assafld - Mathbox for Thierry Arnoux

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Theorem assafld 34203

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 20941	. . 3 ⊢ (𝜑 → 𝐴 ∈ Ring)
3		eqid 2760	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
4		eqid 2760	. . . . . . 7 ⊢ (Unit‘𝐴) = (Unit‘𝐴)
5	3, 4	unitss 20568	. . . . . 6 ⊢ (Unit‘𝐴) ⊆ (Base‘𝐴)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (Unit‘𝐴) ⊆ (Base‘𝐴))
7		eqid 2760	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
8	1	idomdomd 20939	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Domn)
9		domnnzr 20920	. . . . . . . . 9 ⊢ (𝐴 ∈ Domn → 𝐴 ∈ NzRing)
10	8, 9	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ NzRing)
11	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → 𝐴 ∈ NzRing)
12		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → (0_g‘𝐴) ∈ (Unit‘𝐴))
13	4, 7, 11, 12	unitnz 33733	. . . . . 6 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → (0_g‘𝐴) ≠ (0_g‘𝐴))
14		neirr 2964	. . . . . . 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 829	. . . . 5 ⊢ (𝜑 → ¬ (0_g‘𝐴) ∈ (Unit‘𝐴))
17		ssdifsn 4750	. . . . 5 ⊢ ((Unit‘𝐴) ⊆ ((Base‘𝐴) ∖ {(0_g‘𝐴)}) ↔ ((Unit‘𝐴) ⊆ (Base‘𝐴) ∧ ¬ (0_g‘𝐴) ∈ (Unit‘𝐴)))
18	6, 16, 17	sylanbrc 595	. . . 4 ⊢ (𝜑 → (Unit‘𝐴) ⊆ ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
19		eqid 2760	. . . . 5 ⊢ (RLReg‘𝐴) = (RLReg‘𝐴)
20		assafld.k	. . . . 5 ⊢ 𝐾 = (Scalar‘𝐴)
21		assafld.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ AssAlg)
22	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝐴 ∈ AssAlg)
23		assafld.2	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ DivRing)
24	23	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝐾 ∈ DivRing)
25		assafld.3	. . . . . 6 ⊢ (𝜑 → (dim‘𝐴) ∈ ℕ₀)
26	25	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → (dim‘𝐴) ∈ ℕ₀)
27	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝐴 ∈ Domn)
28		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
29	28	eldifad 3910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Base‘𝐴))
30		eldifsni 4752	. . . . . . 7 ⊢ (𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)}) → 𝑥 ≠ (0_g‘𝐴))
31	28, 30	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ≠ (0_g‘𝐴))
32	3, 19, 7	domnrrg 20926	. . . . . 6 ⊢ ((𝐴 ∈ Domn ∧ 𝑥 ∈ (Base‘𝐴) ∧ 𝑥 ≠ (0_g‘𝐴)) → 𝑥 ∈ (RLReg‘𝐴))
33	27, 29, 31, 32	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (RLReg‘𝐴))
34	19, 4, 20, 22, 24, 26, 33	assarrginv 34202	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Unit‘𝐴))
35	18, 34	eqelssd 3951	. . 3 ⊢ (𝜑 → (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
36	3, 4, 7	isdrng 20946	. . 3 ⊢ (𝐴 ∈ DivRing ↔ (𝐴 ∈ Ring ∧ (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)})))
37	2, 35, 36	sylanbrc 595	. 2 ⊢ (𝜑 → 𝐴 ∈ DivRing)
38	1	idomcringd 20940	. 2 ⊢ (𝜑 → 𝐴 ∈ CRing)
39		isfld 20955	. 2 ⊢ (𝐴 ∈ Field ↔ (𝐴 ∈ DivRing ∧ 𝐴 ∈ CRing))
40	37, 38, 39	sylanbrc 595	1 ⊢ (𝜑 → 𝐴 ∈ Field)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∖ cdif 3895 ⊆ wss 3898 {csn 4583 ‘cfv 6527 ℕ₀cn0 12576 Basecbs 17349 Scalarcsca 17393 0_gc0g 17572 Ringcrg 20421 CRingccrg 20422 Unitcui 20547 NzRingcnzr 20724 RLRegcrlreg 20905 Domncdomn 20906 IDomncidom 20907 DivRingcdr 20942 Fieldcfield 20943 AssAlgcasa 22120 dimcldim 34165

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 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-reg 9564 ax-inf2 9620 ax-ac2 10513 ax-cnex 11228 ax-resscn 11229 ax-1cn 11230 ax-icn 11231 ax-addcl 11232 ax-addrcl 11233 ax-mulcl 11234 ax-mulrcl 11235 ax-mulcom 11236 ax-addass 11237 ax-mulass 11238 ax-distr 11239 ax-i2m1 11240 ax-1ne0 11241 ax-1rid 11242 ax-rnegex 11243 ax-rrecex 11244 ax-cnre 11245 ax-pre-lttri 11246 ax-pre-lttrn 11247 ax-pre-ltadd 11248 ax-pre-mulgt0 11249

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7676 df-rpss 7722 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-oadd 8458 df-er 8695 df-map 8827 df-ixp 8904 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-sup 9412 df-oi 9482 df-r1 9746 df-rank 9747 df-scott 9901 df-dju 9954 df-card 9992 df-acn 9995 df-ac 10167 df-pnf 11317 df-mnf 11318 df-xr 11319 df-ltxr 11320 df-le 11321 df-sub 11515 df-neg 11516 df-nn 12306 df-2 12375 df-3 12376 df-4 12377 df-5 12378 df-6 12379 df-7 12380 df-8 12381 df-9 12382 df-n0 12577 df-xnn0 12650 df-z 12664 df-dec 12785 df-uz 12936 df-xadd 13212 df-fz 13610 df-fzo 13758 df-seq 14114 df-hash 14443 df-struct 17287 df-sets 17304 df-slot 17322 df-ndx 17334 df-base 17350 df-ress 17371 df-plusg 17403 df-mulr 17404 df-sca 17406 df-vsca 17407 df-ip 17408 df-tset 17409 df-ple 17410 df-ocomp 17411 df-ds 17412 df-hom 17414 df-cco 17415 df-0g 17574 df-gsum 17575 df-prds 17580 df-pws 17582 df-mre 17718 df-mrc 17719 df-mri 17720 df-acs 17721 df-proset 18430 df-drs 18431 df-poset 18449 df-ipo 18664 df-mgm 18778 df-sgrp 18870 df-mnd 18886 df-mhm 18940 df-submnd 18941 df-grp 19109 df-minusg 19110 df-sbg 19111 df-mulg 19240 df-subg 19295 df-ghm 19390 df-cntz 19493 df-lsm 19812 df-cmn 19958 df-abl 19959 df-mgp 20323 df-rng 20337 df-ur 20370 df-ring 20423 df-cring 20424 df-oppr 20529 df-dvdsr 20549 df-unit 20550 df-invr 20580 df-nzr 20725 df-subrg 20784 df-rlreg 20908 df-domn 20909 df-idom 20910 df-drng 20944 df-field 20945 df-lmod 21099 df-lss 21169 df-lsp 21209 df-lmhm 21259 df-lmim 21260 df-lbs 21312 df-lvec 21340 df-sra 21410 df-rgmod 21411 df-dsmm 22000 df-frlm 22015 df-uvc 22051 df-lindf 22074 df-linds 22075 df-assa 22123 df-dim 34166

This theorem is used by: fldextrspunfld 34242

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