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

Theorem rhmimasubrng 20583
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 20501 . . 3 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 subrngsubg 20569 . . 3 (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3 ghmima 19268 . . 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 2735 . . . 4 (mulGrp‘𝑀) = (mulGrp‘𝑀)
6 eqid 2735 . . . 4 (mulGrp‘𝑁) = (mulGrp‘𝑁)
75, 6rhmmhm 20496 . . 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 2735 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
105, 9mgpbas 20158 . . . . . . . 8 (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
1110eqcomi 2744 . . . . . . 7 (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
1211subrngss 20565 . . . . . 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 2736 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15 eqidd 2736 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16 eqid 2735 . . . . . . . . 9 (.r𝑀) = (.r𝑀)
175, 16mgpplusg 20156 . . . . . . . 8 (.r𝑀) = (+g‘(mulGrp‘𝑀))
1817eqcomi 2744 . . . . . . 7 (+g‘(mulGrp‘𝑀)) = (.r𝑀)
1918subrngmcl 20574 . . . . . 6 ((𝑋 ∈ (SubRng‘𝑀) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
20193adant1l 1175 . . . . 5 (((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
218, 13, 14, 15, 20mhmimalem 18850 . . . 4 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋))
22 eqid 2735 . . . . . . . . 9 (.r𝑁) = (.r𝑁)
236, 22mgpplusg 20156 . . . . . . . 8 (.r𝑁) = (+g‘(mulGrp‘𝑁))
2423eqcomi 2744 . . . . . . 7 (+g‘(mulGrp‘𝑁)) = (.r𝑁)
2524oveqi 7444 . . . . . 6 (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r𝑁)𝑦)
2625eleq1i 2830 . . . . 5 ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋) ↔ (𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
27262ralbii 3126 . . . 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 20494 . . . . 5 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31 ringrng 20299 . . . . 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 2735 . . . 4 (Base‘𝑁) = (Base‘𝑁)
3534, 22issubrng2 20575 . . 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 2106  wral 3059  wss 3963  cima 5692  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299   MndHom cmhm 18807  SubGrpcsubg 19151   GrpHom cghm 19243  mulGrpcmgp 20152  Rngcrng 20170  Ringcrg 20251   RingHom crh 20486  SubRngcsubrng 20562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-subg 19154  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-rhm 20489  df-subrng 20563
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator