0ringnnzr - Metamath Proof Explorer

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Theorem 0ringnnzr 20414

Description: A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)

**Assertion**
Ref	Expression
0ringnnzr	⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

**Proof of Theorem 0ringnnzr**
Step	Hyp	Ref	Expression
1		1re 11218	. . . . . . . 8 ⊢ 1 ∈ ℝ
2	1	ltnri 11327	. . . . . . 7 ⊢ ¬ 1 < 1
3		breq2 5151	. . . . . . 7 ⊢ ((♯‘(Base‘𝑅)) = 1 → (1 < (♯‘(Base‘𝑅)) ↔ 1 < 1))
4	2, 3	mtbiri 326	. . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = 1 → ¬ 1 < (♯‘(Base‘𝑅)))
5	4	adantl 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ 1 < (♯‘(Base‘𝑅)))
6	5	intnand 487	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
7	6	ex 411	. . 3 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 → ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))))
8		ianor 978	. . . . 5 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ (¬ 𝑅 ∈ Ring ∨ ¬ 1 < (♯‘(Base‘𝑅))))
9		pm2.21 123	. . . . . 6 ⊢ (¬ 𝑅 ∈ Ring → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
10		fvex 6903	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
11		hashxrcl 14321	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘(Base‘𝑅)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘(Base‘𝑅)) ∈ ℝ^*
13		1xr 11277	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
14		xrlenlt 11283	. . . . . . . . 9 ⊢ (((♯‘(Base‘𝑅)) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅))))
15	12, 13, 14	mp2an 688	. . . . . . . 8 ⊢ ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅)))
16	15	bicomi 223	. . . . . . 7 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) ↔ (♯‘(Base‘𝑅)) ≤ 1)
17		simpr 483	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ≤ 1)
18		1nn0 12492	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
19		hashbnd 14300	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ∈ V ∧ 1 ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
20	10, 18, 17, 19	mp3an12i 1463	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
21		hashcl 14320	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑅) ∈ Fin → (♯‘(Base‘𝑅)) ∈ ℕ₀)
22		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ₀)
23		hasheq0 14327	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Base‘𝑅) ∈ V → ((♯‘(Base‘𝑅)) = 0 ↔ (Base‘𝑅) = ∅))
24	10, 23	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((♯‘(Base‘𝑅)) = 0 ↔ (Base‘𝑅) = ∅))
25	24	biimpd 228	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((♯‘(Base‘𝑅)) = 0 → (Base‘𝑅) = ∅))
26	25	necon3d 2959	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((Base‘𝑅) ≠ ∅ → (♯‘(Base‘𝑅)) ≠ 0))
27	26	impcom 406	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ≠ 0)
28		elnnne0 12490	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ ↔ ((♯‘(Base‘𝑅)) ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≠ 0))
29	22, 27, 28	sylanbrc 581	. . . . . . . . . . . . . . 15 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ)
30	29	ex 411	. . . . . . . . . . . . . 14 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ∈ ℕ₀ → (♯‘(Base‘𝑅)) ∈ ℕ))
31	30	adantr 479	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → ((♯‘(Base‘𝑅)) ∈ ℕ₀ → (♯‘(Base‘𝑅)) ∈ ℕ))
32	21, 31	syl5com 31	. . . . . . . . . . . 12 ⊢ ((Base‘𝑅) ∈ Fin → (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ∈ ℕ))
33	20, 32	mpcom 38	. . . . . . . . . . 11 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ∈ ℕ)
34		nnle1eq1 12246	. . . . . . . . . . 11 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ → ((♯‘(Base‘𝑅)) ≤ 1 ↔ (♯‘(Base‘𝑅)) = 1))
35	33, 34	syl 17	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → ((♯‘(Base‘𝑅)) ≤ 1 ↔ (♯‘(Base‘𝑅)) = 1))
36	17, 35	mpbid 231	. . . . . . . . 9 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) = 1)
37	36	ex 411	. . . . . . . 8 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ≤ 1 → (♯‘(Base‘𝑅)) = 1))
38		ringgrp 20132	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39	grpbn0 18887	. . . . . . . . 9 ⊢ (𝑅 ∈ Grp → (Base‘𝑅) ≠ ∅)
41	38, 40	syl 17	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ≠ ∅)
42	37, 41	syl11 33	. . . . . . 7 ⊢ ((♯‘(Base‘𝑅)) ≤ 1 → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
43	16, 42	sylbi 216	. . . . . 6 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
44	9, 43	jaoi 853	. . . . 5 ⊢ ((¬ 𝑅 ∈ Ring ∨ ¬ 1 < (♯‘(Base‘𝑅))) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
45	8, 44	sylbi 216	. . . 4 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
46	45	com12 32	. . 3 ⊢ (𝑅 ∈ Ring → (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) → (♯‘(Base‘𝑅)) = 1))
47	7, 46	impbid 211	. 2 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))))
48	39	isnzr2hash 20410	. . . 4 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
49	48	bicomi 223	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ 𝑅 ∈ NzRing)
50	49	notbii 319	. 2 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ ¬ 𝑅 ∈ NzRing)
51	47, 50	bitrdi 286	1 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 Vcvv 3472 ∅c0 4321 class class class wbr 5147 ‘cfv 6542 Fincfn 8941 0cc0 11112 1c1 11113 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 ℕcn 12216 ℕ₀cn0 12476 ♯chash 14294 Basecbs 17148 Grpcgrp 18855 Ringcrg 20127 NzRingcnzr 20403

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-hash 14295 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-plusg 17214 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-nzr 20404

This theorem is referenced by: 0ringdif 20416 0ring1eq0 20422 rng1nnzr 20539 prmidl0 32843 qsdrng 32885 0ringirng 33042 lmod0rng 46908 lindszr 47237

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