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Theorem mgmhmima 44348
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)) → (𝐹𝑋) ∈ (SubMgm‘𝑁))

Proof of Theorem mgmhmima
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 5927 . . 3 (𝐹𝑋) ⊆ ran 𝐹
2 eqid 2824 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2824 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
42, 3mgmhmf 44330 . . . . 5 (𝐹 ∈ (𝑀 MgmHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
54adantr 484 . . . 4 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
65frnd 6510 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
71, 6sstrid 3964 . 2 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹𝑋) ⊆ (Base‘𝑁))
8 simpll 766 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 ∈ (𝑀 MgmHom 𝑁))
92submgmss 44338 . . . . . . . . . . . 12 (𝑋 ∈ (SubMgm‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
109adantl 485 . . . . . . . . . . 11 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝑋 ⊆ (Base‘𝑀))
1110adantr 484 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑋 ⊆ (Base‘𝑀))
12 simprl 770 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧𝑋)
1311, 12sseldd 3954 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧 ∈ (Base‘𝑀))
14 simprr 772 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥𝑋)
1511, 14sseldd 3954 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥 ∈ (Base‘𝑀))
16 eqid 2824 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
17 eqid 2824 . . . . . . . . . 10 (+g𝑁) = (+g𝑁)
182, 16, 17mgmhmlin 44332 . . . . . . . . 9 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
198, 13, 15, 18syl3anc 1368 . . . . . . . 8 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
205ffnd 6504 . . . . . . . . . 10 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝐹 Fn (Base‘𝑀))
2120adantr 484 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 Fn (Base‘𝑀))
2216submgmcl 44340 . . . . . . . . . . 11 ((𝑋 ∈ (SubMgm‘𝑀) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
23223expb 1117 . . . . . . . . . 10 ((𝑋 ∈ (SubMgm‘𝑀) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
2423adantll 713 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
25 fnfvima 6987 . . . . . . . . 9 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧(+g𝑀)𝑥) ∈ 𝑋) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
2621, 11, 24, 25syl3anc 1368 . . . . . . . 8 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
2719, 26eqeltrrd 2917 . . . . . . 7 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
2827anassrs 471 . . . . . 6 ((((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) ∧ 𝑥𝑋) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
2928ralrimiva 3177 . . . . 5 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) → ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
30 oveq2 7157 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
3130eleq1d 2900 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3231ralima 6992 . . . . . . 7 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3320, 10, 32syl2anc 587 . . . . . 6 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3433adantr 484 . . . . 5 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3529, 34mpbird 260 . . . 4 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) → ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
3635ralrimiva 3177 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
37 oveq1 7156 . . . . . . 7 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)𝑦))
3837eleq1d 2900 . . . . . 6 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
3938ralbidv 3192 . . . . 5 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4039ralima 6992 . . . 4 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4120, 10, 40syl2anc 587 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4236, 41mpbird 260 . 2 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))
43 mgmhmrcl 44327 . . . . 5 (𝐹 ∈ (𝑀 MgmHom 𝑁) → (𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm))
4443simprd 499 . . . 4 (𝐹 ∈ (𝑀 MgmHom 𝑁) → 𝑁 ∈ Mgm)
4544adantr 484 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝑁 ∈ Mgm)
463, 17issubmgm 44335 . . 3 (𝑁 ∈ Mgm → ((𝐹𝑋) ∈ (SubMgm‘𝑁) ↔ ((𝐹𝑋) ⊆ (Base‘𝑁) ∧ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))))
4745, 46syl 17 . 2 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ((𝐹𝑋) ∈ (SubMgm‘𝑁) ↔ ((𝐹𝑋) ⊆ (Base‘𝑁) ∧ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))))
487, 42, 47mpbir2and 712 1 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹𝑋) ∈ (SubMgm‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919  ran crn 5543  cima 5545   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  Mgmcmgm 17850   MgmHom cmgmhm 44323  SubMgmcsubmgm 44324
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mgm 17852  df-mgmhm 44325  df-submgm 44326
This theorem is referenced by: (None)
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