fldextrspunfld - Mathbox for Thierry Arnoux

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Theorem fldextrspunfld 34419

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 2763	. . 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 21045	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
5	4	drngringd 21039	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
6		eqidd 2764	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
7		fldextrspunfld.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
8		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝐿) = (Base‘𝐿)
9	8	sdrgss 21101	. . . . . . . . . 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 21101	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
13	11, 12	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
14	10, 13	unssd 4193	. . . . . . . 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 20916	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
20	3, 19	subrfld 33947	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ IDomn)
21	2, 20	eqeltrid 2868	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ IDomn)
22	21	idomcringd 21029	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 21099	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
24	7, 23	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
25	5, 6, 14, 16, 18	rgspnssid 20917	. . . . . 6 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
26	25	unssad 4194	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
27	2	subsubrg 20901	. . . . . 6 ⊢ (𝐶 ∈ (SubRing‘𝐿) → (𝐺 ∈ (SubRing‘𝐸) ↔ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)))
28	27	biimpar 483	. . . . 5 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ (𝐺 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶)) → 𝐺 ∈ (SubRing‘𝐸))
29	19, 24, 26, 28	syl12anc 850	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐸))
30		eqid 2763	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
31	30	sraassa 22232	. . . 4 ⊢ ((𝐸 ∈ CRing ∧ 𝐺 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
32	22, 29, 31	syl2anc 596	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
33		eqid 2763	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
34	8	subrgss 20875	. . . . . . 7 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
35	19, 34	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
36	2, 8	ressbas2 17460	. . . . . 6 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
37	35, 36	syl 18	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
38	26, 37	sseqtrd 3949	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
39	30, 33, 21, 38	sraidom 34326	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ IDomn)
40		ressabs 17470	. . . . . . 7 ⊢ ((𝐶 ∈ (SubRing‘𝐿) ∧ 𝐺 ⊆ 𝐶) → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
41	19, 26, 40	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((𝐿 ↾_s 𝐶) ↾_s 𝐺) = (𝐿 ↾_s 𝐺))
42	2	oveq1i 7461	. . . . . 6 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
43		fldextrspunfld.i	. . . . . 6 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
44	41, 42, 43	3eqtr4g 2823	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
45		eqidd 2764	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
46	45, 38	srasca 21506	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
47	44, 46	eqtr3d 2800	. . . 4 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
48	43	sdrgdrng 21098	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
49	7, 48	syl 18	. . . 4 ⊢ (𝜑 → 𝐼 ∈ DivRing)
50	47, 49	eqeltrrd 2865	. . 3 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
51	30	sralmod 21513	. . . . . . 7 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
52	29, 51	syl 18	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
53	1	islvec 21430	. . . . . 6 ⊢ (((subringAlg ‘𝐸)‘𝐺) ∈ LVec ↔ (((subringAlg ‘𝐸)‘𝐺) ∈ LMod ∧ (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing))
54	52, 50, 53	sylanbrc 595	. . . . 5 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LVec)
55		dimcl 34346	. . . . 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 34418	. . . 4 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
63		xnn0lenn0nn0 13415	. . . 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 1399	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀)
65	1, 32, 39, 50, 64	assafld 34380	. 2 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ Field)
66	45, 38	srabase 21503	. . . 4 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
67	37, 66	eqtrd 2798	. . 3 ⊢ (𝜑 → 𝐶 = (Base‘((subringAlg ‘𝐸)‘𝐺)))
68	45, 38	sraaddg 21504	. . . 4 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘((subringAlg ‘𝐸)‘𝐺)))
69	68	oveqdr 7479	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘𝐸)𝑦) = (𝑥(+_g‘((subringAlg ‘𝐸)‘𝐺))𝑦))
70	45, 38	sramulr 21505	. . . 4 ⊢ (𝜑 → (._r‘𝐸) = (._r‘((subringAlg ‘𝐸)‘𝐺)))
71	70	oveqdr 7479	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐸)𝑦) = (𝑥(._r‘((subringAlg ‘𝐸)‘𝐺))𝑦))
72	37, 67, 69, 71	fldpropd 21079	. 2 ⊢ (𝜑 → (𝐸 ∈ Field ↔ ((subringAlg ‘𝐸)‘𝐺) ∈ Field))
73	65, 72	mpbird 260	1 ⊢ (𝜑 → 𝐸 ∈ Field)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∪ cun 4152 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ≤ cle 11383 ℕ₀cn0 12646 ℕ₀^*cxnn0 12719 Basecbs 17431 ↾_s cress 17452 +_gcplusg 17472 ._rcmulr 17473 Scalarcsca 17475 CRingccrg 20511 SubRingcsubrg 20872 RingSpancrgspn 20913 IDomncidom 20996 DivRingcdr 21031 Fieldcfield 21032 SubDRingcsdrg 21094 LModclmod 21186 LVecclvec 21428 subringAlg csra 21497 AssAlgcasa 22213 dimcldim 34342 [:]cextdg 34383

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-ac2 10580 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-rpss 7771 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-tpos 8281 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-er 8755 df-map 8887 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-acn 10060 df-ac 10234 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-ind 12361 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-rp 13161 df-xadd 13282 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-hash 14516 df-word 14700 df-lsw 14749 df-concat 14757 df-s1 14784 df-substr 14831 df-pfx 14863 df-s2 15041 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-sum 15898 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ocomp 17494 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-0g 17662 df-gsum 17663 df-prds 17668 df-pws 17670 df-mre 17806 df-mrc 17807 df-mri 17808 df-acs 17809 df-proset 18518 df-drs 18519 df-poset 18537 df-ipo 18752 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-mhm 19028 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-ghm 19478 df-cntz 19582 df-cntr 19583 df-lsm 19901 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-cring 20513 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-nzr 20814 df-subrng 20849 df-subrg 20873 df-rgspn 20914 df-rlreg 20997 df-domn 20998 df-idom 20999 df-drng 21033 df-field 21034 df-sdrg 21095 df-lmod 21188 df-lss 21258 df-lsp 21298 df-lmhm 21348 df-lmim 21349 df-lbs 21401 df-lvec 21429 df-sra 21499 df-rgmod 21500 df-cnfld 21730 df-zring 21807 df-dsmm 22093 df-frlm 22108 df-uvc 22144 df-lindf 22167 df-linds 22168 df-assa 22216 df-dim 34343 df-fldext 34384 df-extdg 34385

This theorem is used by: fldextrspunlem2 34420 fldextrspundgdvdslem 34423 fldextrspundgdvds 34424

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