Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  zrrnghm Structured version   Visualization version   GIF version

Theorem zrrnghm 41707
 Description: The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.)
Hypotheses
Ref Expression
zrrhm.b 𝐵 = (Base‘𝑇)
zrrhm.0 0 = (0g𝑆)
zrrhm.h 𝐻 = (𝑥𝐵0 )
Assertion
Ref Expression
zrrnghm ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆   𝑥,𝑇   𝑥, 0
Allowed substitution hint:   𝐻(𝑥)

Proof of Theorem zrrnghm
Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 3694 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2 ringrng 41669 . . . . 5 (𝑇 ∈ Ring → 𝑇 ∈ Rng)
31, 2syl 17 . . . 4 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
43anim1i 590 . . 3 ((𝑇 ∈ (Ring ∖ NzRing) ∧ 𝑆 ∈ Rng) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
54ancoms 468 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
6 rngabl 41667 . . . . . 6 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7 ablgrp 18021 . . . . . 6 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
86, 7syl 17 . . . . 5 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
98adantr 480 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑆 ∈ Grp)
10 ringgrp 18375 . . . . . 6 (𝑇 ∈ Ring → 𝑇 ∈ Grp)
111, 10syl 17 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp)
1211adantl 481 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑇 ∈ Grp)
13 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
14 eqid 2610 . . . . . 6 (0g𝑇) = (0g𝑇)
1513, 140ringbas 41661 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g𝑇)})
1615adantl 481 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐵 = {(0g𝑇)})
17 zrrhm.0 . . . . 5 0 = (0g𝑆)
18 zrrhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
1913, 17, 18, 14c0snghm 41706 . . . 4 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
209, 12, 16, 19syl3anc 1318 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
2118a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 𝐻 = (𝑥𝐵0 ))
22 eqidd 2611 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ 𝑥 = (0g𝑇)) → 0 = 0 )
2313, 14ring0cl 18392 . . . . . . . . . 10 (𝑇 ∈ Ring → (0g𝑇) ∈ 𝐵)
241, 23syl 17 . . . . . . . . 9 (𝑇 ∈ (Ring ∖ NzRing) → (0g𝑇) ∈ 𝐵)
2524ad2antlr 759 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (0g𝑇) ∈ 𝐵)
26 fvex 6113 . . . . . . . . . 10 (0g𝑆) ∈ V
2717, 26eqeltri 2684 . . . . . . . . 9 0 ∈ V
2827a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 0 ∈ V)
2921, 22, 25, 28fvmptd 6197 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘(0g𝑇)) = 0 )
30 eqid 2610 . . . . . . . . . . . . . 14 (Base‘𝑆) = (Base‘𝑆)
3130, 17grpidcl 17273 . . . . . . . . . . . . 13 (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
328, 31syl 17 . . . . . . . . . . . 12 (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
33 eqid 2610 . . . . . . . . . . . . 13 (.r𝑆) = (.r𝑆)
3430, 33, 17rnglz 41674 . . . . . . . . . . . 12 ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r𝑆) 0 ) = 0 )
3532, 34mpdan 699 . . . . . . . . . . 11 (𝑆 ∈ Rng → ( 0 (.r𝑆) 0 ) = 0 )
3635adantr 480 . . . . . . . . . 10 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ( 0 (.r𝑆) 0 ) = 0 )
3736adantr 480 . . . . . . . . 9 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ( 0 (.r𝑆) 0 ) = 0 )
3837adantr 480 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ( 0 (.r𝑆) 0 ) = 0 )
39 simpr 476 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘(0g𝑇)) = 0 )
4039, 39oveq12d 6567 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))) = ( 0 (.r𝑆) 0 ))
41 eqid 2610 . . . . . . . . . . . . . . 15 (.r𝑇) = (.r𝑇)
4213, 41, 14ringlz 18410 . . . . . . . . . . . . . 14 ((𝑇 ∈ Ring ∧ (0g𝑇) ∈ 𝐵) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4323, 42mpdan 699 . . . . . . . . . . . . 13 (𝑇 ∈ Ring → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
441, 43syl 17 . . . . . . . . . . . 12 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4544ad2antlr 759 . . . . . . . . . . 11 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4645adantr 480 . . . . . . . . . 10 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4746fveq2d 6107 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = (𝐻‘(0g𝑇)))
4847, 39eqtrd 2644 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = 0 )
4938, 40, 483eqtr4rd 2655 . . . . . . 7 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
5029, 49mpdan 699 . . . . . 6 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
5123, 23jca 553 . . . . . . . . 9 (𝑇 ∈ Ring → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
521, 51syl 17 . . . . . . . 8 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
5352ad2antlr 759 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
54 oveq1 6556 . . . . . . . . . 10 (𝑎 = (0g𝑇) → (𝑎(.r𝑇)𝑐) = ((0g𝑇)(.r𝑇)𝑐))
5554fveq2d 6107 . . . . . . . . 9 (𝑎 = (0g𝑇) → (𝐻‘(𝑎(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)𝑐)))
56 fveq2 6103 . . . . . . . . . 10 (𝑎 = (0g𝑇) → (𝐻𝑎) = (𝐻‘(0g𝑇)))
5756oveq1d 6564 . . . . . . . . 9 (𝑎 = (0g𝑇) → ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)))
5855, 57eqeq12d 2625 . . . . . . . 8 (𝑎 = (0g𝑇) → ((𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐))))
59 oveq2 6557 . . . . . . . . . 10 (𝑐 = (0g𝑇) → ((0g𝑇)(.r𝑇)𝑐) = ((0g𝑇)(.r𝑇)(0g𝑇)))
6059fveq2d 6107 . . . . . . . . 9 (𝑐 = (0g𝑇) → (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))))
61 fveq2 6103 . . . . . . . . . 10 (𝑐 = (0g𝑇) → (𝐻𝑐) = (𝐻‘(0g𝑇)))
6261oveq2d 6565 . . . . . . . . 9 (𝑐 = (0g𝑇) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
6360, 62eqeq12d 2625 . . . . . . . 8 (𝑐 = (0g𝑇) → ((𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇)))))
6458, 632ralsng 4167 . . . . . . 7 (((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵) → (∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇)))))
6553, 64syl 17 . . . . . 6 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇)))))
6650, 65mpbird 246 . . . . 5 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
67 raleq 3115 . . . . . . 7 (𝐵 = {(0g𝑇)} → (∀𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6867raleqbi1dv 3123 . . . . . 6 (𝐵 = {(0g𝑇)} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6968adantl 481 . . . . 5 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
7066, 69mpbird 246 . . . 4 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
7116, 70mpdan 699 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
7220, 71jca 553 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
7313, 41, 33isrnghm 41682 . 2 (𝐻 ∈ (𝑇 RngHomo 𝑆) ↔ ((𝑇 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))))
745, 72, 73sylanbrc 695 1 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  {csn 4125   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  0gc0g 15923  Grpcgrp 17245   GrpHom cghm 17480  Abelcabl 18017  Ringcrg 18370  NzRingcnzr 19078  Rngcrng 41664   RngHomo crngh 41675 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-ghm 17481  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-nzr 19079  df-mgmhm 41569  df-rng0 41665  df-rnghomo 41677 This theorem is referenced by:  zrinitorngc  41792
 Copyright terms: Public domain W3C validator