0ringnnzr - Metamath Proof Explorer

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Theorem 0ringnnzr 19209

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 9999	. . . . . . . 8 ⊢ 1 ∈ ℝ
2	1	ltnri 10106	. . . . . . 7 ⊢ ¬ 1 < 1
3		breq2 4627	. . . . . . 7 ⊢ ((#‘(Base‘𝑅)) = 1 → (1 < (#‘(Base‘𝑅)) ↔ 1 < 1))
4	2, 3	mtbiri 317	. . . . . 6 ⊢ ((#‘(Base‘𝑅)) = 1 → ¬ 1 < (#‘(Base‘𝑅)))
5	4	adantl 482	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (#‘(Base‘𝑅)) = 1) → ¬ 1 < (#‘(Base‘𝑅)))
6	5	intnand 961	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (#‘(Base‘𝑅)) = 1) → ¬ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅))))
7	6	ex 450	. . 3 ⊢ (𝑅 ∈ Ring → ((#‘(Base‘𝑅)) = 1 → ¬ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅)))))
8		ianor 509	. . . . 5 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅))) ↔ (¬ 𝑅 ∈ Ring ∨ ¬ 1 < (#‘(Base‘𝑅))))
9		pm2.21 120	. . . . . 6 ⊢ (¬ 𝑅 ∈ Ring → (𝑅 ∈ Ring → (#‘(Base‘𝑅)) = 1))
10		fvex 6168	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
11		hashxrcl 13104	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (#‘(Base‘𝑅)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (#‘(Base‘𝑅)) ∈ ℝ^*
13	1	rexri 10057	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
14		xrlenlt 10063	. . . . . . . . 9 ⊢ (((#‘(Base‘𝑅)) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((#‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (#‘(Base‘𝑅))))
15	12, 13, 14	mp2an 707	. . . . . . . 8 ⊢ ((#‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (#‘(Base‘𝑅)))
16	15	bicomi 214	. . . . . . 7 ⊢ (¬ 1 < (#‘(Base‘𝑅)) ↔ (#‘(Base‘𝑅)) ≤ 1)
17		simpr 477	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (#‘(Base‘𝑅)) ≤ 1)
18	10	a1i 11	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ V)
19		1nn0 11268	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → 1 ∈ ℕ₀)
21		hashbnd 13079	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ∈ V ∧ 1 ∈ ℕ₀ ∧ (#‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
22	18, 20, 17, 21	syl3anc 1323	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
23		hashcl 13103	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑅) ∈ Fin → (#‘(Base‘𝑅)) ∈ ℕ₀)
24		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ∈ ℕ₀) → (#‘(Base‘𝑅)) ∈ ℕ₀)
25	10	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ₀ → (Base‘𝑅) ∈ V)
26		hasheq0 13110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Base‘𝑅) ∈ V → ((#‘(Base‘𝑅)) = 0 ↔ (Base‘𝑅) = ∅))
27	25, 26	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ₀ → ((#‘(Base‘𝑅)) = 0 ↔ (Base‘𝑅) = ∅))
28	27	biimpd 219	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ₀ → ((#‘(Base‘𝑅)) = 0 → (Base‘𝑅) = ∅))
29	28	necon3d 2811	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ₀ → ((Base‘𝑅) ≠ ∅ → (#‘(Base‘𝑅)) ≠ 0))
30	29	impcom 446	. . . . . . . . . . . . . . . . 17 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ∈ ℕ₀) → (#‘(Base‘𝑅)) ≠ 0)
31		elnnne0 11266	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ ↔ ((#‘(Base‘𝑅)) ∈ ℕ₀ ∧ (#‘(Base‘𝑅)) ≠ 0))
32	24, 30, 31	sylanbrc 697	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ∈ ℕ₀) → (#‘(Base‘𝑅)) ∈ ℕ)
33	32	ex 450	. . . . . . . . . . . . . . 15 ⊢ ((Base‘𝑅) ≠ ∅ → ((#‘(Base‘𝑅)) ∈ ℕ₀ → (#‘(Base‘𝑅)) ∈ ℕ))
34	33	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → ((#‘(Base‘𝑅)) ∈ ℕ₀ → (#‘(Base‘𝑅)) ∈ ℕ))
35	34	com12 32	. . . . . . . . . . . . 13 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ₀ → (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (#‘(Base‘𝑅)) ∈ ℕ))
36	23, 35	syl 17	. . . . . . . . . . . 12 ⊢ ((Base‘𝑅) ∈ Fin → (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (#‘(Base‘𝑅)) ∈ ℕ))
37	22, 36	mpcom 38	. . . . . . . . . . 11 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (#‘(Base‘𝑅)) ∈ ℕ)
38		nnle1eq1 11008	. . . . . . . . . . 11 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ → ((#‘(Base‘𝑅)) ≤ 1 ↔ (#‘(Base‘𝑅)) = 1))
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → ((#‘(Base‘𝑅)) ≤ 1 ↔ (#‘(Base‘𝑅)) = 1))
40	17, 39	mpbid 222	. . . . . . . . 9 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (#‘(Base‘𝑅)) = 1)
41	40	ex 450	. . . . . . . 8 ⊢ ((Base‘𝑅) ≠ ∅ → ((#‘(Base‘𝑅)) ≤ 1 → (#‘(Base‘𝑅)) = 1))
42		ringgrp 18492	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
43		eqid 2621	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
44	43	grpbn0 17391	. . . . . . . . 9 ⊢ (𝑅 ∈ Grp → (Base‘𝑅) ≠ ∅)
45	42, 44	syl 17	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) ≠ ∅)
46	41, 45	syl11 33	. . . . . . 7 ⊢ ((#‘(Base‘𝑅)) ≤ 1 → (𝑅 ∈ Ring → (#‘(Base‘𝑅)) = 1))
47	16, 46	sylbi 207	. . . . . 6 ⊢ (¬ 1 < (#‘(Base‘𝑅)) → (𝑅 ∈ Ring → (#‘(Base‘𝑅)) = 1))
48	9, 47	jaoi 394	. . . . 5 ⊢ ((¬ 𝑅 ∈ Ring ∨ ¬ 1 < (#‘(Base‘𝑅))) → (𝑅 ∈ Ring → (#‘(Base‘𝑅)) = 1))
49	8, 48	sylbi 207	. . . 4 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅))) → (𝑅 ∈ Ring → (#‘(Base‘𝑅)) = 1))
50	49	com12 32	. . 3 ⊢ (𝑅 ∈ Ring → (¬ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅))) → (#‘(Base‘𝑅)) = 1))
51	7, 50	impbid 202	. 2 ⊢ (𝑅 ∈ Ring → ((#‘(Base‘𝑅)) = 1 ↔ ¬ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅)))))
52	43	isnzr2hash 19204	. . . 4 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅))))
53	52	bicomi 214	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅))) ↔ 𝑅 ∈ NzRing)
54	53	notbii 310	. 2 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (#‘(Base‘𝑅))) ↔ ¬ 𝑅 ∈ NzRing)
55	51, 54	syl6bb 276	1 ⊢ (𝑅 ∈ Ring → ((#‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1480 ∈ wcel 1987 ≠ wne 2790 Vcvv 3190 ∅c0 3897 class class class wbr 4623 ‘cfv 5857 Fincfn 7915 0cc0 9896 1c1 9897 ℝ^*cxr 10033 < clt 10034 ≤ cle 10035 ℕcn 10980 ℕ₀cn0 11252 #chash 13073 Basecbs 15800 Grpcgrp 17362 Ringcrg 18487 NzRingcnzr 19197

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1836 ax-6 1885 ax-7 1932 ax-8 1989 ax-9 1996 ax-10 2016 ax-11 2031 ax-12 2044 ax-13 2245 ax-ext 2601 ax-rep 4741 ax-sep 4751 ax-nul 4759 ax-pow 4813 ax-pr 4877 ax-un 6914 ax-cnex 9952 ax-resscn 9953 ax-1cn 9954 ax-icn 9955 ax-addcl 9956 ax-addrcl 9957 ax-mulcl 9958 ax-mulrcl 9959 ax-mulcom 9960 ax-addass 9961 ax-mulass 9962 ax-distr 9963 ax-i2m1 9964 ax-1ne0 9965 ax-1rid 9966 ax-rnegex 9967 ax-rrecex 9968 ax-cnre 9969 ax-pre-lttri 9970 ax-pre-lttrn 9971 ax-pre-ltadd 9972 ax-pre-mulgt0 9973

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1878 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2750 df-ne 2791 df-nel 2894 df-ral 2913 df-rex 2914 df-reu 2915 df-rmo 2916 df-rab 2917 df-v 3192 df-sbc 3423 df-csb 3520 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-pss 3576 df-nul 3898 df-if 4065 df-pw 4138 df-sn 4156 df-pr 4158 df-tp 4160 df-op 4162 df-uni 4410 df-int 4448 df-iun 4494 df-br 4624 df-opab 4684 df-mpt 4685 df-tr 4723 df-eprel 4995 df-id 4999 df-po 5005 df-so 5006 df-fr 5043 df-we 5045 df-xp 5090 df-rel 5091 df-cnv 5092 df-co 5093 df-dm 5094 df-rn 5095 df-res 5096 df-ima 5097 df-pred 5649 df-ord 5695 df-on 5696 df-lim 5697 df-suc 5698 df-iota 5820 df-fun 5859 df-fn 5860 df-f 5861 df-f1 5862 df-fo 5863 df-f1o 5864 df-fv 5865 df-riota 6576 df-ov 6618 df-oprab 6619 df-mpt2 6620 df-om 7028 df-1st 7128 df-2nd 7129 df-wrecs 7367 df-recs 7428 df-rdg 7466 df-1o 7520 df-oadd 7524 df-er 7702 df-en 7916 df-dom 7917 df-sdom 7918 df-fin 7919 df-card 8725 df-cda 8950 df-pnf 10036 df-mnf 10037 df-xr 10038 df-ltxr 10039 df-le 10040 df-sub 10228 df-neg 10229 df-nn 10981 df-2 11039 df-n0 11253 df-xnn0 11324 df-z 11338 df-uz 11648 df-fz 12285 df-hash 13074 df-ndx 15803 df-slot 15804 df-base 15805 df-sets 15806 df-plusg 15894 df-0g 16042 df-mgm 17182 df-sgrp 17224 df-mnd 17235 df-grp 17365 df-minusg 17366 df-mgp 18430 df-ur 18442 df-ring 18489 df-nzr 19198

This theorem is referenced by: rng1nnzr 19214 lmod0rng 41186 0ringdif 41188 0ring1eq0 41190 lindszr 41576

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