Theorem rhmima 14419

Description: The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)

Assertion

Ref	Expression
rhmima	⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRing‘𝑁))

Proof of Theorem rhmima

Step	Hyp	Ref	Expression
1		rhmghm 14329	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		subrgsubg 14395	. . 3 ⊢ (𝑋 ∈ (SubRing‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3		ghmima 14003	. . 3 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑋 ∈ (SubGrp‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
4	1, 2, 3	syl2an 289	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
5		eqid 2234	. . . 4 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
6		eqid 2234	. . . 4 ⊢ (mulGrp‘𝑁) = (mulGrp‘𝑁)
7	5, 6	rhmmhm 14326	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
8	5	subrgsubm 14402	. . 3 ⊢ (𝑋 ∈ (SubRing‘𝑀) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑀)))
9		mhmima 13725	. . 3 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubMnd‘(mulGrp‘𝑀))) → (𝐹 “ 𝑋) ∈ (SubMnd‘(mulGrp‘𝑁)))
10	7, 8, 9	syl2an 289	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMnd‘(mulGrp‘𝑁)))
11		rhmrcl2 14323	. . . 4 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
12	11	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → 𝑁 ∈ Ring)
13	6	issubrg3 14415	. . 3 ⊢ (𝑁 ∈ Ring → ((𝐹 “ 𝑋) ∈ (SubRing‘𝑁) ↔ ((𝐹 “ 𝑋) ∈ (SubGrp‘𝑁) ∧ (𝐹 “ 𝑋) ∈ (SubMnd‘(mulGrp‘𝑁)))))
14	12, 13	syl 14	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → ((𝐹 “ 𝑋) ∈ (SubRing‘𝑁) ↔ ((𝐹 “ 𝑋) ∈ (SubGrp‘𝑁) ∧ (𝐹 “ 𝑋) ∈ (SubMnd‘(mulGrp‘𝑁)))))
15	4, 10, 14	mpbir2and 953	1 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRing‘𝑁))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2205   “ cima 4754  ‘cfv 5354  (class class class)co 6052   MndHom cmhm 13691  SubMndcsubmnd 13692  SubGrpcsubg 13905   GrpHom cghm 13978  mulGrpcmgp 14085  Ringcrg 14161   RingHom crh 14317  SubRingcsubrg 14385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-lttrn 8246  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-iress 13241  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-mhm 13693  df-submnd 13694  df-grp 13737  df-minusg 13738  df-subg 13908  df-ghm 13979  df-mgp 14086  df-ur 14125  df-ring 14163  df-rhm 14319  df-subrg 14387

This theorem is referenced by:  rnrhmsubrg  14420