Theorem mgmhmima 18911

Description: The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)

Assertion

Ref	Expression
mgmhmima	⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMgm‘𝑁))

Proof of Theorem mgmhmima

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

Step	Hyp	Ref	Expression
1		imassrn 6199	. . 3 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		eqid 2760	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2760	. . . . . 6 ⊢ (Base‘𝑁) = (Base‘𝑁)
4	2, 3	mgmhmf 18893	. . . . 5 ⊢ (𝐹 ∈ (𝑀 MgmHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
5	4	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
6	5	frnd 6721	. . 3 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
7	1, 6	sstrid 3920	. 2 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹 “ 𝑋) ⊆ (Base‘𝑁))
8		simpll 779	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 ∈ (𝑀 MgmHom 𝑁))
9	2	submgmss 18901	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ (SubMgm‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
10	9	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝑋 ⊆ (Base‘𝑀))
11	10	adantr 486	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑀))
12		simprl 783	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
13	11, 12	sseldd 3910	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑧 ∈ (Base‘𝑀))
14		simprr 785	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
15	11, 14	sseldd 3910	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
16		eqid 2760	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
17		eqid 2760	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘𝑁)
18	2, 16, 17	mgmhmlin 18895	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
19	8, 13, 15, 18	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
20	5	ffnd 6713	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝐹 Fn (Base‘𝑀))
21	20	adantr 486	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝐹 Fn (Base‘𝑀))
22	16	submgmcl 18903	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ (SubMgm‘𝑀) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
23	22	3expb 1138	. . . . . . . . . 10 ⊢ ((𝑋 ∈ (SubMgm‘𝑀) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
24	23	adantll 727	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
25		fnfvima 7241	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧(+g‘𝑀)𝑥) ∈ 𝑋) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) ∈ (𝐹 “ 𝑋))
26	21, 11, 24, 25	syl3anc 1398	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → (𝐹‘(𝑧(+g‘𝑀)𝑥)) ∈ (𝐹 “ 𝑋))
27	19, 26	eqeltrrd 2861	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
28	27	anassrs 473	. . . . . 6 ⊢ ((((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
29	28	ralrimiva 3154	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
30		oveq2 7430	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
31	30	eleq1d 2845	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
32	31	ralima 7245	. . . . . . 7 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
33	20, 10, 32	syl2anc 596	. . . . . 6 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
34	33	adantr 486	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧 ∈ 𝑋) → (∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
35	29, 34	mpbird 260	. . . 4 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧 ∈ 𝑋) → ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
36	35	ralrimiva 3154	. . 3 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
37		oveq1 7429	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)𝑦))
38	37	eleq1d 2845	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
39	38	ralbidv 3185	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑧) → (∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
40	39	ralima 7245	. . . 4 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
41	20, 10, 40	syl2anc 596	. . 3 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
42	36, 41	mpbird 260	. 2 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
43		mgmhmrcl 18890	. . . . 5 ⊢ (𝐹 ∈ (𝑀 MgmHom 𝑁) → (𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm))
44	43	simprd 501	. . . 4 ⊢ (𝐹 ∈ (𝑀 MgmHom 𝑁) → 𝑁 ∈ Mgm)
45	44	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝑁 ∈ Mgm)
46	3, 17	issubmgm 18898	. . 3 ⊢ (𝑁 ∈ Mgm → ((𝐹 “ 𝑋) ∈ (SubMgm‘𝑁) ↔ ((𝐹 “ 𝑋) ⊆ (Base‘𝑁) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
47	45, 46	syl 18	. 2 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ((𝐹 “ 𝑋) ∈ (SubMgm‘𝑁) ↔ ((𝐹 “ 𝑋) ⊆ (Base‘𝑁) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
48	7, 42, 47	mpbir2and 726	1 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMgm‘𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3896  ran crn 5651   “ cima 5653   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  +_gcplusg 17435  Mgmcmgm 18821   MgmHom cmgmhm 18886  SubMgmcsubmgm 18887

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-er 8722  df-map 8854  df-en 8979  df-dom 8980  df-sdom 8981  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-sets 17349  df-slot 17367  df-ndx 17379  df-base 17395  df-ress 17416  df-plusg 17448  df-mgm 18823  df-mgmhm 18888  df-submgm 18889

This theorem is used by: (None)