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Theorem nrhmzr 42435
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.
StepHypRef Expression
1 eqid 2802 . . . . . . . . . 10 (Base‘𝑍) = (Base‘𝑍)
2 eqid 2802 . . . . . . . . . 10 (0g𝑍) = (0g𝑍)
3 eqid 2802 . . . . . . . . . 10 (1r𝑍) = (1r𝑍)
41, 2, 30ring1eq0 42434 . . . . . . . . 9 (𝑍 ∈ (Ring ∖ NzRing) → (1r𝑍) = (0g𝑍))
54adantr 468 . . . . . . . 8 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (1r𝑍) = (0g𝑍))
65adantr 468 . . . . . . 7 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (1r𝑍) = (0g𝑍))
76eqcomd 2808 . . . . . 6 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (0g𝑍) = (1r𝑍))
87fveq2d 6406 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g𝑍)) = (𝑓‘(1r𝑍)))
9 eqid 2802 . . . . . . 7 (1r𝑅) = (1r𝑅)
103, 9rhm1 18928 . . . . . 6 (𝑓 ∈ (𝑍 RingHom 𝑅) → (𝑓‘(1r𝑍)) = (1r𝑅))
1110adantl 469 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(1r𝑍)) = (1r𝑅))
128, 11eqtrd 2836 . . . 4 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g𝑍)) = (1r𝑅))
13 rhmghm 18923 . . . . . 6 (𝑓 ∈ (𝑍 RingHom 𝑅) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
1413adantl 469 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
15 eqid 2802 . . . . . 6 (0g𝑅) = (0g𝑅)
162, 15ghmid 17862 . . . . 5 (𝑓 ∈ (𝑍 GrpHom 𝑅) → (𝑓‘(0g𝑍)) = (0g𝑅))
1714, 16syl 17 . . . 4 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g𝑍)) = (0g𝑅))
1812, 17jca 503 . . 3 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
1918ralrimiva 3150 . 2 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
209, 15nzrnz 19463 . . . . . . . . . . . . 13 (𝑅 ∈ NzRing → (1r𝑅) ≠ (0g𝑅))
2120necomd 3029 . . . . . . . . . . . 12 (𝑅 ∈ NzRing → (0g𝑅) ≠ (1r𝑅))
2221adantl 469 . . . . . . . . . . 11 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (0g𝑅) ≠ (1r𝑅))
2322adantr 468 . . . . . . . . . 10 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → (0g𝑅) ≠ (1r𝑅))
24 neeq1 3036 . . . . . . . . . . 11 ((𝑓‘(0g𝑍)) = (0g𝑅) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ↔ (0g𝑅) ≠ (1r𝑅)))
2524adantl 469 . . . . . . . . . 10 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ↔ (0g𝑅) ≠ (1r𝑅)))
2623, 25mpbird 248 . . . . . . . . 9 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → (𝑓‘(0g𝑍)) ≠ (1r𝑅))
2726orcd 891 . . . . . . . 8 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)))
2827expcom 400 . . . . . . 7 ((𝑓‘(0g𝑍)) = (0g𝑅) → ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅))))
29 olc 886 . . . . . . . 8 ((𝑓‘(0g𝑍)) ≠ (0g𝑅) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)))
3029a1d 25 . . . . . . 7 ((𝑓‘(0g𝑍)) ≠ (0g𝑅) → ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅))))
3128, 30pm2.61ine 3057 . . . . . 6 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)))
32 neorian 3068 . . . . . 6 (((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)) ↔ ¬ ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
3331, 32sylib 209 . . . . 5 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ¬ ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
34 con3 150 . . . . 5 ((𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅))) → (¬ ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → ¬ 𝑓 ∈ (𝑍 RingHom 𝑅)))
3533, 34syl5com 31 . . . 4 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅))) → ¬ 𝑓 ∈ (𝑍 RingHom 𝑅)))
3635alimdv 2007 . . 3 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (∀𝑓(𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅))) → ∀𝑓 ¬ 𝑓 ∈ (𝑍 RingHom 𝑅)))
37 df-ral 3097 . . 3 (∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) ↔ ∀𝑓(𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅))))
38 eq0 4124 . . 3 ((𝑍 RingHom 𝑅) = ∅ ↔ ∀𝑓 ¬ 𝑓 ∈ (𝑍 RingHom 𝑅))
3936, 37, 383imtr4g 287 . 2 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → (𝑍 RingHom 𝑅) = ∅))
4019, 39mpd 15 1 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  wal 1635   = wceq 1637  wcel 2155  wne 2974  wral 3092  cdif 3760  c0 4110  cfv 6095  (class class class)co 6868  Basecbs 16062  0gc0g 16299   GrpHom cghm 17853  1rcur 18697  Ringcrg 18743   RingHom crh 18910  NzRingcnzr 19460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-cnex 10271  ax-resscn 10272  ax-1cn 10273  ax-icn 10274  ax-addcl 10275  ax-addrcl 10276  ax-mulcl 10277  ax-mulrcl 10278  ax-mulcom 10279  ax-addass 10280  ax-mulass 10281  ax-distr 10282  ax-i2m1 10283  ax-1ne0 10284  ax-1rid 10285  ax-rnegex 10286  ax-rrecex 10287  ax-cnre 10288  ax-pre-lttri 10289  ax-pre-lttrn 10290  ax-pre-ltadd 10291  ax-pre-mulgt0 10292
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-oadd 7794  df-er 7973  df-map 8088  df-en 8187  df-dom 8188  df-sdom 8189  df-fin 8190  df-card 9042  df-cda 9269  df-pnf 10355  df-mnf 10356  df-xr 10357  df-ltxr 10358  df-le 10359  df-sub 10547  df-neg 10548  df-nn 11300  df-2 11358  df-n0 11554  df-xnn0 11624  df-z 11638  df-uz 11899  df-fz 12544  df-hash 13332  df-ndx 16065  df-slot 16066  df-base 16068  df-sets 16069  df-plusg 16160  df-0g 16301  df-mgm 17441  df-sgrp 17483  df-mnd 17494  df-mhm 17534  df-grp 17624  df-minusg 17625  df-ghm 17854  df-mgp 18686  df-ur 18698  df-ring 18745  df-rnghom 18913  df-nzr 19461
This theorem is referenced by:  zrninitoringc  42633  nzerooringczr  42634
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