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Theorem mgmhmima 18684
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 6079 . . 3 (𝐹𝑋) ⊆ ran 𝐹
2 eqid 2728 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2728 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
42, 3mgmhmf 18666 . . . . 5 (𝐹 ∈ (𝑀 MgmHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
54adantr 479 . . . 4 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
65frnd 6735 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
71, 6sstrid 3993 . 2 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹𝑋) ⊆ (Base‘𝑁))
8 simpll 765 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 ∈ (𝑀 MgmHom 𝑁))
92submgmss 18674 . . . . . . . . . . . 12 (𝑋 ∈ (SubMgm‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
109adantl 480 . . . . . . . . . . 11 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝑋 ⊆ (Base‘𝑀))
1110adantr 479 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑋 ⊆ (Base‘𝑀))
12 simprl 769 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧𝑋)
1311, 12sseldd 3983 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧 ∈ (Base‘𝑀))
14 simprr 771 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥𝑋)
1511, 14sseldd 3983 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥 ∈ (Base‘𝑀))
16 eqid 2728 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
17 eqid 2728 . . . . . . . . . 10 (+g𝑁) = (+g𝑁)
182, 16, 17mgmhmlin 18668 . . . . . . . . 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 6728 . . . . . . . . . 10 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝐹 Fn (Base‘𝑀))
2120adantr 479 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 Fn (Base‘𝑀))
2216submgmcl 18676 . . . . . . . . . . 11 ((𝑋 ∈ (SubMgm‘𝑀) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
23223expb 1117 . . . . . . . . . 10 ((𝑋 ∈ (SubMgm‘𝑀) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
2423adantll 712 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
25 fnfvima 7251 . . . . . . . . 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 2830 . . . . . . 7 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
2827anassrs 466 . . . . . 6 ((((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) ∧ 𝑥𝑋) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
2928ralrimiva 3143 . . . . 5 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) → ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
30 oveq2 7434 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
3130eleq1d 2814 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3231ralima 7256 . . . . . . 7 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3320, 10, 32syl2anc 582 . . . . . 6 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3433adantr 479 . . . . 5 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
3529, 34mpbird 256 . . . 4 (((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) ∧ 𝑧𝑋) → ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
3635ralrimiva 3143 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
37 oveq1 7433 . . . . . . 7 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)𝑦))
3837eleq1d 2814 . . . . . 6 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
3938ralbidv 3175 . . . . 5 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4039ralima 7256 . . . 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 582 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4236, 41mpbird 256 . 2 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))
43 mgmhmrcl 18663 . . . . 5 (𝐹 ∈ (𝑀 MgmHom 𝑁) → (𝑀 ∈ Mgm ∧ 𝑁 ∈ Mgm))
4443simprd 494 . . . 4 (𝐹 ∈ (𝑀 MgmHom 𝑁) → 𝑁 ∈ Mgm)
4544adantr 479 . . 3 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → 𝑁 ∈ Mgm)
463, 17issubmgm 18671 . . 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 711 1 ((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹𝑋) ∈ (SubMgm‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  wss 3949  ran crn 5683  cima 5685   Fn wfn 6548  wf 6549  cfv 6553  (class class class)co 7426  Basecbs 17189  +gcplusg 17242  Mgmcmgm 18607   MgmHom cmgmhm 18659  SubMgmcsubmgm 18660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-sets 17142  df-slot 17160  df-ndx 17172  df-base 17190  df-ress 17219  df-plusg 17255  df-mgm 18609  df-mgmhm 18661  df-submgm 18662
This theorem is referenced by: (None)
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