assafld - Mathbox for Thierry Arnoux

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

Theorem assafld 33802

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 20694	. . 3 ⊢ (𝜑 → 𝐴 ∈ Ring)
3		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐴) = (Base‘𝐴)
4		eqid 2737	. . . . . . 7 ⊢ (Unit‘𝐴) = (Unit‘𝐴)
5	3, 4	unitss 20345	. . . . . 6 ⊢ (Unit‘𝐴) ⊆ (Base‘𝐴)
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (Unit‘𝐴) ⊆ (Base‘𝐴))
7		eqid 2737	. . . . . . 7 ⊢ (0_g‘𝐴) = (0_g‘𝐴)
8	1	idomdomd 20692	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ Domn)
9		domnnzr 20672	. . . . . . . . 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 33320	. . . . . 6 ⊢ ((𝜑 ∧ (0_g‘𝐴) ∈ (Unit‘𝐴)) → (0_g‘𝐴) ≠ (0_g‘𝐴))
14		neirr 2942	. . . . . . 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 817	. . . . 5 ⊢ (𝜑 → ¬ (0_g‘𝐴) ∈ (Unit‘𝐴))
17		ssdifsn 4732	. . . . 5 ⊢ ((Unit‘𝐴) ⊆ ((Base‘𝐴) ∖ {(0_g‘𝐴)}) ↔ ((Unit‘𝐴) ⊆ (Base‘𝐴) ∧ ¬ (0_g‘𝐴) ∈ (Unit‘𝐴)))
18	6, 16, 17	sylanbrc 584	. . . 4 ⊢ (𝜑 → (Unit‘𝐴) ⊆ ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
19		eqid 2737	. . . . 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 3902	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Base‘𝐴))
30		eldifsni 4734	. . . . . . 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 20679	. . . . . 6 ⊢ ((𝐴 ∈ Domn ∧ 𝑥 ∈ (Base‘𝐴) ∧ 𝑥 ≠ (0_g‘𝐴)) → 𝑥 ∈ (RLReg‘𝐴))
33	27, 29, 31, 32	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (RLReg‘𝐴))
34	19, 4, 20, 22, 24, 26, 33	assarrginv 33801	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ((Base‘𝐴) ∖ {(0_g‘𝐴)})) → 𝑥 ∈ (Unit‘𝐴))
35	18, 34	eqelssd 3944	. . 3 ⊢ (𝜑 → (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)}))
36	3, 4, 7	isdrng 20699	. . 3 ⊢ (𝐴 ∈ DivRing ↔ (𝐴 ∈ Ring ∧ (Unit‘𝐴) = ((Base‘𝐴) ∖ {(0_g‘𝐴)})))
37	2, 35, 36	sylanbrc 584	. 2 ⊢ (𝜑 → 𝐴 ∈ DivRing)
38	1	idomcringd 20693	. 2 ⊢ (𝜑 → 𝐴 ∈ CRing)
39		isfld 20706	. 2 ⊢ (𝐴 ∈ Field ↔ (𝐴 ∈ DivRing ∧ 𝐴 ∈ CRing))
40	37, 38, 39	sylanbrc 584	1 ⊢ (𝜑 → 𝐴 ∈ Field)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ⊆ wss 3890 {csn 4568 ‘cfv 6490 ℕ₀cn0 12426 Basecbs 17168 Scalarcsca 17212 0_gc0g 17391 Ringcrg 20203 CRingccrg 20204 Unitcui 20324 NzRingcnzr 20478 RLRegcrlreg 20657 Domncdomn 20658 IDomncidom 20659 DivRingcdr 20695 Fieldcfield 20696 AssAlgcasa 21838 dimcldim 33763

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-reg 9498 ax-inf2 9551 ax-ac2 10374 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-rpss 7668 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-er 8634 df-map 8766 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-r1 9677 df-rank 9678 df-dju 9814 df-card 9852 df-acn 9855 df-ac 10027 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-xnn0 12500 df-z 12514 df-dec 12634 df-uz 12778 df-xadd 13053 df-fz 13451 df-fzo 13598 df-seq 13953 df-hash 14282 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ocomp 17230 df-ds 17231 df-hom 17233 df-cco 17234 df-0g 17393 df-gsum 17394 df-prds 17399 df-pws 17401 df-mre 17537 df-mrc 17538 df-mri 17539 df-acs 17540 df-proset 18249 df-drs 18250 df-poset 18268 df-ipo 18483 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-ghm 19177 df-cntz 19281 df-lsm 19600 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-oppr 20306 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-nzr 20479 df-subrg 20536 df-rlreg 20660 df-domn 20661 df-idom 20662 df-drng 20697 df-field 20698 df-lmod 20846 df-lss 20916 df-lsp 20956 df-lmhm 21007 df-lmim 21008 df-lbs 21060 df-lvec 21088 df-sra 21158 df-rgmod 21159 df-dsmm 21720 df-frlm 21735 df-uvc 21771 df-lindf 21794 df-linds 21795 df-assa 21841 df-dim 33764

This theorem is referenced by: fldextrspunfld 33841

W3C validator