assafld - Mathbox for Thierry Arnoux

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Theorem assafld 33614

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 20695	. . 3 ⊢ (𝜑 → 𝐴 ∈ Ring)
3		eqid 2734	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
4		eqid 2734	. . . . . . 7 ⊢ (Unit‘𝐴) = (Unit‘𝐴)
5	3, 4	unitss 20343	. . . . . 6 ⊢ (Unit‘𝐴) ⊆ (Base‘𝐴)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (Unit‘𝐴) ⊆ (Base‘𝐴))
7		eqid 2734	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
8	1	idomdomd 20693	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Domn)
9		domnnzr 20673	. . . . . . . . 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 33173	. . . . . 6 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → (0_g‘𝐴) ≠ (0_g‘𝐴))
14		neirr 2940	. . . . . . 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 4768	. . . . 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 2734	. . . . 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 3943	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Base‘𝐴))
30		eldifsni 4770	. . . . . . 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 20680	. . . . . 6 ⊢ ((𝐴 ∈ Domn ∧ 𝑥 ∈ (Base‘𝐴) ∧ 𝑥 ≠ (0_g‘𝐴)) → 𝑥 ∈ (RLReg‘𝐴))
33	27, 29, 31, 32	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (RLReg‘𝐴))
34	19, 4, 20, 22, 24, 26, 33	assarrginv 33613	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Unit‘𝐴))
35	18, 34	eqelssd 3985	. . 3 ⊢ (𝜑 → (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
36	3, 4, 7	isdrng 20700	. . 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 20694	. 2 ⊢ (𝜑 → 𝐴 ∈ CRing)
39		isfld 20707	. 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 1539 ∈ wcel 2107 ≠ wne 2931 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 ‘cfv 6540 ℕ₀cn0 12508 Basecbs 17228 Scalarcsca 17275 0_gc0g 17454 Ringcrg 20197 CRingccrg 20198 Unitcui 20322 NzRingcnzr 20479 RLRegcrlreg 20658 Domncdomn 20659 IDomncidom 20660 DivRingcdr 20696 Fieldcfield 20697 AssAlgcasa 21823 dimcldim 33575

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7736 ax-reg 9613 ax-inf2 9662 ax-ac2 10484 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6493 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7678 df-rpss 7724 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8726 df-map 8849 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9383 df-sup 9463 df-oi 9531 df-r1 9785 df-rank 9786 df-dju 9922 df-card 9960 df-acn 9963 df-ac 10137 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11475 df-neg 11476 df-nn 12248 df-2 12310 df-3 12311 df-4 12312 df-5 12313 df-6 12314 df-7 12315 df-8 12316 df-9 12317 df-n0 12509 df-xnn0 12582 df-z 12596 df-dec 12716 df-uz 12860 df-xadd 13136 df-fz 13529 df-fzo 13676 df-seq 14024 df-hash 14351 df-struct 17165 df-sets 17182 df-slot 17200 df-ndx 17212 df-base 17229 df-ress 17252 df-plusg 17285 df-mulr 17286 df-sca 17288 df-vsca 17289 df-ip 17290 df-tset 17291 df-ple 17292 df-ocomp 17293 df-ds 17294 df-hom 17296 df-cco 17297 df-0g 17456 df-gsum 17457 df-prds 17462 df-pws 17464 df-mre 17599 df-mrc 17600 df-mri 17601 df-acs 17602 df-proset 18309 df-drs 18310 df-poset 18328 df-ipo 18541 df-mgm 18621 df-sgrp 18700 df-mnd 18716 df-mhm 18764 df-submnd 18765 df-grp 18922 df-minusg 18923 df-sbg 18924 df-mulg 19054 df-subg 19109 df-ghm 19199 df-cntz 19303 df-lsm 19621 df-cmn 19767 df-abl 19768 df-mgp 20105 df-rng 20117 df-ur 20146 df-ring 20199 df-cring 20200 df-oppr 20301 df-dvdsr 20324 df-unit 20325 df-invr 20355 df-nzr 20480 df-subrg 20537 df-rlreg 20661 df-domn 20662 df-idom 20663 df-drng 20698 df-field 20699 df-lmod 20827 df-lss 20897 df-lsp 20937 df-lmhm 20988 df-lmim 20989 df-lbs 21041 df-lvec 21069 df-sra 21139 df-rgmod 21140 df-dsmm 21705 df-frlm 21720 df-uvc 21756 df-lindf 21779 df-linds 21780 df-assa 21826 df-dim 33576

This theorem is referenced by: fldextrspunfld 33654

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