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Theorem rhmimasubrng 20502
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 20422 . . 3 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2 subrngsubg 20488 . . 3 (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3 ghmima 19190 . . 3 ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑋 ∈ (SubGrp‘𝑀)) → (𝐹𝑋) ∈ (SubGrp‘𝑁))
41, 2, 3syl2an 594 . 2 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹𝑋) ∈ (SubGrp‘𝑁))
5 eqid 2725 . . . 4 (mulGrp‘𝑀) = (mulGrp‘𝑀)
6 eqid 2725 . . . 4 (mulGrp‘𝑁) = (mulGrp‘𝑁)
75, 6rhmmhm 20417 . . 3 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
8 simpl 481 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
9 eqid 2725 . . . . . . . . 9 (Base‘𝑀) = (Base‘𝑀)
105, 9mgpbas 20079 . . . . . . . 8 (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
1110eqcomi 2734 . . . . . . 7 (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
1211subrngss 20484 . . . . . 6 (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
1312adantl 480 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
14 eqidd 2726 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15 eqidd 2726 . . . . 5 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16 eqid 2725 . . . . . . . . 9 (.r𝑀) = (.r𝑀)
175, 16mgpplusg 20077 . . . . . . . 8 (.r𝑀) = (+g‘(mulGrp‘𝑀))
1817eqcomi 2734 . . . . . . 7 (+g‘(mulGrp‘𝑀)) = (.r𝑀)
1918subrngmcl 20493 . . . . . 6 ((𝑋 ∈ (SubRng‘𝑀) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
20193adant1l 1173 . . . . 5 (((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
218, 13, 14, 15, 20mhmimalem 18775 . . . 4 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋))
22 eqid 2725 . . . . . . . . 9 (.r𝑁) = (.r𝑁)
236, 22mgpplusg 20077 . . . . . . . 8 (.r𝑁) = (+g‘(mulGrp‘𝑁))
2423eqcomi 2734 . . . . . . 7 (+g‘(mulGrp‘𝑁)) = (.r𝑁)
2524oveqi 7426 . . . . . 6 (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r𝑁)𝑦)
2625eleq1i 2816 . . . . 5 ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋) ↔ (𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
27262ralbii 3118 . . . 4 (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
2821, 27sylib 217 . . 3 ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
297, 28sylan 578 . 2 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(.r𝑁)𝑦) ∈ (𝐹𝑋))
30 rhmrcl2 20415 . . . . 5 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31 ringrng 20220 . . . . 5 (𝑁 ∈ Ring → 𝑁 ∈ Rng)
3230, 31syl 17 . . . 4 (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Rng)
3332adantr 479 . . 3 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑁 ∈ Rng)
34 eqid 2725 . . . 4 (Base‘𝑁) = (Base‘𝑁)
3534, 22issubrng2 20494 . . 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 711 1 ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹𝑋) ∈ (SubRng‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wral 3051  wss 3941  cima 5676  cfv 6543  (class class class)co 7413  Basecbs 17174  +gcplusg 17227  .rcmulr 17228   MndHom cmhm 18732  SubGrpcsubg 19074   GrpHom cghm 19166  mulGrpcmgp 20073  Rngcrng 20091  Ringcrg 20172   RingHom crh 20407  SubRngcsubrng 20481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-0g 17417  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-mhm 18734  df-grp 18892  df-minusg 18893  df-subg 19077  df-ghm 19167  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-ring 20174  df-rhm 20410  df-subrng 20482
This theorem is referenced by: (None)
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