Theorem rhmimasubrng 20818

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 20714	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ (𝑀 GrpHom 𝑁))
2		subrngsubg 20804	. . 3 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ∈ (SubGrp‘𝑀))
3		ghmima 19451	. . 3 ⊢ ((𝐹 ∈ (𝑀 GrpHom 𝑁) ∧ 𝑋 ∈ (SubGrp‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubGrp‘𝑁))
5		eqid 2761	. . . 4 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
6		eqid 2761	. . . 4 ⊢ (mulGrp‘𝑁) = (mulGrp‘𝑁)
7	5, 6	rhmmhm 20710	. . 3 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
8		simpl 488	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)))
9		eqid 2761	. . . . . . . . 9 ⊢ (Base‘𝑀) = (Base‘𝑀)
10	5, 9	mgpbas 20365	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘(mulGrp‘𝑀))
11	10	eqcomi 2770	. . . . . . 7 ⊢ (Base‘(mulGrp‘𝑀)) = (Base‘𝑀)
12	11	subrngss 20800	. . . . . 6 ⊢ (𝑋 ∈ (SubRng‘𝑀) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
13	12	adantl 487	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑋 ⊆ (Base‘(mulGrp‘𝑀)))
14		eqidd 2762	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑀)) = (+g‘(mulGrp‘𝑀)))
15		eqidd 2762	. . . . 5 ⊢ ((𝐹 ∈ ((mulGrp‘𝑀) MndHom (mulGrp‘𝑁)) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (+g‘(mulGrp‘𝑁)) = (+g‘(mulGrp‘𝑁)))
16		eqid 2761	. . . . . . . . 9 ⊢ (.r‘𝑀) = (.r‘𝑀)
17	5, 16	mgpplusg 20364	. . . . . . . 8 ⊢ (.r‘𝑀) = (+g‘(mulGrp‘𝑀))
18	17	eqcomi 2770	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑀)) = (.r‘𝑀)
19	18	subrngmcl 20809	. . . . . 6 ⊢ ((𝑋 ∈ (SubRng‘𝑀) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘(mulGrp‘𝑀))𝑥) ∈ 𝑋)
20	19	3adant1l 1195	. . . . 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 19020	. . . 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 2761	. . . . . . . . 9 ⊢ (.r‘𝑁) = (.r‘𝑁)
23	6, 22	mgpplusg 20364	. . . . . . . 8 ⊢ (.r‘𝑁) = (+g‘(mulGrp‘𝑁))
24	23	eqcomi 2770	. . . . . . 7 ⊢ (+g‘(mulGrp‘𝑁)) = (.r‘𝑁)
25	24	oveqi 7433	. . . . . 6 ⊢ (𝑥(+g‘(mulGrp‘𝑁))𝑦) = (𝑥(.r‘𝑁)𝑦)
26	25	eleq1i 2852	. . . . 5 ⊢ ((𝑥(+g‘(mulGrp‘𝑁))𝑦) ∈ (𝐹 “ 𝑋) ↔ (𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
27	26	2ralbii 3138	. . . 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 221	. . 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 592	. 2 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(.r‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
30		rhmrcl2 20708	. . . . 5 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Ring)
31		ringrng 20514	. . . . 5 ⊢ (𝑁 ∈ Ring → 𝑁 ∈ Rng)
32	30, 31	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → 𝑁 ∈ Rng)
33	32	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → 𝑁 ∈ Rng)
34		eqid 2761	. . . 4 ⊢ (Base‘𝑁) = (Base‘𝑁)
35	34, 22	issubrng2 20810	. . 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 18	. 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 726	1 ⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899   “ cima 5654  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  ._rcmulr 17429   MndHom cmhm 18976  SubGrpcsubg 19330   GrpHom cghm 19427  mulGrpcmgp 20360  Rngcrng 20374  Ringcrg 20459   RingHom crh 20699  SubRngcsubrng 20797

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-mhm 18978  df-grp 19147  df-minusg 19148  df-subg 19333  df-ghm 19428  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-rhm 20702  df-subrng 20798

This theorem is used by: (None)