Theorem rhmimasubrng 20538

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)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))

Proof of Theorem rhmimasubrng

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmghm 20454	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		subrngsubg 20524	. . 3 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3		ghmima 19203	. . 3 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑋 ∈ (SubGrp‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
4	1, 2, 3	syl2an 602	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
5		eqid 2739	. . . 4 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
6		eqid 2739	. . . 4 ⊢ (mulGrp‘𝑁) = (mulGrp‘𝑁)
7	5, 6	rhmmhm 20450	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
8		simpl 483	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
9		eqid 2739	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
10	5, 9	mgpbas 20117	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
11	10	eqcomi 2748	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
12	11	subrngss 20520	. . . . . 6 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
13	12	adantl 482	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
14		eqidd 2740	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15		eqidd 2740	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16		eqid 2739	. . . . . . . . 9 ⊢ (.r‘𝑀) = (.r‘𝑀)
17	5, 16	mgpplusg 20116	. . . . . . . 8 ⊢ (.r‘𝑀) = (+g‘(mulGrp‘𝑀))
18	17	eqcomi 2748	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑀)) = (.r‘𝑀)
19	18	subrngmcl 20529	. . . . . 6 ⊢ ((𝑋 ∈ (SubRng‘𝑀) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
20	19	3adant1l 1183	. . . . 5 ⊢ (((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
21	8, 13, 14, 15, 20	mhmimalem 18783	. . . 4 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋))
22		eqid 2739	. . . . . . . . 9 ⊢ (.r‘𝑁) = (.r‘𝑁)
23	6, 22	mgpplusg 20116	. . . . . . . 8 ⊢ (.r‘𝑁) = (+g‘(mulGrp‘𝑁))
24	23	eqcomi 2748	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑁)) = (.r‘𝑁)
25	24	oveqi 7369	. . . . . 6 ⊢ (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r‘𝑁)𝑦)
26	25	eleq1i 2830	. . . . 5 ⊢ ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋) ↔ (𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
27	26	2ralbii 3114	. . . 4 ⊢ (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
28	21, 27	sylib 219	. . 3 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
29	7, 28	sylan 586	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
30		rhmrcl2 20448	. . . . 5 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31		ringrng 20257	. . . . 5 ⊢ (𝑁 ∈ Ring → 𝑁 ∈ Rng)
32	30, 31	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Rng)
33	32	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑁 ∈ Rng)
34		eqid 2739	. . . 4 ⊢ (Base‘𝑁) = (Base‘𝑁)
35	34, 22	issubrng2 20530	. . 3 ⊢ (𝑁 ∈ Rng → ((𝐹 “ 𝑋) ∈ (SubRng‘𝑁) ↔ ((𝐹 “ 𝑋) ∈ (SubGrp‘𝑁) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
36	33, 35	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ((𝐹 “ 𝑋) ∈ (SubRng‘𝑁) ↔ ((𝐹 “ 𝑋) ∈ (SubGrp‘𝑁) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
37	4, 29, 36	mpbir2and 719	1 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883   “ cima 5621  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  +_gcplusg 17211  ._rcmulr 17212   MndHom cmhm 18740  SubGrpcsubg 19087   GrpHom cghm 19178  mulGrpcmgp 20112  Rngcrng 20124  Ringcrg 20205   RingHom crh 20440  SubRngcsubrng 20517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-grp 18903  df-minusg 18904  df-subg 19090  df-ghm 19179  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-rhm 20443  df-subrng 20518

This theorem is referenced by: (None)