fldextrspunfld - Mathbox for Thierry Arnoux

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Theorem fldextrspunfld 34301

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 2761	. . 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 20987	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
5	4	drngringd 20981	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
6		eqidd 2762	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
7		fldextrspunfld.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
8		eqid 2761	. . . . . . . . . . 11 ⊢ (Base‘𝐿) = (Base‘𝐿)
9	8	sdrgss 21043	. . . . . . . . . 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 21043	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
13	11, 12	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
14	10, 13	unssd 4138	. . . . . . . 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 20858	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
20	3, 19	subrfld 33841	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ IDomn)
21	2, 20	eqeltrid 2865	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ IDomn)
22	21	idomcringd 20971	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 21041	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
24	7, 23	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
25	5, 6, 14, 16, 18	rgspnssid 20859	. . . . . 6 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
26	25	unssad 4139	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
27	2	subsubrg 20843	. . . . . 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 2761	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
31	30	sraassa 22170	. . . 4 ⊢ ((𝐸 ∈ CRing ∧ 𝐺 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
32	22, 29, 31	syl2anc 596	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
33		eqid 2761	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
34	8	subrgss 20817	. . . . . . 7 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
35	19, 34	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
36	2, 8	ressbas2 17409	. . . . . 6 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
37	35, 36	syl 18	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
38	26, 37	sseqtrd 3967	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
39	30, 33, 21, 38	sraidom 34208	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ IDomn)
40		ressabs 17419	. . . . . . 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 7428	. . . . . 6 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
43		fldextrspunfld.i	. . . . . 6 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
44	41, 42, 43	3eqtr4g 2821	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
45		eqidd 2762	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
46	45, 38	srasca 21448	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
47	44, 46	eqtr3d 2798	. . . 4 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
48	43	sdrgdrng 21040	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
49	7, 48	syl 18	. . . 4 ⊢ (𝜑 → 𝐼 ∈ DivRing)
50	47, 49	eqeltrrd 2862	. . 3 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
51	30	sralmod 21455	. . . . . . 7 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
52	29, 51	syl 18	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
53	1	islvec 21372	. . . . . 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 34228	. . . . 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 34300	. . . 4 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
63		xnn0lenn0nn0 13368	. . . 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 1398	. . 3 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ∈ ℕ₀)
65	1, 32, 39, 50, 64	assafld 34262	. 2 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ Field)
66	45, 38	srabase 21445	. . . 4 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
67	37, 66	eqtrd 2796	. . 3 ⊢ (𝜑 → 𝐶 = (Base‘((subringAlg ‘𝐸)‘𝐺)))
68	45, 38	sraaddg 21446	. . . 4 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘((subringAlg ‘𝐸)‘𝐺)))
69	68	oveqdr 7446	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘𝐸)𝑦) = (𝑥(+_g‘((subringAlg ‘𝐸)‘𝐺))𝑦))
70	45, 38	sramulr 21447	. . . 4 ⊢ (𝜑 → (._r‘𝐸) = (._r‘((subringAlg ‘𝐸)‘𝐺)))
71	70	oveqdr 7446	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐸)𝑦) = (𝑥(._r‘((subringAlg ‘𝐸)‘𝐺))𝑦))
72	37, 67, 69, 71	fldpropd 21021	. 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 2145 ∪ cun 3897 ⊆ wss 3899 class class class wbr 5103 ‘cfv 6537 (class class class)co 7418 ≤ cle 11337 ℕ₀cn0 12599 ℕ₀^*cxnn0 12672 Basecbs 17380 ↾_s cress 17401 +_gcplusg 17421 ._rcmulr 17422 Scalarcsca 17424 CRingccrg 20453 SubRingcsubrg 20814 RingSpancrgspn 20855 IDomncidom 20938 DivRingcdr 20973 Fieldcfield 20974 SubDRingcsdrg 21036 LModclmod 21128 LVecclvec 21370 subringAlg csra 21439 AssAlgcasa 22151 dimcldim 34224 [:]cextdg 34265

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-reg 9579 ax-inf2 9635 ax-ac2 10534 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-rpss 7737 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-tpos 8236 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-oadd 8473 df-er 8710 df-map 8842 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-sup 9427 df-inf 9428 df-oi 9497 df-r1 9761 df-rank 9762 df-scott 9922 df-dju 9975 df-card 10013 df-acn 10016 df-ac 10188 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-ind 12314 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-xnn0 12673 df-z 12687 df-dec 12808 df-uz 12959 df-rp 13114 df-xadd 13235 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-hash 14468 df-word 14652 df-lsw 14701 df-concat 14709 df-s1 14736 df-substr 14782 df-pfx 14814 df-s2 14992 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-sum 15847 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ocomp 17442 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-0g 17605 df-gsum 17606 df-prds 17611 df-pws 17613 df-mre 17749 df-mrc 17750 df-mri 17751 df-acs 17752 df-proset 18461 df-drs 18462 df-poset 18480 df-ipo 18695 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-mhm 18971 df-submnd 18972 df-grp 19140 df-minusg 19141 df-sbg 19142 df-mulg 19271 df-subg 19326 df-ghm 19421 df-cntz 19524 df-cntr 19525 df-lsm 19843 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-cring 20455 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-nzr 20756 df-subrng 20791 df-subrg 20815 df-rgspn 20856 df-rlreg 20939 df-domn 20940 df-idom 20941 df-drng 20975 df-field 20976 df-sdrg 21037 df-lmod 21130 df-lss 21200 df-lsp 21240 df-lmhm 21290 df-lmim 21291 df-lbs 21343 df-lvec 21371 df-sra 21441 df-rgmod 21442 df-cnfld 21672 df-zring 21746 df-dsmm 22031 df-frlm 22046 df-uvc 22082 df-lindf 22105 df-linds 22106 df-assa 22154 df-dim 34225 df-fldext 34266 df-extdg 34267

This theorem is used by: fldextrspunlem2 34302 fldextrspundgdvdslem 34305 fldextrspundgdvds 34306

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