Theorem mhmima 18784

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 6030	. . 3 ⊢ (𝐹 “ 𝑋) ⊆ ran 𝐹
2		eqid 2737	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2737	. . . . . 6 ⊢ (Base‘𝑁) = (Base‘𝑁)
4	2, 3	mhmf 18748	. . . . 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 6670	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
7	1, 6	sstrid 3934	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹 “ 𝑋) ⊆ (Base‘𝑁))
8		eqid 2737	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
9		eqid 2737	. . . . 5 ⊢ (0g‘𝑁) = (0g‘𝑁)
10	8, 9	mhm0 18753	. . . 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 6663	. . . 4 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝐹 Fn (Base‘𝑀))
13	2	submss 18768	. . . . 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 18770	. . . . 5 ⊢ (𝑋 ∈ (SubMnd‘𝑀) → (0g‘𝑀) ∈ 𝑋)
16	15	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (0g‘𝑀) ∈ 𝑋)
17		fnfvima 7181	. . . 4 ⊢ ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑋) → (𝐹‘(0g‘𝑀)) ∈ (𝐹 “ 𝑋))
18	12, 14, 16, 17	syl3anc 1374	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹‘(0g‘𝑀)) ∈ (𝐹 “ 𝑋))
19	11, 18	eqeltrrd 2838	. 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 2738	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (+g‘𝑀) = (+g‘𝑀))
22		eqidd 2738	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (+g‘𝑁) = (+g‘𝑁))
23		eqid 2737	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
24	23	submcl 18771	. . . 4 ⊢ ((𝑋 ∈ (SubMnd‘𝑀) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
25	24	3adant1l 1178	. . 3 ⊢ (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ 𝑧 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑧(+g‘𝑀)𝑥) ∈ 𝑋)
26	20, 14, 21, 22, 25	mhmimalem 18783	. 2 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
27		mhmrcl2 18747	. . . 4 ⊢ (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝑁 ∈ Mnd)
28	27	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝑁 ∈ Mnd)
29		eqid 2737	. . . 4 ⊢ (+g‘𝑁) = (+g‘𝑁)
30	3, 9, 29	issubm 18762	. . 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 1344	1 ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubMnd‘𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ran crn 5625   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  +_gcplusg 17211  0_gc0g 17393  Mndcmnd 18693   MndHom cmhm 18740  SubMndcsubmnd 18741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743

This theorem is referenced by:  rhmima  20572