Theorem mhmima 18850

Description: The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.) (Proof shortened by AV, 8-Mar-2025.)

Assertion

Ref	Expression
mhmima	⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMnd‘𝑁))

Proof of Theorem mhmima

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

Step	Hyp	Ref	Expression
1		imassrn 6090	. . 3 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		eqid 2734	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2734	. . . . . 6 ⊢ (Base‘𝑁) = (Base‘𝑁)
4	2, 3	mhmf 18814	. . . . 5 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
5	4	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
6	5	frnd 6744	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
7	1, 6	sstrid 4006	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹 “ 𝑋) ⊆ (Base‘𝑁))
8		eqid 2734	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
9		eqid 2734	. . . . 5 ⊢ (0g‘𝑁) = (0g‘𝑁)
10	8, 9	mhm0 18819	. . . 4 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → (𝐹‘(0g‘𝑀)) = (0g‘𝑁))
11	10	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹‘(0g‘𝑀)) = (0g‘𝑁))
12	5	ffnd 6737	. . . 4 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝐹 Fn (Base‘𝑀))
13	2	submss 18834	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
14	13	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝑋 ⊆ (Base‘𝑀))
15	8	subm0cl 18836	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑀) → (0g‘𝑀) ∈ 𝑋)
16	15	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (0g‘𝑀) ∈ 𝑋)
17		fnfvima 7252	. . . 4 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑋) → (𝐹‘(0g‘𝑀)) ∈ (𝐹 “ 𝑋))
18	12, 14, 16, 17	syl3anc 1370	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹‘(0g‘𝑀)) ∈ (𝐹 “ 𝑋))
19	11, 18	eqeltrrd 2839	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (0g‘𝑁) ∈ (𝐹 “ 𝑋))
20		simpl 482	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
21		eqidd 2735	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (+g‘𝑀) = (+g‘𝑀))
22		eqidd 2735	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (+g‘𝑁) = (+g‘𝑁))
23		eqid 2734	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
24	23	submcl 18837	. . . 4 ⊢ ((𝑋 ∈ (SubMnd‘𝑀) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
25	24	3adant1l 1175	. . 3 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
26	20, 14, 21, 22, 25	mhmimalem 18849	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
27		mhmrcl2 18813	. . . 4 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝑁 ∈ Mnd)
28	27	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝑁 ∈ Mnd)
29		eqid 2734	. . . 4 ⊢ (+g‘𝑁) = (+g‘𝑁)
30	3, 9, 29	issubm 18828	. . 3 ⊢ (𝑁 ∈ Mnd → ((𝐹 “ 𝑋) ∈ (SubMnd‘𝑁) ↔ ((𝐹 “ 𝑋) ⊆ (Base‘𝑁) ∧ (0g‘𝑁) ∈ (𝐹 “ 𝑋) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
31	28, 30	syl 17	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ((𝐹 “ 𝑋) ∈ (SubMnd‘𝑁) ↔ ((𝐹 “ 𝑋) ⊆ (Base‘𝑁) ∧ (0g‘𝑁) ∈ (𝐹 “ 𝑋) ∧ ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))))
32	7, 19, 26, 31	mpbir3and 1341	1 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMnd‘𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058   ⊆ wss 3962  ran crn 5689   “ cima 5691   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  +_gcplusg 17297  0_gc0g 17485  Mndcmnd 18759   MndHom cmhm 18806  SubMndcsubmnd 18807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809

This theorem is referenced by:  rhmima  20620