0ringnnzr - Metamath Proof Explorer

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Theorem 0ringnnzr 20470

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 11144	. . . . . . . 8 ⊢ 1 ∈ ℝ
2	1	ltnri 11254	. . . . . . 7 ⊢ ¬ 1 < 1
3		breq2 5104	. . . . . . 7 ⊢ ((♯‘(Base‘𝑅)) = 1 → (1 < (♯‘(Base‘𝑅)) ↔ 1 < 1))
4	2, 3	mtbiri 327	. . . . . 6 ⊢ ((♯‘(Base‘𝑅)) = 1 → ¬ 1 < (♯‘(Base‘𝑅)))
5	4	adantl 481	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ 1 < (♯‘(Base‘𝑅)))
6	5	intnand 488	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (♯‘(Base‘𝑅)) = 1) → ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
7	6	ex 412	. . 3 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 → ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))))
8		ianor 984	. . . . 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 6855	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
11		hashxrcl 14292	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘(Base‘𝑅)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘(Base‘𝑅)) ∈ ℝ^*
13		1xr 11203	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
14		xrlenlt 11209	. . . . . . . . 9 ⊢ (((♯‘(Base‘𝑅)) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅))))
15	12, 13, 14	mp2an 693	. . . . . . . 8 ⊢ ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅)))
16	15	bicomi 224	. . . . . . 7 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) ↔ (♯‘(Base‘𝑅)) ≤ 1)
17		simpr 484	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ≤ 1)
18		1nn0 12429	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
19		hashbnd 14271	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ∈ V ∧ 1 ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
20	10, 18, 17, 19	mp3an12i 1468	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
21		hashcl 14291	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑅) ∈ Fin → (♯‘(Base‘𝑅)) ∈ ℕ₀)
22		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ₀)
23		hasheq0 14298	. . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((♯‘(Base‘𝑅)) = 0 → (Base‘𝑅) = ∅))
26	25	necon3d 2954	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((Base‘𝑅) ≠ ∅ → (♯‘(Base‘𝑅)) ≠ 0))
27	26	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ≠ 0)
28		elnnne0 12427	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ ↔ ((♯‘(Base‘𝑅)) ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≠ 0))
29	22, 27, 28	sylanbrc 584	. . . . . . . . . . . . . . 15 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ)
30	29	ex 412	. . . . . . . . . . . . . 14 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ∈ ℕ₀ → (♯‘(Base‘𝑅)) ∈ ℕ))
31	30	adantr 480	. . . . . . . . . . . . 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 12187	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) = 1)
37	36	ex 412	. . . . . . . 8 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ≤ 1 → (♯‘(Base‘𝑅)) = 1))
38		ringgrp 20185	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39	grpbn0 18908	. . . . . . . . 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 217	. . . . . 6 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
44	9, 43	jaoi 858	. . . . 5 ⊢ ((¬ 𝑅 ∈ Ring ∨ ¬ 1 < (♯‘(Base‘𝑅))) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
45	8, 44	sylbi 217	. . . 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 212	. 2 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))))
48	39	isnzr2hash 20464	. . . 4 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
49	48	bicomi 224	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ 𝑅 ∈ NzRing)
50	49	notbii 320	. 2 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ ¬ 𝑅 ∈ NzRing)
51	47, 50	bitrdi 287	1 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ∅c0 4287 class class class wbr 5100 ‘cfv 6500 Fincfn 8895 0cc0 11038 1c1 11039 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 ℕcn 12157 ℕ₀cn0 12413 ♯chash 14265 Basecbs 17148 Grpcgrp 18875 Ringcrg 20180 NzRingcnzr 20457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-fz 13436 df-hash 14266 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-0g 17373 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-grp 18878 df-minusg 18879 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-nzr 20458

This theorem is referenced by: 0ringdif 20472 0ring1eq0 20478 rng1nnzr 20720 prmidl0 33542 qsdrng 33589 0ringirng 33866 lmod0rng 48586 lindszr 48826

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