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Theorem zrrnghm 46289
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 4091 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2 ringrng 46251 . . . . 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 46249 . . . . . 6 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7 ablgrp 19574 . . . . . 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 19976 . . . . . 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 2737 . . . . . 6 (0g𝑇) = (0g𝑇)
1513, 140ringbas 46243 . . . . 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 46288 . . . 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 2738 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ 𝑥 = (0g𝑇)) → 0 = 0 )
2313, 14ring0cl 19997 . . . . . . . . . 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 6861 . . . . . . . . 9 0 ∈ V
2726a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 0 ∈ V)
2821, 22, 25, 27fvmptd 6960 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘(0g𝑇)) = 0 )
29 eqid 2737 . . . . . . . . . . . . . 14 (Base‘𝑆) = (Base‘𝑆)
3029, 17grpidcl 18785 . . . . . . . . . . . . 13 (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
318, 30syl 17 . . . . . . . . . . . 12 (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32 eqid 2737 . . . . . . . . . . . . 13 (.r𝑆) = (.r𝑆)
3329, 32, 17rnglz 46256 . . . . . . . . . . . 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 7380 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))) = ( 0 (.r𝑆) 0 ))
40 eqid 2737 . . . . . . . . . . . . . 14 (.r𝑇) = (.r𝑇)
4113, 40, 14ringlz 20018 . . . . . . . . . . . . 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 6851 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = (𝐻‘(0g𝑇)))
4645, 38eqtrd 2777 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = 0 )
4737, 39, 463eqtr4rd 2788 . . . . . . 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 7385 . . . . . . . . 9 (𝑎 = (0g𝑇) → (𝐻‘(𝑎(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)𝑐)))
53 fveq2 6847 . . . . . . . . . 10 (𝑎 = (0g𝑇) → (𝐻𝑎) = (𝐻‘(0g𝑇)))
5453oveq1d 7377 . . . . . . . . 9 (𝑎 = (0g𝑇) → ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)))
5552, 54eqeq12d 2753 . . . . . . . 8 (𝑎 = (0g𝑇) → ((𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐))))
56 oveq2 7370 . . . . . . . . . 10 (𝑐 = (0g𝑇) → ((0g𝑇)(.r𝑇)𝑐) = ((0g𝑇)(.r𝑇)(0g𝑇)))
5756fveq2d 6851 . . . . . . . . 9 (𝑐 = (0g𝑇) → (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))))
58 fveq2 6847 . . . . . . . . . 10 (𝑐 = (0g𝑇) → (𝐻𝑐) = (𝐻‘(0g𝑇)))
5958oveq2d 7378 . . . . . . . . 9 (𝑐 = (0g𝑇) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
6057, 59eqeq12d 2753 . . . . . . . 8 (𝑐 = (0g𝑇) → ((𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇)))))
6155, 602ralsng 4642 . . . . . . 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 3312 . . . . . . 7 (𝐵 = {(0g𝑇)} → (∀𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6564raleqbi1dv 3310 . . . . . 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 46264 . 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 3065  Vcvv 3448  cdif 3912  {csn 4591  cmpt 5193  cfv 6501  (class class class)co 7362  Basecbs 17090  .rcmulr 17141  0gc0g 17328  Grpcgrp 18755   GrpHom cghm 19012  Abelcabl 19570  Ringcrg 19971  NzRingcnzr 20743  Rngcrng 46246   RngHomo crngh 46257
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-hash 14238  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-plusg 17153  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-mhm 18608  df-grp 18758  df-minusg 18759  df-ghm 19013  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-nzr 20744  df-mgmhm 46147  df-rng 46247  df-rnghomo 46259
This theorem is referenced by:  zrinitorngc  46372
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