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Theorem zrrnghm 46716
Description: The constant mapping to zero is a non-unital ring homomorphism from the zero ring to any non-unital 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 4127 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2 ringrng 46655 . . . . 5 (𝑇 ∈ Ring → 𝑇 ∈ Rng)
31, 2syl 17 . . . 4 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
43anim1i 616 . . 3 ((𝑇 ∈ (Ring ∖ NzRing) ∧ 𝑆 ∈ Rng) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
54ancoms 460 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
6 rngabl 46651 . . . . . 6 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7 ablgrp 19653 . . . . . 6 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
86, 7syl 17 . . . . 5 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
98adantr 482 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑆 ∈ Grp)
10 ringgrp 20061 . . . . . 6 (𝑇 ∈ Ring → 𝑇 ∈ Grp)
111, 10syl 17 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp)
1211adantl 483 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑇 ∈ Grp)
13 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
14 eqid 2733 . . . . . 6 (0g𝑇) = (0g𝑇)
1513, 140ringbas 46645 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g𝑇)})
1615adantl 483 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐵 = {(0g𝑇)})
17 zrrhm.0 . . . . 5 0 = (0g𝑆)
18 zrrhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
1913, 17, 18, 14c0snghm 46715 . . . 4 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
209, 12, 16, 19syl3anc 1372 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
2118a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 𝐻 = (𝑥𝐵0 ))
22 eqidd 2734 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ 𝑥 = (0g𝑇)) → 0 = 0 )
2313, 14ring0cl 20084 . . . . . . . . . 10 (𝑇 ∈ Ring → (0g𝑇) ∈ 𝐵)
241, 23syl 17 . . . . . . . . 9 (𝑇 ∈ (Ring ∖ NzRing) → (0g𝑇) ∈ 𝐵)
2524ad2antlr 726 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (0g𝑇) ∈ 𝐵)
2617fvexi 6906 . . . . . . . . 9 0 ∈ V
2726a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 0 ∈ V)
2821, 22, 25, 27fvmptd 7006 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘(0g𝑇)) = 0 )
29 eqid 2733 . . . . . . . . . . . . . 14 (Base‘𝑆) = (Base‘𝑆)
3029, 17grpidcl 18850 . . . . . . . . . . . . 13 (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
318, 30syl 17 . . . . . . . . . . . 12 (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32 eqid 2733 . . . . . . . . . . . . 13 (.r𝑆) = (.r𝑆)
3329, 32, 17rnglz 46664 . . . . . . . . . . . 12 ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r𝑆) 0 ) = 0 )
3431, 33mpdan 686 . . . . . . . . . . 11 (𝑆 ∈ Rng → ( 0 (.r𝑆) 0 ) = 0 )
3534adantr 482 . . . . . . . . . 10 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ( 0 (.r𝑆) 0 ) = 0 )
3635adantr 482 . . . . . . . . 9 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ( 0 (.r𝑆) 0 ) = 0 )
3736adantr 482 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ( 0 (.r𝑆) 0 ) = 0 )
38 simpr 486 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘(0g𝑇)) = 0 )
3938, 38oveq12d 7427 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))) = ( 0 (.r𝑆) 0 ))
40 eqid 2733 . . . . . . . . . . . . . 14 (.r𝑇) = (.r𝑇)
4113, 40, 14ringlz 20107 . . . . . . . . . . . . 13 ((𝑇 ∈ Ring ∧ (0g𝑇) ∈ 𝐵) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
421, 23, 41syl2anc2 586 . . . . . . . . . . . 12 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4342ad2antlr 726 . . . . . . . . . . 11 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4443adantr 482 . . . . . . . . . 10 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4544fveq2d 6896 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = (𝐻‘(0g𝑇)))
4645, 38eqtrd 2773 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = 0 )
4737, 39, 463eqtr4rd 2784 . . . . . . 7 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
4828, 47mpdan 686 . . . . . 6 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
4923, 23jca 513 . . . . . . . . 9 (𝑇 ∈ Ring → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
501, 49syl 17 . . . . . . . 8 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
5150ad2antlr 726 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
52 fvoveq1 7432 . . . . . . . . 9 (𝑎 = (0g𝑇) → (𝐻‘(𝑎(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)𝑐)))
53 fveq2 6892 . . . . . . . . . 10 (𝑎 = (0g𝑇) → (𝐻𝑎) = (𝐻‘(0g𝑇)))
5453oveq1d 7424 . . . . . . . . 9 (𝑎 = (0g𝑇) → ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)))
5552, 54eqeq12d 2749 . . . . . . . 8 (𝑎 = (0g𝑇) → ((𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐))))
56 oveq2 7417 . . . . . . . . . 10 (𝑐 = (0g𝑇) → ((0g𝑇)(.r𝑇)𝑐) = ((0g𝑇)(.r𝑇)(0g𝑇)))
5756fveq2d 6896 . . . . . . . . 9 (𝑐 = (0g𝑇) → (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))))
58 fveq2 6892 . . . . . . . . . 10 (𝑐 = (0g𝑇) → (𝐻𝑐) = (𝐻‘(0g𝑇)))
5958oveq2d 7425 . . . . . . . . 9 (𝑐 = (0g𝑇) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
6057, 59eqeq12d 2749 . . . . . . . 8 (𝑐 = (0g𝑇) → ((𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇)))))
6155, 602ralsng 4681 . . . . . . 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)))))
6251, 61syl 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)))))
6348, 62mpbird 257 . . . . 5 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
64 raleq 3323 . . . . . . 7 (𝐵 = {(0g𝑇)} → (∀𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6564raleqbi1dv 3334 . . . . . 6 (𝐵 = {(0g𝑇)} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6665adantl 483 . . . . 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))))
6763, 66mpbird 257 . . . 4 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
6816, 67mpdan 686 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
6920, 68jca 513 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
7013, 40, 32isrnghm 46690 . 2 (𝐻 ∈ (𝑇 RngHomo 𝑆) ↔ ((𝑇 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))))
715, 69, 70sylanbrc 584 1 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cdif 3946  {csn 4629  cmpt 5232  cfv 6544  (class class class)co 7409  Basecbs 17144  .rcmulr 17198  0gc0g 17385  Grpcgrp 18819   GrpHom cghm 19089  Abelcabl 19649  Ringcrg 20056  NzRingcnzr 20291  Rngcrng 46648   RngHomo crngh 46683
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-hash 14291  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-grp 18822  df-minusg 18823  df-ghm 19090  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-nzr 20292  df-mgmhm 46549  df-rng 46649  df-rnghomo 46685
This theorem is referenced by:  zrinitorngc  46898
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