fldextrspunfld - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fldextrspunfld		Structured version Visualization version GIF version

Theorem fldextrspunfld 33677

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 2730	. . 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 20656	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
5	4	drngringd 20652	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
6		eqidd 2731	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐿) = (Base‘𝐿))
7		fldextrspunfld.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
8		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝐿) = (Base‘𝐿)
9	8	sdrgss 20708	. . . . . . . . . 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 20708	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
14	10, 13	unssd 4157	. . . . . . . 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 20528	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝐿))
20	3, 19	subrfld 33243	. . . . . 6 ⊢ (𝜑 → (𝐿 ↾_s 𝐶) ∈ IDomn)
21	2, 20	eqeltrid 2833	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ IDomn)
22	21	idomcringd 20642	. . . 4 ⊢ (𝜑 → 𝐸 ∈ CRing)
23		sdrgsubrg 20706	. . . . . 6 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
24	7, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
25	5, 6, 14, 16, 18	rgspnssid 20529	. . . . . 6 ⊢ (𝜑 → (𝐺 ∪ 𝐻) ⊆ 𝐶)
26	25	unssad 4158	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
27	2	subsubrg 20513	. . . . . 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 2730	. . . . 5 ⊢ ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺)
31	30	sraassa 21784	. . . 4 ⊢ ((𝐸 ∈ CRing ∧ 𝐺 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
32	22, 29, 31	syl2anc 584	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ AssAlg)
33		eqid 2730	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
34	8	subrgss 20487	. . . . . . 7 ⊢ (𝐶 ∈ (SubRing‘𝐿) → 𝐶 ⊆ (Base‘𝐿))
35	19, 34	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ (Base‘𝐿))
36	2, 8	ressbas2 17214	. . . . . 6 ⊢ (𝐶 ⊆ (Base‘𝐿) → 𝐶 = (Base‘𝐸))
37	35, 36	syl 17	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘𝐸))
38	26, 37	sseqtrd 3985	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐸))
39	30, 33, 21, 38	sraidom 33585	. . 3 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ IDomn)
40		ressabs 17224	. . . . . . 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 7399	. . . . . 6 ⊢ (𝐸 ↾_s 𝐺) = ((𝐿 ↾_s 𝐶) ↾_s 𝐺)
43		fldextrspunfld.i	. . . . . 6 ⊢ 𝐼 = (𝐿 ↾_s 𝐺)
44	41, 42, 43	3eqtr4g 2790	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = 𝐼)
45		eqidd 2731	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) = ((subringAlg ‘𝐸)‘𝐺))
46	45, 38	srasca 21093	. . . . 5 ⊢ (𝜑 → (𝐸 ↾_s 𝐺) = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
47	44, 46	eqtr3d 2767	. . . 4 ⊢ (𝜑 → 𝐼 = (Scalar‘((subringAlg ‘𝐸)‘𝐺)))
48	43	sdrgdrng 20705	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐼 ∈ DivRing)
49	7, 48	syl 17	. . . 4 ⊢ (𝜑 → 𝐼 ∈ DivRing)
50	47, 49	eqeltrrd 2830	. . 3 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐺)) ∈ DivRing)
51	30	sralmod 21100	. . . . . . 7 ⊢ (𝐺 ∈ (SubRing‘𝐸) → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
52	29, 51	syl 17	. . . . . 6 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ LMod)
53	1	islvec 21017	. . . . . 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 33604	. . . . 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 33676	. . . 4 ⊢ (𝜑 → (dim‘((subringAlg ‘𝐸)‘𝐺)) ≤ (𝐽[:]𝐾))
63		xnn0lenn0nn0 13211	. . . 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 33639	. 2 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐺) ∈ Field)
66	45, 38	srabase 21090	. . . 4 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐺)))
67	37, 66	eqtrd 2765	. . 3 ⊢ (𝜑 → 𝐶 = (Base‘((subringAlg ‘𝐸)‘𝐺)))
68	45, 38	sraaddg 21091	. . . 4 ⊢ (𝜑 → (+_g‘𝐸) = (+_g‘((subringAlg ‘𝐸)‘𝐺)))
69	68	oveqdr 7417	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+_g‘𝐸)𝑦) = (𝑥(+_g‘((subringAlg ‘𝐸)‘𝐺))𝑦))
70	45, 38	sramulr 21092	. . . 4 ⊢ (𝜑 → (._r‘𝐸) = (._r‘((subringAlg ‘𝐸)‘𝐺)))
71	70	oveqdr 7417	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(._r‘𝐸)𝑦) = (𝑥(._r‘((subringAlg ‘𝐸)‘𝐺))𝑦))
72	37, 67, 69, 71	fldpropd 20685	. 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 1540 ∈ wcel 2109 ∪ cun 3914 ⊆ wss 3916 class class class wbr 5109 ‘cfv 6513 (class class class)co 7389 ≤ cle 11215 ℕ₀cn0 12448 ℕ₀^*cxnn0 12521 Basecbs 17185 ↾_s cress 17206 +_gcplusg 17226 ._rcmulr 17227 Scalarcsca 17229 CRingccrg 20149 SubRingcsubrg 20484 RingSpancrgspn 20525 IDomncidom 20608 DivRingcdr 20644 Fieldcfield 20645 SubDRingcsdrg 20701 LModclmod 20772 LVecclvec 21015 subringAlg csra 21084 AssAlgcasa 21765 dimcldim 33600 [:]cextdg 33642

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-reg 9551 ax-inf2 9600 ax-ac2 10422 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-rpss 7701 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-er 8673 df-map 8803 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-sup 9399 df-inf 9400 df-oi 9469 df-r1 9723 df-rank 9724 df-dju 9860 df-card 9898 df-acn 9901 df-ac 10075 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-rp 12958 df-xadd 13079 df-fz 13475 df-fzo 13622 df-seq 13973 df-exp 14033 df-hash 14302 df-word 14485 df-lsw 14534 df-concat 14542 df-s1 14567 df-substr 14612 df-pfx 14642 df-s2 14820 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-clim 15460 df-sum 15659 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ocomp 17247 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17410 df-gsum 17411 df-prds 17416 df-pws 17418 df-mre 17553 df-mrc 17554 df-mri 17555 df-acs 17556 df-proset 18261 df-drs 18262 df-poset 18280 df-ipo 18493 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-ghm 19151 df-cntz 19255 df-cntr 19256 df-lsm 19572 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rgspn 20526 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-sdrg 20702 df-lmod 20774 df-lss 20844 df-lsp 20884 df-lmhm 20935 df-lmim 20936 df-lbs 20988 df-lvec 21016 df-sra 21086 df-rgmod 21087 df-cnfld 21271 df-zring 21363 df-dsmm 21647 df-frlm 21662 df-uvc 21698 df-lindf 21721 df-linds 21722 df-assa 21768 df-ind 32780 df-dim 33601 df-fldext 33643 df-extdg 33644

This theorem is referenced by: fldextrspunlem2 33678 fldextrspundgdvdslem 33681 fldextrspundgdvds 33682

W3C validator