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Theorem zrrnghm 46205
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 4086 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2 ringrng 46167 . . . . 5 (𝑇 ∈ Ring → 𝑇 ∈ Rng)
31, 2syl 17 . . . 4 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
43anim1i 615 . . 3 ((𝑇 ∈ (Ring ∖ NzRing) ∧ 𝑆 ∈ Rng) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
54ancoms 459 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
6 rngabl 46165 . . . . . 6 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7 ablgrp 19567 . . . . . 6 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
86, 7syl 17 . . . . 5 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
98adantr 481 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑆 ∈ Grp)
10 ringgrp 19969 . . . . . 6 (𝑇 ∈ Ring → 𝑇 ∈ Grp)
111, 10syl 17 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp)
1211adantl 482 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑇 ∈ Grp)
13 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
14 eqid 2736 . . . . . 6 (0g𝑇) = (0g𝑇)
1513, 140ringbas 46159 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g𝑇)})
1615adantl 482 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐵 = {(0g𝑇)})
17 zrrhm.0 . . . . 5 0 = (0g𝑆)
18 zrrhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
1913, 17, 18, 14c0snghm 46204 . . . 4 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
209, 12, 16, 19syl3anc 1371 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
2118a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 𝐻 = (𝑥𝐵0 ))
22 eqidd 2737 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ 𝑥 = (0g𝑇)) → 0 = 0 )
2313, 14ring0cl 19990 . . . . . . . . . 10 (𝑇 ∈ Ring → (0g𝑇) ∈ 𝐵)
241, 23syl 17 . . . . . . . . 9 (𝑇 ∈ (Ring ∖ NzRing) → (0g𝑇) ∈ 𝐵)
2524ad2antlr 725 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (0g𝑇) ∈ 𝐵)
2617fvexi 6856 . . . . . . . . 9 0 ∈ V
2726a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 0 ∈ V)
2821, 22, 25, 27fvmptd 6955 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘(0g𝑇)) = 0 )
29 eqid 2736 . . . . . . . . . . . . . 14 (Base‘𝑆) = (Base‘𝑆)
3029, 17grpidcl 18778 . . . . . . . . . . . . 13 (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
318, 30syl 17 . . . . . . . . . . . 12 (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
32 eqid 2736 . . . . . . . . . . . . 13 (.r𝑆) = (.r𝑆)
3329, 32, 17rnglz 46172 . . . . . . . . . . . 12 ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r𝑆) 0 ) = 0 )
3431, 33mpdan 685 . . . . . . . . . . 11 (𝑆 ∈ Rng → ( 0 (.r𝑆) 0 ) = 0 )
3534adantr 481 . . . . . . . . . 10 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ( 0 (.r𝑆) 0 ) = 0 )
3635adantr 481 . . . . . . . . 9 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ( 0 (.r𝑆) 0 ) = 0 )
3736adantr 481 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ( 0 (.r𝑆) 0 ) = 0 )
38 simpr 485 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘(0g𝑇)) = 0 )
3938, 38oveq12d 7375 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))) = ( 0 (.r𝑆) 0 ))
40 eqid 2736 . . . . . . . . . . . . . 14 (.r𝑇) = (.r𝑇)
4113, 40, 14ringlz 20011 . . . . . . . . . . . . 13 ((𝑇 ∈ Ring ∧ (0g𝑇) ∈ 𝐵) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
421, 23, 41syl2anc2 585 . . . . . . . . . . . 12 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4342ad2antlr 725 . . . . . . . . . . 11 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4443adantr 481 . . . . . . . . . 10 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4544fveq2d 6846 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = (𝐻‘(0g𝑇)))
4645, 38eqtrd 2776 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = 0 )
4737, 39, 463eqtr4rd 2787 . . . . . . 7 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
4828, 47mpdan 685 . . . . . 6 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
4923, 23jca 512 . . . . . . . . 9 (𝑇 ∈ Ring → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
501, 49syl 17 . . . . . . . 8 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
5150ad2antlr 725 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
52 fvoveq1 7380 . . . . . . . . 9 (𝑎 = (0g𝑇) → (𝐻‘(𝑎(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)𝑐)))
53 fveq2 6842 . . . . . . . . . 10 (𝑎 = (0g𝑇) → (𝐻𝑎) = (𝐻‘(0g𝑇)))
5453oveq1d 7372 . . . . . . . . 9 (𝑎 = (0g𝑇) → ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)))
5552, 54eqeq12d 2752 . . . . . . . 8 (𝑎 = (0g𝑇) → ((𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐))))
56 oveq2 7365 . . . . . . . . . 10 (𝑐 = (0g𝑇) → ((0g𝑇)(.r𝑇)𝑐) = ((0g𝑇)(.r𝑇)(0g𝑇)))
5756fveq2d 6846 . . . . . . . . 9 (𝑐 = (0g𝑇) → (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))))
58 fveq2 6842 . . . . . . . . . 10 (𝑐 = (0g𝑇) → (𝐻𝑐) = (𝐻‘(0g𝑇)))
5958oveq2d 7373 . . . . . . . . 9 (𝑐 = (0g𝑇) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
6057, 59eqeq12d 2752 . . . . . . . 8 (𝑐 = (0g𝑇) → ((𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇)))))
6155, 602ralsng 4637 . . . . . . 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 256 . . . . 5 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
64 raleq 3309 . . . . . . 7 (𝐵 = {(0g𝑇)} → (∀𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6564raleqbi1dv 3307 . . . . . 6 (𝐵 = {(0g𝑇)} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6665adantl 482 . . . . 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 256 . . . 4 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
6816, 67mpdan 685 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
6920, 68jca 512 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
7013, 40, 32isrnghm 46180 . 2 (𝐻 ∈ (𝑇 RngHomo 𝑆) ↔ ((𝑇 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))))
715, 69, 70sylanbrc 583 1 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cdif 3907  {csn 4586  cmpt 5188  cfv 6496  (class class class)co 7357  Basecbs 17083  .rcmulr 17134  0gc0g 17321  Grpcgrp 18748   GrpHom cghm 19005  Abelcabl 19563  Ringcrg 19964  NzRingcnzr 20727  Rngcrng 46162   RngHomo crngh 46173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-minusg 18752  df-ghm 19006  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-nzr 20728  df-mgmhm 46063  df-rng 46163  df-rnghomo 46175
This theorem is referenced by:  zrinitorngc  46288
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