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Theorem nrhmzr 46257
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 2733 . . . . . . . . . 10 (Base‘𝑍) = (Base‘𝑍)
2 eqid 2733 . . . . . . . . . 10 (0g𝑍) = (0g𝑍)
3 eqid 2733 . . . . . . . . . 10 (1r𝑍) = (1r𝑍)
41, 2, 30ring1eq0 46256 . . . . . . . . 9 (𝑍 ∈ (Ring ∖ NzRing) → (1r𝑍) = (0g𝑍))
54adantr 482 . . . . . . . 8 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (1r𝑍) = (0g𝑍))
65adantr 482 . . . . . . 7 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (1r𝑍) = (0g𝑍))
76eqcomd 2739 . . . . . 6 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (0g𝑍) = (1r𝑍))
87fveq2d 6847 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g𝑍)) = (𝑓‘(1r𝑍)))
9 eqid 2733 . . . . . . 7 (1r𝑅) = (1r𝑅)
103, 9rhm1 20169 . . . . . 6 (𝑓 ∈ (𝑍 RingHom 𝑅) → (𝑓‘(1r𝑍)) = (1r𝑅))
1110adantl 483 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(1r𝑍)) = (1r𝑅))
128, 11eqtrd 2773 . . . 4 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g𝑍)) = (1r𝑅))
13 rhmghm 20164 . . . . . 6 (𝑓 ∈ (𝑍 RingHom 𝑅) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
1413adantl 483 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
15 eqid 2733 . . . . . 6 (0g𝑅) = (0g𝑅)
162, 15ghmid 19019 . . . . 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 513 . . 3 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
1918ralrimiva 3140 . 2 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
209, 15nzrnz 20746 . . . . . . . . . . . . 13 (𝑅 ∈ NzRing → (1r𝑅) ≠ (0g𝑅))
2120necomd 2996 . . . . . . . . . . . 12 (𝑅 ∈ NzRing → (0g𝑅) ≠ (1r𝑅))
2221adantl 483 . . . . . . . . . . 11 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (0g𝑅) ≠ (1r𝑅))
2322adantr 482 . . . . . . . . . 10 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → (0g𝑅) ≠ (1r𝑅))
24 neeq1 3003 . . . . . . . . . . 11 ((𝑓‘(0g𝑍)) = (0g𝑅) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ↔ (0g𝑅) ≠ (1r𝑅)))
2524adantl 483 . . . . . . . . . 10 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ↔ (0g𝑅) ≠ (1r𝑅)))
2623, 25mpbird 257 . . . . . . . . 9 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → (𝑓‘(0g𝑍)) ≠ (1r𝑅))
2726orcd 872 . . . . . . . 8 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)))
2827expcom 415 . . . . . . 7 ((𝑓‘(0g𝑍)) = (0g𝑅) → ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅))))
29 olc 867 . . . . . . . 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 3025 . . . . . 6 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)))
32 neorian 3036 . . . . . 6 (((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)) ↔ ¬ ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
3331, 32sylib 217 . . . . 5 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ¬ ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
34 con3 153 . . . . 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 1920 . . 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 3062 . . 3 (∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) ↔ ∀𝑓(𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅))))
38 eq0 4304 . . 3 ((𝑍 RingHom 𝑅) = ∅ ↔ ∀𝑓 ¬ 𝑓 ∈ (𝑍 RingHom 𝑅))
3936, 37, 383imtr4g 296 . 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 205  wa 397  wo 846  wal 1540   = wceq 1542  wcel 2107  wne 2940  wral 3061  cdif 3908  c0 4283  cfv 6497  (class class class)co 7358  Basecbs 17088  0gc0g 17326   GrpHom cghm 19010  1rcur 19918  Ringcrg 19969   RingHom crh 20150  NzRingcnzr 20743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-dju 9842  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-n0 12419  df-xnn0 12491  df-z 12505  df-uz 12769  df-fz 13431  df-hash 14237  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-grp 18756  df-minusg 18757  df-ghm 19011  df-mgp 19902  df-ur 19919  df-ring 19971  df-rnghom 20153  df-nzr 20744
This theorem is referenced by:  zrninitoringc  46455  nzerooringczr  46456
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