 df-ghm 19280 df-cntz 19383 df-cntr 19384 df-lsm 19702 df-cmn 19848 df-abl 19849 df-mgp 20213 df-rng 20227 df-ur 20260 df-ring 20313 df-cring 20314 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-nzr 20592 df-subrng 20627 df-subrg 20651 df-rgspn 20692 df-rlreg 20775 df-domn 20776 df-idom 20777 df-drng 20811 df-field 20812 df-sdrg 20864 df-lmod 20957 df-lss 21027 df-lsp 21067 df-lmhm 21117 df-lmim 21118 df-lbs 21170 df-lvec 21198 df-sra 21268 df-rgmod 21269 df-cnfld 21488 df-zring 21562 df-dsmm 21847 df-frlm 21862 df-uvc 21898 df-lindf 21921 df-linds 21922 df-assa 21968 df-dim 33931 df-fldext 33972 df-extdg 33973

This theorem is referenced by: fldextrspunlem2 34008 fldextrspundgdvdslem 34011 fldextrspundgdvds 34012

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