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Theorem mhmima 12752
Description: The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
Assertion
Ref Expression
mhmima ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹𝑋) ∈ (SubMnd‘𝑁))

Proof of Theorem mhmima
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 4976 . . 3 (𝐹𝑋) ⊆ ran 𝐹
2 eqid 2177 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2177 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
42, 3mhmf 12733 . . . . 5 (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
54adantr 276 . . . 4 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝐹:(Base‘𝑀)⟶(Base‘𝑁))
65frnd 5370 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
71, 6sstrid 3166 . 2 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹𝑋) ⊆ (Base‘𝑁))
8 eqid 2177 . . . . 5 (0g𝑀) = (0g𝑀)
9 eqid 2177 . . . . 5 (0g𝑁) = (0g𝑁)
108, 9mhm0 12736 . . . 4 (𝐹 ∈ (𝑀 MndHom 𝑁) → (𝐹‘(0g𝑀)) = (0g𝑁))
1110adantr 276 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹‘(0g𝑀)) = (0g𝑁))
125ffnd 5361 . . . 4 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝐹 Fn (Base‘𝑀))
132submss 12744 . . . . 5 (𝑋 ∈ (SubMnd‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
1413adantl 277 . . . 4 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝑋 ⊆ (Base‘𝑀))
158subm0cl 12746 . . . . 5 (𝑋 ∈ (SubMnd‘𝑀) → (0g𝑀) ∈ 𝑋)
1615adantl 277 . . . 4 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (0g𝑀) ∈ 𝑋)
17 fnfvima 5745 . . . 4 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑋) → (𝐹‘(0g𝑀)) ∈ (𝐹𝑋))
1812, 14, 16, 17syl3anc 1238 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹‘(0g𝑀)) ∈ (𝐹𝑋))
1911, 18eqeltrrd 2255 . 2 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (0g𝑁) ∈ (𝐹𝑋))
20 simpll 527 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 ∈ (𝑀 MndHom 𝑁))
2114adantr 276 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑋 ⊆ (Base‘𝑀))
22 simprl 529 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧𝑋)
2321, 22sseldd 3156 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧 ∈ (Base‘𝑀))
24 simprr 531 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥𝑋)
2521, 24sseldd 3156 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥 ∈ (Base‘𝑀))
26 eqid 2177 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
27 eqid 2177 . . . . . . . . . 10 (+g𝑁) = (+g𝑁)
282, 26, 27mhmlin 12735 . . . . . . . . 9 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
2920, 23, 25, 28syl3anc 1238 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
3012adantr 276 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝐹 Fn (Base‘𝑀))
3126submcl 12747 . . . . . . . . . . 11 ((𝑋 ∈ (SubMnd‘𝑀) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
32313expb 1204 . . . . . . . . . 10 ((𝑋 ∈ (SubMnd‘𝑀) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
3332adantll 476 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
34 fnfvima 5745 . . . . . . . . 9 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧(+g𝑀)𝑥) ∈ 𝑋) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
3530, 21, 33, 34syl3anc 1238 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
3629, 35eqeltrrd 2255 . . . . . . 7 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
3736anassrs 400 . . . . . 6 ((((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ 𝑧𝑋) ∧ 𝑥𝑋) → ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
3837ralrimiva 2550 . . . . 5 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ 𝑧𝑋) → ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
39 oveq2 5876 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
4039eleq1d 2246 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
4140ralima 5750 . . . . . . 7 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
4212, 14, 41syl2anc 411 . . . . . 6 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
4342adantr 276 . . . . 5 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ 𝑧𝑋) → (∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
4438, 43mpbird 167 . . . 4 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ 𝑧𝑋) → ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
4544ralrimiva 2550 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
46 oveq1 5875 . . . . . . 7 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)𝑦))
4746eleq1d 2246 . . . . . 6 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4847ralbidv 2477 . . . . 5 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4948ralima 5750 . . . 4 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
5012, 14, 49syl2anc 411 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
5145, 50mpbird 167 . 2 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))
52 mhmrcl2 12732 . . . 4 (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝑁 ∈ Mnd)
5352adantr 276 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝑁 ∈ Mnd)
543, 9, 27issubm 12740 . . 3 (𝑁 ∈ Mnd → ((𝐹𝑋) ∈ (SubMnd‘𝑁) ↔ ((𝐹𝑋) ⊆ (Base‘𝑁) ∧ (0g𝑁) ∈ (𝐹𝑋) ∧ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))))
5553, 54syl 14 . 2 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ((𝐹𝑋) ∈ (SubMnd‘𝑁) ↔ ((𝐹𝑋) ⊆ (Base‘𝑁) ∧ (0g𝑁) ∈ (𝐹𝑋) ∧ ∀𝑥 ∈ (𝐹𝑋)∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋))))
567, 19, 51, 55mpbir3and 1180 1 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹𝑋) ∈ (SubMnd‘𝑁))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  wral 2455  wss 3129  ran crn 4623  cima 4625   Fn wfn 5206  wf 5207  cfv 5211  (class class class)co 5868  Basecbs 12432  +gcplusg 12505  0gc0g 12640  Mndcmnd 12696   MndHom cmhm 12726  SubMndcsubmnd 12727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-map 6643  df-inn 8896  df-ndx 12435  df-slot 12436  df-base 12438  df-mhm 12728  df-submnd 12729
This theorem is referenced by: (None)
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