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Theorem zrrnghm 41729
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 3693 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Ring)
2 ringrng 41691 . . . . 5 (𝑇 ∈ Ring → 𝑇 ∈ Rng)
31, 2syl 17 . . . 4 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Rng)
43anim1i 589 . . 3 ((𝑇 ∈ (Ring ∖ NzRing) ∧ 𝑆 ∈ Rng) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
54ancoms 467 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝑇 ∈ Rng ∧ 𝑆 ∈ Rng))
6 rngabl 41689 . . . . . 6 (𝑆 ∈ Rng → 𝑆 ∈ Abel)
7 ablgrp 17970 . . . . . 6 (𝑆 ∈ Abel → 𝑆 ∈ Grp)
86, 7syl 17 . . . . 5 (𝑆 ∈ Rng → 𝑆 ∈ Grp)
98adantr 479 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑆 ∈ Grp)
10 ringgrp 18324 . . . . . 6 (𝑇 ∈ Ring → 𝑇 ∈ Grp)
111, 10syl 17 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝑇 ∈ Grp)
1211adantl 480 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝑇 ∈ Grp)
13 zrrhm.b . . . . . 6 𝐵 = (Base‘𝑇)
14 eqid 2609 . . . . . 6 (0g𝑇) = (0g𝑇)
1513, 140ringbas 41683 . . . . 5 (𝑇 ∈ (Ring ∖ NzRing) → 𝐵 = {(0g𝑇)})
1615adantl 480 . . . 4 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐵 = {(0g𝑇)})
17 zrrhm.0 . . . . 5 0 = (0g𝑆)
18 zrrhm.h . . . . 5 𝐻 = (𝑥𝐵0 )
1913, 17, 18, 14c0snghm 41728 . . . 4 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {(0g𝑇)}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
209, 12, 16, 19syl3anc 1317 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 GrpHom 𝑆))
2118a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 𝐻 = (𝑥𝐵0 ))
22 eqidd 2610 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ 𝑥 = (0g𝑇)) → 0 = 0 )
2313, 14ring0cl 18341 . . . . . . . . . 10 (𝑇 ∈ Ring → (0g𝑇) ∈ 𝐵)
241, 23syl 17 . . . . . . . . 9 (𝑇 ∈ (Ring ∖ NzRing) → (0g𝑇) ∈ 𝐵)
2524ad2antlr 758 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (0g𝑇) ∈ 𝐵)
26 fvex 6098 . . . . . . . . . 10 (0g𝑆) ∈ V
2717, 26eqeltri 2683 . . . . . . . . 9 0 ∈ V
2827a1i 11 . . . . . . . 8 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → 0 ∈ V)
2921, 22, 25, 28fvmptd 6182 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘(0g𝑇)) = 0 )
30 eqid 2609 . . . . . . . . . . . . . 14 (Base‘𝑆) = (Base‘𝑆)
3130, 17grpidcl 17222 . . . . . . . . . . . . 13 (𝑆 ∈ Grp → 0 ∈ (Base‘𝑆))
328, 31syl 17 . . . . . . . . . . . 12 (𝑆 ∈ Rng → 0 ∈ (Base‘𝑆))
33 eqid 2609 . . . . . . . . . . . . 13 (.r𝑆) = (.r𝑆)
3430, 33, 17rnglz 41696 . . . . . . . . . . . 12 ((𝑆 ∈ Rng ∧ 0 ∈ (Base‘𝑆)) → ( 0 (.r𝑆) 0 ) = 0 )
3532, 34mpdan 698 . . . . . . . . . . 11 (𝑆 ∈ Rng → ( 0 (.r𝑆) 0 ) = 0 )
3635adantr 479 . . . . . . . . . 10 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ( 0 (.r𝑆) 0 ) = 0 )
3736adantr 479 . . . . . . . . 9 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ( 0 (.r𝑆) 0 ) = 0 )
3837adantr 479 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ( 0 (.r𝑆) 0 ) = 0 )
39 simpr 475 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘(0g𝑇)) = 0 )
4039, 39oveq12d 6545 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))) = ( 0 (.r𝑆) 0 ))
41 eqid 2609 . . . . . . . . . . . . . . 15 (.r𝑇) = (.r𝑇)
4213, 41, 14ringlz 18359 . . . . . . . . . . . . . 14 ((𝑇 ∈ Ring ∧ (0g𝑇) ∈ 𝐵) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4323, 42mpdan 698 . . . . . . . . . . . . 13 (𝑇 ∈ Ring → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
441, 43syl 17 . . . . . . . . . . . 12 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4544ad2antlr 758 . . . . . . . . . . 11 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4645adantr 479 . . . . . . . . . 10 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → ((0g𝑇)(.r𝑇)(0g𝑇)) = (0g𝑇))
4746fveq2d 6092 . . . . . . . . 9 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = (𝐻‘(0g𝑇)))
4847, 39eqtrd 2643 . . . . . . . 8 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = 0 )
4938, 40, 483eqtr4rd 2654 . . . . . . 7 ((((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) ∧ (𝐻‘(0g𝑇)) = 0 ) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
5029, 49mpdan 698 . . . . . 6 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
5123, 23jca 552 . . . . . . . . 9 (𝑇 ∈ Ring → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
521, 51syl 17 . . . . . . . 8 (𝑇 ∈ (Ring ∖ NzRing) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
5352ad2antlr 758 . . . . . . 7 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ((0g𝑇) ∈ 𝐵 ∧ (0g𝑇) ∈ 𝐵))
54 oveq1 6534 . . . . . . . . . 10 (𝑎 = (0g𝑇) → (𝑎(.r𝑇)𝑐) = ((0g𝑇)(.r𝑇)𝑐))
5554fveq2d 6092 . . . . . . . . 9 (𝑎 = (0g𝑇) → (𝐻‘(𝑎(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)𝑐)))
56 fveq2 6088 . . . . . . . . . 10 (𝑎 = (0g𝑇) → (𝐻𝑎) = (𝐻‘(0g𝑇)))
5756oveq1d 6542 . . . . . . . . 9 (𝑎 = (0g𝑇) → ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)))
5855, 57eqeq12d 2624 . . . . . . . 8 (𝑎 = (0g𝑇) → ((𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐))))
59 oveq2 6535 . . . . . . . . . 10 (𝑐 = (0g𝑇) → ((0g𝑇)(.r𝑇)𝑐) = ((0g𝑇)(.r𝑇)(0g𝑇)))
6059fveq2d 6092 . . . . . . . . 9 (𝑐 = (0g𝑇) → (𝐻‘((0g𝑇)(.r𝑇)𝑐)) = (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))))
61 fveq2 6088 . . . . . . . . . 10 (𝑐 = (0g𝑇) → (𝐻𝑐) = (𝐻‘(0g𝑇)))
6261oveq2d 6543 . . . . . . . . 9 (𝑐 = (0g𝑇) → ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇))))
6360, 62eqeq12d 2624 . . . . . . . 8 (𝑐 = (0g𝑇) → ((𝐻‘((0g𝑇)(.r𝑇)𝑐)) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻𝑐)) ↔ (𝐻‘((0g𝑇)(.r𝑇)(0g𝑇))) = ((𝐻‘(0g𝑇))(.r𝑆)(𝐻‘(0g𝑇)))))
6458, 632ralsng 4166 . . . . . . 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 245 . . . . 5 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
67 raleq 3114 . . . . . . 7 (𝐵 = {(0g𝑇)} → (∀𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6867raleqbi1dv 3122 . . . . . 6 (𝐵 = {(0g𝑇)} → (∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)) ↔ ∀𝑎 ∈ {(0g𝑇)}∀𝑐 ∈ {(0g𝑇)} (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
6968adantl 480 . . . . 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 245 . . . 4 (((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) ∧ 𝐵 = {(0g𝑇)}) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
7116, 70mpdan 698 . . 3 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))
7220, 71jca 552 . 2 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐))))
7313, 41, 33isrnghm 41704 . 2 (𝐻 ∈ (𝑇 RngHomo 𝑆) ↔ ((𝑇 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐻 ∈ (𝑇 GrpHom 𝑆) ∧ ∀𝑎𝐵𝑐𝐵 (𝐻‘(𝑎(.r𝑇)𝑐)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑐)))))
745, 72, 73sylanbrc 694 1 ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  cdif 3536  {csn 4124  cmpt 4637  cfv 5790  (class class class)co 6527  Basecbs 15644  .rcmulr 15718  0gc0g 15872  Grpcgrp 17194   GrpHom cghm 17429  Abelcabl 17966  Ringcrg 18319  NzRingcnzr 19027  Rngcrng 41686   RngHomo crngh 41697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-card 8626  df-cda 8851  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-hash 12938  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-plusg 15730  df-0g 15874  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-mhm 17107  df-grp 17197  df-minusg 17198  df-ghm 17430  df-cmn 17967  df-abl 17968  df-mgp 18262  df-ur 18274  df-ring 18321  df-nzr 19028  df-mgmhm 41591  df-rng0 41687  df-rnghomo 41699
This theorem is referenced by:  zrinitorngc  41814
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