assafld - Mathbox for Thierry Arnoux

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Theorem assafld 33642

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 20745	. . 3 ⊢ (𝜑 → 𝐴 ∈ Ring)
3		eqid 2734	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
4		eqid 2734	. . . . . . 7 ⊢ (Unit‘𝐴) = (Unit‘𝐴)
5	3, 4	unitss 20397	. . . . . 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 20743	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Domn)
9		domnnzr 20723	. . . . . . . . 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 33211	. . . . . 6 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → (0_g‘𝐴) ≠ (0_g‘𝐴))
14		neirr 2951	. . . . . . 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 4813	. . . . 5 ⊢ ((Unit‘𝐴) ⊆ ((Base‘𝐴) ∖ {(0_g‘𝐴)}) ↔ ((Unit‘𝐴) ⊆ (Base‘𝐴) ∧ ¬ (0_g‘𝐴) ∈ (Unit‘𝐴)))
18	6, 16, 17	sylanbrc 582	. . . 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 3982	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Base‘𝐴))
30		eldifsni 4815	. . . . . . 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 20730	. . . . . 6 ⊢ ((𝐴 ∈ Domn ∧ 𝑥 ∈ (Base‘𝐴) ∧ 𝑥 ≠ (0_g‘𝐴)) → 𝑥 ∈ (RLReg‘𝐴))
33	27, 29, 31, 32	syl3anc 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (RLReg‘𝐴))
34	19, 4, 20, 22, 24, 26, 33	assarrginv 33641	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Unit‘𝐴))
35	18, 34	eqelssd 4024	. . 3 ⊢ (𝜑 → (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
36	3, 4, 7	isdrng 20750	. . 3 ⊢ (𝐴 ∈ DivRing ↔ (𝐴 ∈ Ring ∧ (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)})))
37	2, 35, 36	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐴 ∈ DivRing)
38	1	idomcringd 20744	. 2 ⊢ (𝜑 → 𝐴 ∈ CRing)
39		isfld 20757	. 2 ⊢ (𝐴 ∈ Field ↔ (𝐴 ∈ DivRing ∧ 𝐴 ∈ CRing))
40	37, 38, 39	sylanbrc 582	1 ⊢ (𝜑 → 𝐴 ∈ Field)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ∖ cdif 3967 ⊆ wss 3970 {csn 4648 ‘cfv 6572 ℕ₀cn0 12549 Basecbs 17253 Scalarcsca 17309 0_gc0g 17494 Ringcrg 20255 CRingccrg 20256 Unitcui 20376 NzRingcnzr 20533 RLRegcrlreg 20708 Domncdomn 20709 IDomncidom 20710 DivRingcdr 20746 Fieldcfield 20747 AssAlgcasa 21888 dimcldim 33603

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-reg 9657 ax-inf2 9706 ax-ac2 10528 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-rpss 7754 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-oadd 8522 df-er 8759 df-map 8882 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-sup 9507 df-oi 9575 df-r1 9829 df-rank 9830 df-dju 9966 df-card 10004 df-acn 10007 df-ac 10181 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-xnn0 12622 df-z 12636 df-dec 12755 df-uz 12900 df-xadd 13172 df-fz 13564 df-fzo 13708 df-seq 14049 df-hash 14376 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ocomp 17327 df-ds 17328 df-hom 17330 df-cco 17331 df-0g 17496 df-gsum 17497 df-prds 17502 df-pws 17504 df-mre 17639 df-mrc 17640 df-mri 17641 df-acs 17642 df-proset 18360 df-drs 18361 df-poset 18378 df-ipo 18593 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-submnd 18814 df-grp 18971 df-minusg 18972 df-sbg 18973 df-mulg 19103 df-subg 19158 df-ghm 19248 df-cntz 19352 df-lsm 19673 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-nzr 20534 df-subrg 20592 df-rlreg 20711 df-domn 20712 df-idom 20713 df-drng 20748 df-field 20749 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lmhm 21039 df-lmim 21040 df-lbs 21092 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-uvc 21821 df-lindf 21844 df-linds 21845 df-assa 21891 df-dim 33604

This theorem is referenced by: (None)

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