0ringnnzr - Metamath Proof Explorer

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Theorem 0ringnnzr 20660

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 11226	. . . . . . . 8 ⊢ 1 ∈ ℝ
2	1	ltnri 11337	. . . . . . 7 ⊢ ¬ 1 < 1
3		breq2 5118	. . . . . . 7 ⊢ ((♯‘(Base‘𝑅)) = 1 → (1 < (♯‘(Base‘𝑅)) ↔ 1 < 1))
4	2, 3	mtbiri 330	. . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = 1 → ¬ 1 < (♯‘(Base‘𝑅)))
5	4	adantl 487	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ 1 < (♯‘(Base‘𝑅)))
6	5	intnand 494	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
7	6	ex 418	. . 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 6901	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
11		hashxrcl 14413	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘(Base‘𝑅)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘(Base‘𝑅)) ∈ ℝ^*
13		1xr 11286	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
14		xrlenlt 11292	. . . . . . . . 9 ⊢ (((♯‘(Base‘𝑅)) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅))))
15	12, 13, 14	mp2an 705	. . . . . . . 8 ⊢ ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅)))
16	15	bicomi 227	. . . . . . 7 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) ↔ (♯‘(Base‘𝑅)) ≤ 1)
17		simpr 490	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ≤ 1)
18		1nn0 12538	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
19		hashbnd 14392	. . . . . . . . . . . . 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 14412	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑅) ∈ Fin → (♯‘(Base‘𝑅)) ∈ ℕ₀)
22		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ₀)
23		hasheq0 14419	. . . . . . . . . . . . . . . . . . . 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 2982	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((Base‘𝑅) ≠ ∅ → (♯‘(Base‘𝑅)) ≠ 0))
27	26	impcom 413	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ≠ 0)
28		elnnne0 12536	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ ↔ ((♯‘(Base‘𝑅)) ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≠ 0))
29	22, 27, 28	sylanbrc 595	. . . . . . . . . . . . . . 15 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ)
30	29	ex 418	. . . . . . . . . . . . . 14 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ∈ ℕ₀ → (♯‘(Base‘𝑅)) ∈ ℕ))
31	30	adantr 486	. . . . . . . . . . . . 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 12284	. . . . . . . . . . 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 418	. . . . . . . 8 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ≤ 1 → (♯‘(Base‘𝑅)) = 1))
38		ringgrp 20351	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39		eqid 2766	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39	grpbn0 19064	. . . . . . . . 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 871	. . . . 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 20654	. . . 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 Vcvv 3458 ∅c0 4289 class class class wbr 5114 ‘cfv 6543 Fincfn 8952 0cc0 11118 1c1 11119 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262 ℕcn 12251 ℕ₀cn0 12522 ♯chash 14386 Basecbs 17294 Grpcgrp 19031 Ringcrg 20346 NzRingcnzr 20646

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-oadd 8466 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-n0 12523 df-xnn0 12596 df-z 12610 df-uz 12881 df-fz 13554 df-hash 14387 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-plusg 17348 df-0g 17519 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-grp 19034 df-minusg 19035 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-nzr 20647

This theorem is used by: 0ringdif 20662 0ring1eq0 20669 rng1nnzr 20916 prmidl0 21515 qsdrng 33810 0ringirng 34110 lmod0rng 49035 prmringnzring 49143 lindszr 49290

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