0ringnnzr - Metamath Proof Explorer

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Theorem 0ringnnzr 19269

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 10039	. . . . . . . 8 ⊢ 1 ∈ ℝ
2	1	ltnri 10146	. . . . . . 7 ⊢ ¬ 1 < 1
3		breq2 4657	. . . . . . 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 962	. . . 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 6201	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
11		hashxrcl 13148	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (#‘(Base‘𝑅)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (#‘(Base‘𝑅)) ∈ ℝ^*
13	1	rexri 10097	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
14		xrlenlt 10103	. . . . . . . . 9 ⊢ (((#‘(Base‘𝑅)) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((#‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (#‘(Base‘𝑅))))
15	12, 13, 14	mp2an 708	. . . . . . . 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 11308	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ₀
20	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → 1 ∈ ℕ₀)
21		hashbnd 13123	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ∈ V ∧ 1 ∈ ℕ₀ ∧ (#‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
22	18, 20, 17, 21	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
23		hashcl 13147	. . . . . . . . . . . . 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 13154	. . . . . . . . . . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ₀ → ((Base‘𝑅) ≠ ∅ → (#‘(Base‘𝑅)) ≠ 0))
30	29	impcom 446	. . . . . . . . . . . . . . . . 17 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (#‘(Base‘𝑅)) ∈ ℕ₀) → (#‘(Base‘𝑅)) ≠ 0)
31		elnnne0 11306	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘(Base‘𝑅)) ∈ ℕ ↔ ((#‘(Base‘𝑅)) ∈ ℕ₀ ∧ (#‘(Base‘𝑅)) ≠ 0))
32	24, 30, 31	sylanbrc 698	. . . . . . . . . . . . . . . 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 11048	. . . . . . . . . . 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 18552	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
43		eqid 2622	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
44	43	grpbn0 17451	. . . . . . . . 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 19264	. . . 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 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 Fincfn 7955 0cc0 9936 1c1 9937 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 ℕ₀cn0 11292 #chash 13117 Basecbs 15857 Grpcgrp 17422 Ringcrg 18547 NzRingcnzr 19257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-hash 13118 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mgp 18490 df-ur 18502 df-ring 18549 df-nzr 19258

This theorem is referenced by: rng1nnzr 19274 lmod0rng 41868 0ringdif 41870 0ring1eq0 41872 lindszr 42258

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