0ringnnzr - Metamath Proof Explorer

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Theorem 0ringnnzr 20415

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 11219	. . . . . . . 8 ⊢ 1 ∈ ℝ
2	1	ltnri 11328	. . . . . . 7 ⊢ ¬ 1 < 1
3		breq2 5153	. . . . . . 7 ⊢ ((♯‘(Base‘𝑅)) = 1 → (1 < (♯‘(Base‘𝑅)) ↔ 1 < 1))
4	2, 3	mtbiri 326	. . . . . 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 979	. . . . 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 6905	. . . . . . . . . 10 ⊢ (Base‘𝑅) ∈ V
11		hashxrcl 14322	. . . . . . . . . 10 ⊢ ((Base‘𝑅) ∈ V → (♯‘(Base‘𝑅)) ∈ ℝ^*)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ (♯‘(Base‘𝑅)) ∈ ℝ^*
13		1xr 11278	. . . . . . . . 9 ⊢ 1 ∈ ℝ^*
14		xrlenlt 11284	. . . . . . . . 9 ⊢ (((♯‘(Base‘𝑅)) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅))))
15	12, 13, 14	mp2an 689	. . . . . . . 8 ⊢ ((♯‘(Base‘𝑅)) ≤ 1 ↔ ¬ 1 < (♯‘(Base‘𝑅)))
16	15	bicomi 223	. . . . . . 7 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) ↔ (♯‘(Base‘𝑅)) ≤ 1)
17		simpr 484	. . . . . . . . . 10 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) ≤ 1)
18		1nn0 12493	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℕ₀
19		hashbnd 14301	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑅) ∈ V ∧ 1 ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
20	10, 18, 17, 19	mp3an12i 1464	. . . . . . . . . . . 12 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (Base‘𝑅) ∈ Fin)
21		hashcl 14321	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑅) ∈ Fin → (♯‘(Base‘𝑅)) ∈ ℕ₀)
22		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ∈ ℕ₀)
23		hasheq0 14328	. . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((♯‘(Base‘𝑅)) = 0 → (Base‘𝑅) = ∅))
26	25	necon3d 2960	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ₀ → ((Base‘𝑅) ≠ ∅ → (♯‘(Base‘𝑅)) ≠ 0))
27	26	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ∈ ℕ₀) → (♯‘(Base‘𝑅)) ≠ 0)
28		elnnne0 12491	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘(Base‘𝑅)) ∈ ℕ ↔ ((♯‘(Base‘𝑅)) ∈ ℕ₀ ∧ (♯‘(Base‘𝑅)) ≠ 0))
29	22, 27, 28	sylanbrc 582	. . . . . . . . . . . . . . 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 12247	. . . . . . . . . . 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 231	. . . . . . . . 9 ⊢ (((Base‘𝑅) ≠ ∅ ∧ (♯‘(Base‘𝑅)) ≤ 1) → (♯‘(Base‘𝑅)) = 1)
37	36	ex 412	. . . . . . . 8 ⊢ ((Base‘𝑅) ≠ ∅ → ((♯‘(Base‘𝑅)) ≤ 1 → (♯‘(Base‘𝑅)) = 1))
38		ringgrp 20133	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
39		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
40	39	grpbn0 18888	. . . . . . . . 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 216	. . . . . 6 ⊢ (¬ 1 < (♯‘(Base‘𝑅)) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
44	9, 43	jaoi 854	. . . . 5 ⊢ ((¬ 𝑅 ∈ Ring ∨ ¬ 1 < (♯‘(Base‘𝑅))) → (𝑅 ∈ Ring → (♯‘(Base‘𝑅)) = 1))
45	8, 44	sylbi 216	. . . 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 211	. 2 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅)))))
48	39	isnzr2hash 20411	. . . 4 ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))))
49	48	bicomi 223	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ 𝑅 ∈ NzRing)
50	49	notbii 319	. 2 ⊢ (¬ (𝑅 ∈ Ring ∧ 1 < (♯‘(Base‘𝑅))) ↔ ¬ 𝑅 ∈ NzRing)
51	47, 50	bitrdi 286	1 ⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 ∅c0 4323 class class class wbr 5149 ‘cfv 6544 Fincfn 8942 0cc0 11113 1c1 11114 ℝ^*cxr 11252 < clt 11253 ≤ cle 11254 ℕcn 12217 ℕ₀cn0 12477 ♯chash 14295 Basecbs 17149 Grpcgrp 18856 Ringcrg 20128 NzRingcnzr 20404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-oadd 8473 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-dju 9899 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-fz 13490 df-hash 14296 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-grp 18859 df-minusg 18860 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-nzr 20405

This theorem is referenced by: 0ringdif 20417 0ring1eq0 20423 rng1nnzr 20540 prmidl0 32840 qsdrng 32882 0ringirng 33039 lmod0rng 46910 lindszr 47239

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