Theorem nrhmzr 20454

Description: There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.)

Assertion

Ref	Expression
nrhmzr	⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅)

Proof of Theorem nrhmzr

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍)
2		eqid 2733	. . . . . . . . . 10 ⊢ (0g‘𝑍) = (0g‘𝑍)
3		eqid 2733	. . . . . . . . . 10 ⊢ (1r‘𝑍) = (1r‘𝑍)
4	1, 2, 3	0ring1eq0 20450	. . . . . . . . 9 ⊢ (𝑍 ∈ (Ring ∖ NzRing) → (1r‘𝑍) = (0g‘𝑍))
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (1r‘𝑍) = (0g‘𝑍))
6	5	adantr 480	. . . . . . 7 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (1r‘𝑍) = (0g‘𝑍))
7	6	eqcomd 2739	. . . . . 6 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (0g‘𝑍) = (1r‘𝑍))
8	7	fveq2d 6832	. . . . 5 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g‘𝑍)) = (𝑓‘(1r‘𝑍)))
9		eqid 2733	. . . . . . 7 ⊢ (1r‘𝑅) = (1r‘𝑅)
10	3, 9	rhm1 20408	. . . . . 6 ⊢ (𝑓 ∈ (𝑍 RingHom 𝑅) → (𝑓‘(1r‘𝑍)) = (1r‘𝑅))
11	10	adantl 481	. . . . 5 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(1r‘𝑍)) = (1r‘𝑅))
12	8, 11	eqtrd 2768	. . . 4 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g‘𝑍)) = (1r‘𝑅))
13		rhmghm 20403	. . . . . 6 ⊢ (𝑓 ∈ (𝑍 RingHom 𝑅) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
14	13	adantl 481	. . . . 5 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
15		eqid 2733	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	2, 15	ghmid 19136	. . . . 5 ⊢ (𝑓 ∈ (𝑍 GrpHom 𝑅) → (𝑓‘(0g‘𝑍)) = (0g‘𝑅))
17	14, 16	syl 17	. . . 4 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g‘𝑍)) = (0g‘𝑅))
18	12, 17	jca 511	. . 3 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)))
19	18	ralrimiva 3125	. 2 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)))
20	9, 15	nzrnz 20432	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅))
21	20	necomd 2984	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ NzRing → (0g‘𝑅) ≠ (1r‘𝑅))
22	21	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (0g‘𝑅) ≠ (1r‘𝑅))
23	22	adantr 480	. . . . . . . . . 10 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)) → (0g‘𝑅) ≠ (1r‘𝑅))
24		neeq1 2991	. . . . . . . . . . 11 ⊢ ((𝑓‘(0g‘𝑍)) = (0g‘𝑅) → ((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ↔ (0g‘𝑅) ≠ (1r‘𝑅)))
25	24	adantl 481	. . . . . . . . . 10 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)) → ((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ↔ (0g‘𝑅) ≠ (1r‘𝑅)))
26	23, 25	mpbird 257	. . . . . . . . 9 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)) → (𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅))
27	26	orcd 873	. . . . . . . 8 ⊢ (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)) → ((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ∨ (𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅)))
28	27	expcom 413	. . . . . . 7 ⊢ ((𝑓‘(0g‘𝑍)) = (0g‘𝑅) → ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ∨ (𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅))))
29		olc 868	. . . . . . . 8 ⊢ ((𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅) → ((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ∨ (𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅)))
30	29	a1d 25	. . . . . . 7 ⊢ ((𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅) → ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ∨ (𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅))))
31	28, 30	pm2.61ine 3012	. . . . . 6 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ∨ (𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅)))
32		neorian 3024	. . . . . 6 ⊢ (((𝑓‘(0g‘𝑍)) ≠ (1r‘𝑅) ∨ (𝑓‘(0g‘𝑍)) ≠ (0g‘𝑅)) ↔ ¬ ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)))
33	31, 32	sylib 218	. . . . 5 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ¬ ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)))
34		con3 153	. . . . 5 ⊢ ((𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅))) → (¬ ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)) → ¬ 𝑓 ∈ (𝑍 RingHom 𝑅)))
35	33, 34	syl5com 31	. . . 4 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅))) → ¬ 𝑓 ∈ (𝑍 RingHom 𝑅)))
36	35	alimdv 1917	. . 3 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (∀𝑓(𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅))) → ∀𝑓 ¬ 𝑓 ∈ (𝑍 RingHom 𝑅)))
37		df-ral 3049	. . 3 ⊢ (∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)) ↔ ∀𝑓(𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅))))
38		eq0 4299	. . 3 ⊢ ((𝑍 RingHom 𝑅) = ∅ ↔ ∀𝑓 ¬ 𝑓 ∈ (𝑍 RingHom 𝑅))
39	36, 37, 38	3imtr4g 296	. 2 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g‘𝑍)) = (1r‘𝑅) ∧ (𝑓‘(0g‘𝑍)) = (0g‘𝑅)) → (𝑍 RingHom 𝑅) = ∅))
40	19, 39	mpd 15	1 ⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048   ∖ cdif 3895  ∅c0 4282  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  0_gc0g 17345   GrpHom cghm 19126  1_rcur 20101  Ringcrg 20153   RingHom crh 20389  NzRingcnzr 20429

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-oadd 8395  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-dju 9801  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-n0 12389  df-xnn0 12462  df-z 12476  df-uz 12739  df-fz 13410  df-hash 14240  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-plusg 17176  df-0g 17347  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-grp 18851  df-minusg 18852  df-ghm 19127  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-rhm 20392  df-nzr 20430

This theorem is referenced by:  zrninitoringc  20593  nzerooringczr  21419