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Theorem nrhmzr 43976
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 2826 . . . . . . . . . 10 (Base‘𝑍) = (Base‘𝑍)
2 eqid 2826 . . . . . . . . . 10 (0g𝑍) = (0g𝑍)
3 eqid 2826 . . . . . . . . . 10 (1r𝑍) = (1r𝑍)
41, 2, 30ring1eq0 43975 . . . . . . . . 9 (𝑍 ∈ (Ring ∖ NzRing) → (1r𝑍) = (0g𝑍))
54adantr 481 . . . . . . . 8 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (1r𝑍) = (0g𝑍))
65adantr 481 . . . . . . 7 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (1r𝑍) = (0g𝑍))
76eqcomd 2832 . . . . . 6 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (0g𝑍) = (1r𝑍))
87fveq2d 6671 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g𝑍)) = (𝑓‘(1r𝑍)))
9 eqid 2826 . . . . . . 7 (1r𝑅) = (1r𝑅)
103, 9rhm1 19402 . . . . . 6 (𝑓 ∈ (𝑍 RingHom 𝑅) → (𝑓‘(1r𝑍)) = (1r𝑅))
1110adantl 482 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(1r𝑍)) = (1r𝑅))
128, 11eqtrd 2861 . . . 4 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → (𝑓‘(0g𝑍)) = (1r𝑅))
13 rhmghm 19397 . . . . . 6 (𝑓 ∈ (𝑍 RingHom 𝑅) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
1413adantl 482 . . . . 5 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → 𝑓 ∈ (𝑍 GrpHom 𝑅))
15 eqid 2826 . . . . . 6 (0g𝑅) = (0g𝑅)
162, 15ghmid 18294 . . . . 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 512 . . 3 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ 𝑓 ∈ (𝑍 RingHom 𝑅)) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
1918ralrimiva 3187 . 2 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
209, 15nzrnz 19952 . . . . . . . . . . . . 13 (𝑅 ∈ NzRing → (1r𝑅) ≠ (0g𝑅))
2120necomd 3076 . . . . . . . . . . . 12 (𝑅 ∈ NzRing → (0g𝑅) ≠ (1r𝑅))
2221adantl 482 . . . . . . . . . . 11 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (0g𝑅) ≠ (1r𝑅))
2322adantr 481 . . . . . . . . . 10 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → (0g𝑅) ≠ (1r𝑅))
24 neeq1 3083 . . . . . . . . . . 11 ((𝑓‘(0g𝑍)) = (0g𝑅) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ↔ (0g𝑅) ≠ (1r𝑅)))
2524adantl 482 . . . . . . . . . 10 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ↔ (0g𝑅) ≠ (1r𝑅)))
2623, 25mpbird 258 . . . . . . . . 9 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → (𝑓‘(0g𝑍)) ≠ (1r𝑅))
2726orcd 871 . . . . . . . 8 (((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)))
2827expcom 414 . . . . . . 7 ((𝑓‘(0g𝑍)) = (0g𝑅) → ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅))))
29 olc 864 . . . . . . . 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 3105 . . . . . 6 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)))
32 neorian 3116 . . . . . 6 (((𝑓‘(0g𝑍)) ≠ (1r𝑅) ∨ (𝑓‘(0g𝑍)) ≠ (0g𝑅)) ↔ ¬ ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
3331, 32sylib 219 . . . . 5 ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → ¬ ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)))
34 con3 156 . . . . 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 1910 . . 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 3148 . . 3 (∀𝑓 ∈ (𝑍 RingHom 𝑅)((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅)) ↔ ∀𝑓(𝑓 ∈ (𝑍 RingHom 𝑅) → ((𝑓‘(0g𝑍)) = (1r𝑅) ∧ (𝑓‘(0g𝑍)) = (0g𝑅))))
38 eq0 4312 . . 3 ((𝑍 RingHom 𝑅) = ∅ ↔ ∀𝑓 ¬ 𝑓 ∈ (𝑍 RingHom 𝑅))
3936, 37, 383imtr4g 297 . 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 207  wa 396  wo 843  wal 1528   = wceq 1530  wcel 2107  wne 3021  wral 3143  cdif 3937  c0 4295  cfv 6352  (class class class)co 7148  Basecbs 16473  0gc0g 16703   GrpHom cghm 18285  1rcur 19171  Ringcrg 19217   RingHom crh 19384  NzRingcnzr 19949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-hash 13681  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-0g 16705  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-mhm 17944  df-grp 18036  df-minusg 18037  df-ghm 18286  df-mgp 19160  df-ur 19172  df-ring 19219  df-rnghom 19387  df-nzr 19950
This theorem is referenced by:  zrninitoringc  44174  nzerooringczr  44175
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