fldextrspunfld - Mathbox for Thierry Arnoux

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Theorem fldextrspunfld 33681

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 2731	. . 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 20651	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
5	4	drngringd 20647	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
6		eqidd 2732	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
7		fldextrspunfld.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
8		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐿) = (Base‘𝐿)
9	8	sdrgss 20703	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
11		fldextrspunfld.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
12	8	sdrgss 20703	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
14	10, 13	unssd 4137	. . . . . . . 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 20523	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
20	3, 19	subrfld 33245	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ IDomn)
21	2, 20	eqeltrid 2835	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ IDomn)
22	21	idomcringd 20637	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 20701	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
24	7, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
25	5, 6, 14, 16, 18	rgspnssid 20524	. . . . . 6 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
26	25	unssad 4138	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
27	2	subsubrg 20508	. . . . . 6 ⊢ (𝐶 ∈ (SubRing‘𝐿) → (𝐺 ∈ (SubRing‘𝐸) ↔ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)))
28	27	biimpar 477	. . . . 5 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)) → 𝐺 ∈ (SubRing‘𝐸))
29	19, 24, 26, 28	syl12anc 836	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐸))
30		eqid 2731	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
31	30	sraassa 21801	. . . 4 ⊢ ((𝐸 ∈ CRing ∧ 𝐺 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
32	22, 29, 31	syl2anc 584	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
33		eqid 2731	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
34	8	subrgss 20482	. . . . . . 7 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
35	19, 34	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
36	2, 8	ressbas2 17144	. . . . . 6 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
37	35, 36	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
38	26, 37	sseqtrd 3966	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
39	30, 33, 21, 38	sraidom 33587	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ IDomn)
40		ressabs 17154	. . . . . . 7 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶) → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
41	19, 26, 40	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
42	2	oveq1i 7351	. . . . . 6 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
43		fldextrspunfld.i	. . . . . 6 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
44	41, 42, 43	3eqtr4g 2791	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
45		eqidd 2732	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
46	45, 38	srasca 21109	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
47	44, 46	eqtr3d 2768	. . . 4 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
48	43	sdrgdrng 20700	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
49	7, 48	syl 17	. . . 4 ⊢ (𝜑 → 𝐼 ∈ DivRing)
50	47, 49	eqeltrrd 2832	. . 3 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
51	30	sralmod 21116	. . . . . . 7 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
52	29, 51	syl 17	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
53	1	islvec 21033	. . . . . 6 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec ↔ (((subringAlg ‘𝐸)‘𝐺) ∈ LMod ∧ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing))
54	52, 50, 53	sylanbrc 583	. . . . 5 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LVec)
55		dimcl 33607	. . . . 5 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀^*)
56	54, 55	syl 17	. . . 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 33680	. . . 4 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
63		xnn0lenn0nn0 13139	. . . 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 1373	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀)
65	1, 32, 39, 50, 64	assafld 33642	. 2 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ Field)
66	45, 38	srabase 21106	. . . 4 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
67	37, 66	eqtrd 2766	. . 3 ⊢ (𝜑 → 𝐶 = (Base‘((subringAlg ‘𝐸)‘𝐺)))
68	45, 38	sraaddg 21107	. . . 4 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘((subringAlg ‘𝐸)‘𝐺)))
69	68	oveqdr 7369	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘𝐸)𝑦) = (𝑥(+_g‘((subringAlg ‘𝐸)‘𝐺))𝑦))
70	45, 38	sramulr 21108	. . . 4 ⊢ (𝜑 → (._r‘𝐸) = (._r‘((subringAlg ‘𝐸)‘𝐺)))
71	70	oveqdr 7369	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐸)𝑦) = (𝑥(._r‘((subringAlg ‘𝐸)‘𝐺))𝑦))
72	37, 67, 69, 71	fldpropd 20680	. 2 ⊢ (𝜑 → (𝐸 ∈ Field ↔ ((subringAlg ‘𝐸)‘𝐺) ∈ Field))
73	65, 72	mpbird 257	1 ⊢ (𝜑 → 𝐸 ∈ Field)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∪ cun 3895 ⊆ wss 3897 class class class wbr 5086 ‘cfv 6476 (class class class)co 7341 ≤ cle 11142 ℕ₀cn0 12376 ℕ₀^*cxnn0 12449 Basecbs 17115 ↾_s cress 17136 +_gcplusg 17156 ._rcmulr 17157 Scalarcsca 17159 CRingccrg 20147 SubRingcsubrg 20479 RingSpancrgspn 20520 IDomncidom 20603 DivRingcdr 20639 Fieldcfield 20640 SubDRingcsdrg 20696 LModclmod 20788 LVecclvec 21031 subringAlg csra 21100 AssAlgcasa 21782 dimcldim 33603 [:]cextdg 33645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-reg 9473 ax-inf2 9526 ax-ac2 10349 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-rpss 7651 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-oadd 8384 df-er 8617 df-map 8747 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-sup 9321 df-inf 9322 df-oi 9391 df-r1 9652 df-rank 9653 df-dju 9789 df-card 9827 df-acn 9830 df-ac 10002 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-xnn0 12450 df-z 12464 df-dec 12584 df-uz 12728 df-rp 12886 df-xadd 13007 df-fz 13403 df-fzo 13550 df-seq 13904 df-exp 13964 df-hash 14233 df-word 14416 df-lsw 14465 df-concat 14473 df-s1 14499 df-substr 14544 df-pfx 14574 df-s2 14750 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-clim 15390 df-sum 15589 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ocomp 17177 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-0g 17340 df-gsum 17341 df-prds 17346 df-pws 17348 df-mre 17483 df-mrc 17484 df-mri 17485 df-acs 17486 df-proset 18195 df-drs 18196 df-poset 18214 df-ipo 18429 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-submnd 18687 df-grp 18844 df-minusg 18845 df-sbg 18846 df-mulg 18976 df-subg 19031 df-ghm 19120 df-cntz 19224 df-cntr 19225 df-lsm 19543 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-cring 20149 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-nzr 20423 df-subrng 20456 df-subrg 20480 df-rgspn 20521 df-rlreg 20604 df-domn 20605 df-idom 20606 df-drng 20641 df-field 20642 df-sdrg 20697 df-lmod 20790 df-lss 20860 df-lsp 20900 df-lmhm 20951 df-lmim 20952 df-lbs 21004 df-lvec 21032 df-sra 21102 df-rgmod 21103 df-cnfld 21287 df-zring 21379 df-dsmm 21664 df-frlm 21679 df-uvc 21715 df-lindf 21738 df-linds 21739 df-assa 21785 df-ind 32824 df-dim 33604 df-fldext 33646 df-extdg 33647

This theorem is referenced by: fldextrspunlem2 33682 fldextrspundgdvdslem 33685 fldextrspundgdvds 33686

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