Theorem rhmimasubrng 20612

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 20528	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		subrngsubg 20598	. . 3 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3		ghmima 19277	. . 3 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑋 ∈ (SubGrp‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
4	1, 2, 3	syl2an 605	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
5		eqid 2762	. . . 4 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
6		eqid 2762	. . . 4 ⊢ (mulGrp‘𝑁) = (mulGrp‘𝑁)
7	5, 6	rhmmhm 20524	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
8		simpl 486	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
9		eqid 2762	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
10	5, 9	mgpbas 20191	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
11	10	eqcomi 2771	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
12	11	subrngss 20594	. . . . . 6 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
13	12	adantl 485	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
14		eqidd 2763	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15		eqidd 2763	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16		eqid 2762	. . . . . . . . 9 ⊢ (.r‘𝑀) = (.r‘𝑀)
17	5, 16	mgpplusg 20190	. . . . . . . 8 ⊢ (.r‘𝑀) = (+g‘(mulGrp‘𝑀))
18	17	eqcomi 2771	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑀)) = (.r‘𝑀)
19	18	subrngmcl 20603	. . . . . 6 ⊢ ((𝑋 ∈ (SubRng‘𝑀) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
20	19	3adant1l 1190	. . . . 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 18858	. . . 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 2762	. . . . . . . . 9 ⊢ (.r‘𝑁) = (.r‘𝑁)
23	6, 22	mgpplusg 20190	. . . . . . . 8 ⊢ (.r‘𝑁) = (+g‘(mulGrp‘𝑁))
24	23	eqcomi 2771	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑁)) = (.r‘𝑁)
25	24	oveqi 7409	. . . . . 6 ⊢ (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r‘𝑁)𝑦)
26	25	eleq1i 2853	. . . . 5 ⊢ ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋) ↔ (𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
27	26	2ralbii 3137	. . . 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 220	. . 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 589	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
30		rhmrcl2 20522	. . . . 5 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31		ringrng 20331	. . . . 5 ⊢ (𝑁 ∈ Ring → 𝑁 ∈ Rng)
32	30, 31	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Rng)
33	32	adantr 484	. . 3 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑁 ∈ Rng)
34		eqid 2762	. . . 4 ⊢ (Base‘𝑁) = (Base‘𝑁)
35	34, 22	issubrng2 20604	. . 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 723	1 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  ∀wral 3076   ⊆ wss 3904   “ cima 5650  ‘cfv 6521  (class class class)co 7396  Basecbs 17245  +_gcplusg 17286  ._rcmulr 17287   MndHom cmhm 18815  SubGrpcsubg 19162   GrpHom cghm 19253  mulGrpcmgp 20186  Rngcrng 20198  Ringcrg 20279   RingHom crh 20514  SubRngcsubrng 20591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-0g 17470  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-mhm 18817  df-grp 18978  df-minusg 18979  df-subg 19165  df-ghm 19254  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20228  df-ring 20281  df-rhm 20517  df-subrng 20592

This theorem is referenced by: (None)