MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rhmimasubrng Structured version   Visualization version   GIF version

Theorem rhmimasubrng 20566
Description: The homomorphic image of a subring is a subring. (Contributed by AV, 16-Feb-2025.)
Assertion
Ref Expression
rhmimasubrng ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹𝑋) ∈ (SubRng‘𝑁))

Proof of Theorem rhmimasubrng
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rhmghm 20484 . . 3 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 subrngsubg 20552 . . 3 (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3 ghmima 19255 . . 3 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑋 ∈ (SubGrp‘𝑀)) → (𝐹𝑋) ∈ (SubGrp‘𝑁))
41, 2, 3syl2an 596 . 2 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹𝑋) ∈ (SubGrp‘𝑁))
5 eqid 2737 . . . 4 (mulGrp‘𝑀) = (mulGrp‘𝑀)
6 eqid 2737 . . . 4 (mulGrp‘𝑁) = (mulGrp‘𝑁)
75, 6rhmmhm 20479 . . 3 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
8 simpl 482 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
9 eqid 2737 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
105, 9mgpbas 20142 . . . . . . . 8 (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
1110eqcomi 2746 . . . . . . 7 (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
1211subrngss 20548 . . . . . 6 (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
1312adantl 481 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
14 eqidd 2738 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15 eqidd 2738 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16 eqid 2737 . . . . . . . . 9 (.r𝑀) = (.r𝑀)
175, 16mgpplusg 20141 . . . . . . . 8 (.r𝑀) = (+g‘(mulGrp‘𝑀))
1817eqcomi 2746 . . . . . . 7 (+g‘(mulGrp‘𝑀)) = (.r𝑀)
1918subrngmcl 20557 . . . . . 6 ((𝑋 ∈ (SubRng‘𝑀) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
20193adant1l 1177 . . . . 5 (((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
218, 13, 14, 15, 20mhmimalem 18837 . . . 4 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋))
22 eqid 2737 . . . . . . . . 9 (.r𝑁) = (.r𝑁)
236, 22mgpplusg 20141 . . . . . . . 8 (.r𝑁) = (+g‘(mulGrp‘𝑁))
2423eqcomi 2746 . . . . . . 7 (+g‘(mulGrp‘𝑁)) = (.r𝑁)
2524oveqi 7444 . . . . . 6 (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r𝑁)𝑦)
2625eleq1i 2832 . . . . 5 ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋) ↔ (𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
27262ralbii 3128 . . . 4 (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
2821, 27sylib 218 . . 3 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
297, 28sylan 580 . 2 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
30 rhmrcl2 20477 . . . . 5 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31 ringrng 20282 . . . . 5 (𝑁 ∈ Ring → 𝑁 ∈ Rng)
3230, 31syl 17 . . . 4 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Rng)
3332adantr 480 . . 3 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑁 ∈ Rng)
34 eqid 2737 . . . 4 (Base‘𝑁) = (Base‘𝑁)
3534, 22issubrng2 20558 . . 3 (𝑁 ∈ Rng → ((𝐹𝑋) ∈ (SubRng‘𝑁) ↔ ((𝐹𝑋) ∈ (SubGrp‘𝑁) ∧ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))))
3633, 35syl 17 . 2 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ((𝐹𝑋) ∈ (SubRng‘𝑁) ↔ ((𝐹𝑋) ∈ (SubGrp‘𝑁) ∧ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))))
374, 29, 36mpbir2and 713 1 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹𝑋) ∈ (SubRng‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wral 3061  wss 3951  cima 5688  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298   MndHom cmhm 18794  SubGrpcsubg 19138   GrpHom cghm 19230  mulGrpcmgp 20137  Rngcrng 20149  Ringcrg 20230   RingHom crh 20469  SubRngcsubrng 20545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-minusg 18955  df-subg 19141  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-rhm 20472  df-subrng 20546
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator