prmidl0 - Mathbox for Thierry Arnoux

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Theorem prmidl0 32600

Description: The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.)

**Hypothesis**
Ref	Expression
prmidl0.1	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
prmidl0	⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn)

Proof of Theorem prmidl0 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 1090	. . . 4 ⊢ (({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 }))) ↔ (({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 }))))
2		crngring 20068	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3	2	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ { 0 } ∈ (LIdeal‘𝑅)) ∧ ¬ 𝑅 ∈ NzRing) → 𝑅 ∈ Ring)
4		0ringnnzr 20302	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))
5	4	biimpar 479	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ ¬ 𝑅 ∈ NzRing) → (♯‘(Base‘𝑅)) = 1)
6	3, 5	sylancom 589	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ { 0 } ∈ (LIdeal‘𝑅)) ∧ ¬ 𝑅 ∈ NzRing) → (♯‘(Base‘𝑅)) = 1)
7		eqid 2733	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		prmidl0.1	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑅)
9	7, 8	0ring 20303	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → (Base‘𝑅) = { 0 })
10	3, 6, 9	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑅 ∈ CRing ∧ { 0 } ∈ (LIdeal‘𝑅)) ∧ ¬ 𝑅 ∈ NzRing) → (Base‘𝑅) = { 0 })
11	10	eqcomd 2739	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ { 0 } ∈ (LIdeal‘𝑅)) ∧ ¬ 𝑅 ∈ NzRing) → { 0 } = (Base‘𝑅))
12	11	ex 414	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (LIdeal‘𝑅)) → (¬ 𝑅 ∈ NzRing → { 0 } = (Base‘𝑅)))
13	12	necon1ad 2958	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (LIdeal‘𝑅)) → ({ 0 } ≠ (Base‘𝑅) → 𝑅 ∈ NzRing))
14	13	impr 456	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅))) → 𝑅 ∈ NzRing)
15		nzrring 20295	. . . . . . . . 9 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
16		eqid 2733	. . . . . . . . . 10 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
17	16, 8	lidl0 20844	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → { 0 } ∈ (LIdeal‘𝑅))
18	15, 17	syl 17	. . . . . . . 8 ⊢ (𝑅 ∈ NzRing → { 0 } ∈ (LIdeal‘𝑅))
19	8	fvexi 6906	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
20		hashsng 14329	. . . . . . . . . . . . 13 ⊢ ( 0 ∈ V → (♯‘{ 0 }) = 1)
21	19, 20	ax-mp 5	. . . . . . . . . . . 12 ⊢ (♯‘{ 0 }) = 1
22		1re 11214	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
23	21, 22	eqeltri 2830	. . . . . . . . . . 11 ⊢ (♯‘{ 0 }) ∈ ℝ
24	23	a1i 11	. . . . . . . . . 10 ⊢ (𝑅 ∈ NzRing → (♯‘{ 0 }) ∈ ℝ)
25	7	isnzr2hash 20298	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
26	25	simprbi 498	. . . . . . . . . . 11 ⊢ (𝑅 ∈ NzRing → 1 < (♯‘(Base‘𝑅)))
27	21, 26	eqbrtrid 5184	. . . . . . . . . 10 ⊢ (𝑅 ∈ NzRing → (♯‘{ 0 }) < (♯‘(Base‘𝑅)))
28	24, 27	ltned 11350	. . . . . . . . 9 ⊢ (𝑅 ∈ NzRing → (♯‘{ 0 }) ≠ (♯‘(Base‘𝑅)))
29		fveq2 6892	. . . . . . . . . 10 ⊢ ({ 0 } = (Base‘𝑅) → (♯‘{ 0 }) = (♯‘(Base‘𝑅)))
30	29	necon3i 2974	. . . . . . . . 9 ⊢ ((♯‘{ 0 }) ≠ (♯‘(Base‘𝑅)) → { 0 } ≠ (Base‘𝑅))
31	28, 30	syl 17	. . . . . . . 8 ⊢ (𝑅 ∈ NzRing → { 0 } ≠ (Base‘𝑅))
32	18, 31	jca 513	. . . . . . 7 ⊢ (𝑅 ∈ NzRing → ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅)))
33	32	adantl 483	. . . . . 6 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ NzRing) → ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅)))
34	14, 33	impbida 800	. . . . 5 ⊢ (𝑅 ∈ CRing → (({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅)) ↔ 𝑅 ∈ NzRing))
35	19	elsn2 4668	. . . . . . . 8 ⊢ ((𝑥(._r‘𝑅)𝑦) ∈ { 0 } ↔ (𝑥(._r‘𝑅)𝑦) = 0 )
36		velsn 4645	. . . . . . . . 9 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
37		velsn 4645	. . . . . . . . 9 ⊢ (𝑦 ∈ { 0 } ↔ 𝑦 = 0 )
38	36, 37	orbi12i 914	. . . . . . . 8 ⊢ ((𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 }) ↔ (𝑥 = 0 ∨ 𝑦 = 0 ))
39	35, 38	imbi12i 351	. . . . . . 7 ⊢ (((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 })) ↔ ((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
40	39	2ralbii 3129	. . . . . 6 ⊢ (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 })) ↔ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))
41	40	a1i 11	. . . . 5 ⊢ (𝑅 ∈ CRing → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 })) ↔ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
42	34, 41	anbi12d 632	. . . 4 ⊢ (𝑅 ∈ CRing → ((({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 }))) ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))))
43	1, 42	bitrid 283	. . 3 ⊢ (𝑅 ∈ CRing → (({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 }))) ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))))
44	43	pm5.32i 576	. 2 ⊢ ((𝑅 ∈ CRing ∧ ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 })))) ↔ (𝑅 ∈ CRing ∧ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))))
45		eqid 2733	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
46	7, 45	isprmidlc 32597	. . 3 ⊢ (𝑅 ∈ CRing → ({ 0 } ∈ (PrmIdeal‘𝑅) ↔ ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 })))))
47	46	pm5.32i 576	. 2 ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ (𝑅 ∈ CRing ∧ ({ 0 } ∈ (LIdeal‘𝑅) ∧ { 0 } ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) ∈ { 0 } → (𝑥 ∈ { 0 } ∨ 𝑦 ∈ { 0 })))))
48		df-idom 20901	. . . 4 ⊢ IDomn = (CRing ∩ Domn)
49	48	eleq2i 2826	. . 3 ⊢ (𝑅 ∈ IDomn ↔ 𝑅 ∈ (CRing ∩ Domn))
50		elin 3965	. . 3 ⊢ (𝑅 ∈ (CRing ∩ Domn) ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
51	7, 45, 8	isdomn 20910	. . . 4 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
52	51	anbi2i 624	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑅 ∈ Domn) ↔ (𝑅 ∈ CRing ∧ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))))
53	49, 50, 52	3bitri 297	. 2 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(._r‘𝑅)𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 )))))
54	44, 47, 53	3bitr4i 303	1 ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∩ cin 3948 {csn 4629 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 1c1 11111 < clt 11248 ♯chash 14290 Basecbs 17144 ._rcmulr 17198 0_gc0g 17385 Ringcrg 20056 CRingccrg 20057 NzRingcnzr 20291 LIdealclidl 20783 Domncdomn 20896 IDomncidom 20897 PrmIdealcprmidl 32584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-subg 19003 df-cmn 19650 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-nzr 20292 df-subrg 20317 df-lmod 20473 df-lss 20543 df-lsp 20583 df-sra 20785 df-rgmod 20786 df-lidl 20787 df-rsp 20788 df-domn 20900 df-idom 20901 df-prmidl 32585

This theorem is referenced by: ply1annprmidl 32799

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