Theorem isrnghmmul 20370

Description: A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
isrnghmmul.m	⊢ 𝑀 = (mulGrp‘𝑅)
isrnghmmul.n	⊢ 𝑁 = (mulGrp‘𝑆)

Assertion

Ref	Expression
isrnghmmul	⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))

Proof of Theorem isrnghmmul

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2733	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2733	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	isrnghm 20369	. 2 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))))
5		isrnghmmul.m	. . . . . . . . . . 11 ⊢ 𝑀 = (mulGrp‘𝑅)
6	5	rngmgp 20084	. . . . . . . . . 10 ⊢ (𝑅 ∈ Rng → 𝑀 ∈ Smgrp)
7		sgrpmgm 18642	. . . . . . . . . 10 ⊢ (𝑀 ∈ Smgrp → 𝑀 ∈ Mgm)
8	6, 7	syl 17	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑀 ∈ Mgm)
9		isrnghmmul.n	. . . . . . . . . . 11 ⊢ 𝑁 = (mulGrp‘𝑆)
10	9	rngmgp 20084	. . . . . . . . . 10 ⊢ (𝑆 ∈ Rng → 𝑁 ∈ Smgrp)
11		sgrpmgm 18642	. . . . . . . . . 10 ⊢ (𝑁 ∈ Smgrp → 𝑁 ∈ Mgm)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑆 ∈ Rng → 𝑁 ∈ Mgm)
13	8, 12	anim12i 613	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm))
14		eqid 2733	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
15	1, 14	ghmf 19142	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝐹:(Base‘𝑅)⟶(Base‘𝑆))
16	13, 15	anim12i 613	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)))
17	16	biantrurd 532	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)) ↔ (((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))))
18		anass 468	. . . . . 6 ⊢ ((((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ 𝐹:(Base‘𝑅)⟶(Base‘𝑆)) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))))
19	17, 18	bitrdi 287	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))))))
20	5, 1	mgpbas 20073	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑀)
21	9, 14	mgpbas 20073	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑁)
22	5, 2	mgpplusg 20072	. . . . . 6 ⊢ (.r‘𝑅) = (+g‘𝑀)
23	9, 3	mgpplusg 20072	. . . . . 6 ⊢ (.r‘𝑆) = (+g‘𝑁)
24	20, 21, 22, 23	ismgmhm 18614	. . . . 5 ⊢ (𝐹 ∈ (𝑀 MgmHom 𝑁) ↔ ((𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm) ∧ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))))
25	19, 24	bitr4di 289	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ 𝐹 ∈ (𝑅 GrpHom 𝑆)) → (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)) ↔ 𝐹 ∈ (𝑀 MgmHom 𝑁)))
26	25	pm5.32da 579	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦))) ↔ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))
27	26	pm5.32i 574	. 2 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)(𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥)(.r‘𝑆)(𝐹‘𝑦)))) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))
28	4, 27	bitri 275	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  Basecbs 17130  ._rcmulr 17172  Mgmcmgm 18556   MgmHom cmgmhm 18608  Smgrpcsgrp 18636   GrpHom cghm 19134  mulGrpcmgp 20068  Rngcrng 20080   RngHom crnghm 20362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-er 8631  df-map 8761  df-en 8879  df-dom 8880  df-sdom 8881  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-nn 12136  df-2 12198  df-sets 17085  df-slot 17103  df-ndx 17115  df-base 17131  df-plusg 17184  df-mgmhm 18610  df-sgrp 18637  df-ghm 19135  df-abl 19705  df-mgp 20069  df-rng 20081  df-rnghm 20364

This theorem is referenced by:  rnghmmgmhm  20371  rnghmval2  20372  rnghmf1o  20380  rnghmco  20385  idrnghm  20386  rhmisrnghm  20408  c0rnghm  20460