Theorem isrnghmmul 46181

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 RngHomo 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 2736	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2736	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
3		eqid 2736	. . 3 ⊢ (.r‘𝑆) = (.r‘𝑆)
4	1, 2, 3	isrnghm 46180	. 2 ⊢ (𝐹 ∈ (𝑅 RngHomo 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 46166	. . . . . . . . . 10 ⊢ (𝑅 ∈ Rng → 𝑀 ∈ Smgrp)
7		sgrpmgm 18551	. . . . . . . . . 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 46166	. . . . . . . . . 10 ⊢ (𝑆 ∈ Rng → 𝑁 ∈ Smgrp)
11		sgrpmgm 18551	. . . . . . . . . 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 2736	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
15	1, 14	ghmf 19012	. . . . . . . 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 533	. . . . . 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 469	. . . . . 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 286	. . . . 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 19902	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑀)
21	9, 14	mgpbas 19902	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑁)
22	5, 2	mgpplusg 19900	. . . . . 6 ⊢ (.r‘𝑅) = (+g‘𝑀)
23	9, 3	mgpplusg 19900	. . . . . 6 ⊢ (.r‘𝑆) = (+g‘𝑁)
24	20, 21, 22, 23	ismgmhm 46067	. . . . 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 288	. . . 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 575	. 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 274	1 ⊢ (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  ._rcmulr 17134  Mgmcmgm 18495  Smgrpcsgrp 18545   GrpHom cghm 19005  mulGrpcmgp 19896   MgmHom cmgmhm 46061  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-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-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-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  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-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-sgrp 18546  df-ghm 19006  df-abl 19565  df-mgp 19897  df-mgmhm 46063  df-rng 46163  df-rnghomo 46175

This theorem is referenced by:  rnghmmgmhm  46182  rnghmval2  46183  rnghmf1o  46191  rnghmco  46195  idrnghm  46196  c0rnghm  46201  rhmisrnghm  46208