assafld - Mathbox for Thierry Arnoux

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Theorem assafld 33641

Description: If an algebra 𝐴 of finite degree over a division ring 𝐾 is an integral domain, then it is a field. Corollary of Proposition 2. in Chapter 5. of [BourbakiAlg2] p. 113. (Contributed by Thierry Arnoux, 3-Aug-2025.)

**Hypotheses**
Ref	Expression
assafld.k	⊢ 𝐾 = (Scalar‘𝐴)
assafld.a	⊢ (𝜑 → 𝐴 ∈ AssAlg)
assafld.1	⊢ (𝜑 → 𝐴 ∈ IDomn)
assafld.2	⊢ (𝜑 → 𝐾 ∈ DivRing)
assafld.3	⊢ (𝜑 → (dim‘𝐴) ∈ ℕ₀)

**Assertion**
Ref	Expression
assafld	⊢ (𝜑 → 𝐴 ∈ Field)

Proof of Theorem assafld Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		assafld.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ IDomn)
2	1	idomringd 20643	. . 3 ⊢ (𝜑 → 𝐴 ∈ Ring)
3		eqid 2730	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
4		eqid 2730	. . . . . . 7 ⊢ (Unit‘𝐴) = (Unit‘𝐴)
5	3, 4	unitss 20291	. . . . . 6 ⊢ (Unit‘𝐴) ⊆ (Base‘𝐴)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (Unit‘𝐴) ⊆ (Base‘𝐴))
7		eqid 2730	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
8	1	idomdomd 20641	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Domn)
9		domnnzr 20621	. . . . . . . . 9 ⊢ (𝐴 ∈ Domn → 𝐴 ∈ NzRing)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ NzRing)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → 𝐴 ∈ NzRing)
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → (0_g‘𝐴) ∈ (Unit‘𝐴))
13	4, 7, 11, 12	unitnz 33198	. . . . . 6 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → (0_g‘𝐴) ≠ (0_g‘𝐴))
14		neirr 2936	. . . . . . 7 ⊢ ¬ (0_g‘𝐴) ≠ (0_g‘𝐴)
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → ¬ (0_g‘𝐴) ≠ (0_g‘𝐴))
16	13, 15	pm2.65da 816	. . . . 5 ⊢ (𝜑 → ¬ (0_g‘𝐴) ∈ (Unit‘𝐴))
17		ssdifsn 4760	. . . . 5 ⊢ ((Unit‘𝐴) ⊆ ((Base‘𝐴) ∖ {(0_g‘𝐴)}) ↔ ((Unit‘𝐴) ⊆ (Base‘𝐴) ∧ ¬ (0_g‘𝐴) ∈ (Unit‘𝐴)))
18	6, 16, 17	sylanbrc 583	. . . 4 ⊢ (𝜑 → (Unit‘𝐴) ⊆ ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
19		eqid 2730	. . . . 5 ⊢ (RLReg‘𝐴) = (RLReg‘𝐴)
20		assafld.k	. . . . 5 ⊢ 𝐾 = (Scalar‘𝐴)
21		assafld.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ AssAlg)
22	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝐴 ∈ AssAlg)
23		assafld.2	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ DivRing)
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝐾 ∈ DivRing)
25		assafld.3	. . . . . 6 ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ₀)
26	25	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → (dim‘𝐴) ∈ ℕ₀)
27	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝐴 ∈ Domn)
28		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
29	28	eldifad 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Base‘𝐴))
30		eldifsni 4762	. . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)}) → 𝑥 ≠ (0_g‘𝐴))
31	28, 30	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ≠ (0_g‘𝐴))
32	3, 19, 7	domnrrg 20628	. . . . . 6 ⊢ ((𝐴 ∈ Domn ∧ 𝑥 ∈ (Base‘𝐴) ∧ 𝑥 ≠ (0_g‘𝐴)) → 𝑥 ∈ (RLReg‘𝐴))
33	27, 29, 31, 32	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (RLReg‘𝐴))
34	19, 4, 20, 22, 24, 26, 33	assarrginv 33640	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Unit‘𝐴))
35	18, 34	eqelssd 3976	. . 3 ⊢ (𝜑 → (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
36	3, 4, 7	isdrng 20648	. . 3 ⊢ (𝐴 ∈ DivRing ↔ (𝐴 ∈ Ring ∧ (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)})))
37	2, 35, 36	sylanbrc 583	. 2 ⊢ (𝜑 → 𝐴 ∈ DivRing)
38	1	idomcringd 20642	. 2 ⊢ (𝜑 → 𝐴 ∈ CRing)
39		isfld 20655	. 2 ⊢ (𝐴 ∈ Field ↔ (𝐴 ∈ DivRing ∧ 𝐴 ∈ CRing))
40	37, 38, 39	sylanbrc 583	1 ⊢ (𝜑 → 𝐴 ∈ Field)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2927 ∖ cdif 3919 ⊆ wss 3922 {csn 4597 ‘cfv 6519 ℕ₀cn0 12458 Basecbs 17185 Scalarcsca 17229 0_gc0g 17408 Ringcrg 20148 CRingccrg 20149 Unitcui 20270 NzRingcnzr 20427 RLRegcrlreg 20606 Domncdomn 20607 IDomncidom 20608 DivRingcdr 20644 Fieldcfield 20645 AssAlgcasa 21765 dimcldim 33602

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 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-reg 9563 ax-inf2 9612 ax-ac2 10434 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163

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 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-int 4919 df-iun 4965 df-iin 4966 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-se 5600 df-we 5601 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-isom 6528 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7660 df-rpss 7706 df-om 7851 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-2o 8444 df-oadd 8447 df-er 8682 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9331 df-sup 9411 df-oi 9481 df-r1 9735 df-rank 9736 df-dju 9872 df-card 9910 df-acn 9913 df-ac 10087 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-nn 12198 df-2 12260 df-3 12261 df-4 12262 df-5 12263 df-6 12264 df-7 12265 df-8 12266 df-9 12267 df-n0 12459 df-xnn0 12532 df-z 12546 df-dec 12666 df-uz 12810 df-xadd 13086 df-fz 13482 df-fzo 13629 df-seq 13977 df-hash 14306 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-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ocomp 17247 df-ds 17248 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-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-subrg 20485 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 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-dsmm 21647 df-frlm 21662 df-uvc 21698 df-lindf 21721 df-linds 21722 df-assa 21768 df-dim 33603

This theorem is referenced by: fldextrspunfld 33679

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