Theorem mgmhmima 18897

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 6196	. . 3 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		eqid 2761	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2761	. . . . . 6 ⊢ (Base‘𝑁) = (Base‘𝑁)
4	2, 3	mgmhmf 18879	. . . . 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 6716	. . 3 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
7	1, 6	sstrid 3942	. 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 18887	. . . . . . . . . . . 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 3932	. . . . . . . . 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 3932	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑀))
16		eqid 2761	. . . . . . . . . 10 ⊢ (+g‘𝑀) = (+g‘𝑀)
17		eqid 2761	. . . . . . . . . 10 ⊢ (+g‘𝑁) = (+g‘𝑁)
18	2, 16, 17	mgmhmlin 18881	. . . . . . . . 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 6708	. . . . . . . . . 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 18889	. . . . . . . . . . 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 7237	. . . . . . . . 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 2862	. . . . . . 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 3155	. . . . 5 ⊢ (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋))
30		oveq2 7426	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘𝑧)(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)))
31	30	eleq1d 2846	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)(𝐹‘𝑥)) ∈ (𝐹 “ 𝑋)))
32	31	ralima 7241	. . . . . . 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 3155	. . 3 ⊢ ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ∀𝑧 ∈ 𝑋 ∀𝑦 ∈ (𝐹 “ 𝑋)((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
37		oveq1 7425	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑧) → (𝑥(+g‘𝑁)𝑦) = ((𝐹‘𝑧)(+g‘𝑁)𝑦))
38	37	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑧) → ((𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋) ↔ ((𝐹‘𝑧)(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋)))
39	38	ralbidv 3186	. . . . 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 7241	. . . 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 18876	. . . . 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 18884	. . 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 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  ran crn 5652   “ cima 5654   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  +_gcplusg 17421  Mgmcmgm 18807   MgmHom cmgmhm 18872  SubMgmcsubmgm 18873

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 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

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-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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mgm 18809  df-mgmhm 18874  df-submgm 18875

This theorem is used by: (None)