0ringnnzr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > 0ringnnzr		Structured version Visualization version GIF version

Theorem 0ringnnzr 20610

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 11209	. . . . . . . 8 ⊢ 1 ∈ ℝ
2	1	ltnri 11320	. . . . . . 7 ⊢ ¬ 1 < 1
3		breq2 5114	. . . . . . 7 ⊢ ((♯‘(Base‘𝑅)) = 1 → (1 < (♯‘(Base‘𝑅)) ↔ 1 < 1))
4	2, 3	mtbiri 330	. . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = 1 → ¬ 1 < (♯‘(Base‘𝑅)))
5	4	adantl 486	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ 1 < (♯‘(Base‘𝑅)))
6	5	intnand 493	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
7	6	ex 417	. . 3 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 → ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))))
8		ianor 997	. . . . 5 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ (¬ 𝑅 ∈ Ring ∨ ¬ 1 < (♯‘(Base‘𝑅))))
9		pm2.21 124	. . . . . 6 ⊢ (¬ 𝑅 ∈ Ring → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
10		fvex 6896	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
11		hashxrcl 14395	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘(Base‘𝑅)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘(Base‘𝑅)) ∈ ℝ^*
13		1xr 11269	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
14		xrlenlt 11275	. . . . . . . . 9 ⊢ (((♯‘(Base‘𝑅)) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅))))
15	12, 13, 14	mp2an 704	. . . . . . . 8 ⊢ ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅)))
16	15	bicomi 227	. . . . . . 7 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) ↔ (♯‘(Base‘𝑅)) ≤ 1)
17		simpr 489	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ≤ 1)
18		1nn0 12521	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
19		hashbnd 14374	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ∈ V ∧ 1 ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
20	10, 18, 17, 19	mp3an12i 1494	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
21		hashcl 14394	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑅) ∈ Fin → (♯‘(Base‘𝑅)) ∈ ℕ₀)
22		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ₀)
23		hasheq0 14401	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Base‘𝑅) ∈ V → ((♯‘(Base‘𝑅)) = 0 ↔ (Base‘𝑅) = ∅))
24	10, 23	mp1i 14	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((♯‘(Base‘𝑅)) = 0 ↔ (Base‘𝑅) = ∅))
25	24	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((♯‘(Base‘𝑅)) = 0 → (Base‘𝑅) = ∅))
26	25	necon3d 2979	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((Base‘𝑅) ≠ ∅ → (♯‘(Base‘𝑅)) ≠ 0))
27	26	impcom 412	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ≠ 0)
28		elnnne0 12519	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ ↔ ((♯‘(Base‘𝑅)) ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≠ 0))
29	22, 27, 28	sylanbrc 594	. . . . . . . . . . . . . . 15 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ)
30	29	ex 417	. . . . . . . . . . . . . 14 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ∈ ℕ₀ → (♯‘(Base‘𝑅)) ∈ ℕ))
31	30	adantr 485	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → ((♯‘(Base‘𝑅)) ∈ ℕ₀ → (♯‘(Base‘𝑅)) ∈ ℕ))
32	21, 31	syl5com 32	. . . . . . . . . . . 12 ⊢ ((Base‘𝑅) ∈ Fin → (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ∈ ℕ))
33	20, 32	mpcom 39	. . . . . . . . . . 11 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ∈ ℕ)
34		nnle1eq1 12267	. . . . . . . . . . 11 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ → ((♯‘(Base‘𝑅)) ≤ 1 ↔ (♯‘(Base‘𝑅)) = 1))
35	33, 34	syl 18	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → ((♯‘(Base‘𝑅)) ≤ 1 ↔ (♯‘(Base‘𝑅)) = 1))
36	17, 35	mpbid 235	. . . . . . . . 9 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) = 1)
37	36	ex 417	. . . . . . . 8 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ≤ 1 → (♯‘(Base‘𝑅)) = 1))
38		ringgrp 20321	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39	grpbn0 19034	. . . . . . . . 9 ⊢ (𝑅 ∈ Grp → (Base‘𝑅) ≠ ∅)
41	38, 40	syl 18	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ≠ ∅)
42	37, 41	syl11 34	. . . . . . 7 ⊢ ((♯‘(Base‘𝑅)) ≤ 1 → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
43	16, 42	sylbi 220	. . . . . 6 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
44	9, 43	jaoi 870	. . . . 5 ⊢ ((¬ 𝑅 ∈ Ring ∨ ¬ 1 < (♯‘(Base‘𝑅))) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
45	8, 44	sylbi 220	. . . 4 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
46	45	com12 33	. . 3 ⊢ (𝑅 ∈ Ring → (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) → (♯‘(Base‘𝑅)) = 1))
47	7, 46	impbid 215	. 2 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))))
48	39	isnzr2hash 20604	. . . 4 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
49	48	bicomi 227	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ 𝑅 ∈ NzRing)
50	49	notbii 323	. 2 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ ¬ 𝑅 ∈ NzRing)
51	47, 50	bitrdi 290	1 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 ∅c0 4287 class class class wbr 5110 ‘cfv 6538 Fincfn 8944 0cc0 11101 1c1 11102 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℕcn 12234 ℕ₀cn0 12505 ♯chash 14368 Basecbs 17270 Grpcgrp 19001 Ringcrg 20316 NzRingcnzr 20596

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-oadd 8458 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-n0 12506 df-xnn0 12579 df-z 12593 df-uz 12864 df-fz 13537 df-hash 14369 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-plusg 17324 df-0g 17495 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-grp 19004 df-minusg 19005 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-nzr 20597

This theorem is referenced by: 0ringdif 20612 0ring1eq0 20619 rng1nnzr 20860 prmidl0 21459 qsdrng 33760 0ringirng 34060 lmod0rng 48977 prmringnzring 49085 lindszr 49232

W3C validator