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Theorem mhmima 13576
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 5087 . . 3 (𝐹𝑋) ⊆ ran 𝐹
2 eqid 2231 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2231 . . . . . 6 (Base‘𝑁) = (Base‘𝑁)
42, 3mhmf 13550 . . . . 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 5492 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ran 𝐹 ⊆ (Base‘𝑁))
71, 6sstrid 3238 . 2 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹𝑋) ⊆ (Base‘𝑁))
8 eqid 2231 . . . . 5 (0g𝑀) = (0g𝑀)
9 eqid 2231 . . . . 5 (0g𝑁) = (0g𝑁)
108, 9mhm0 13553 . . . 4 (𝐹 ∈ (𝑀 MndHom 𝑁) → (𝐹‘(0g𝑀)) = (0g𝑁))
1110adantr 276 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹‘(0g𝑀)) = (0g𝑁))
125ffnd 5483 . . . 4 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝐹 Fn (Base‘𝑀))
132submss 13561 . . . . 5 (𝑋 ∈ (SubMnd‘𝑀) → 𝑋 ⊆ (Base‘𝑀))
1413adantl 277 . . . 4 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝑋 ⊆ (Base‘𝑀))
158subm0cl 13563 . . . . 5 (𝑋 ∈ (SubMnd‘𝑀) → (0g𝑀) ∈ 𝑋)
1615adantl 277 . . . 4 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (0g𝑀) ∈ 𝑋)
17 fnfvima 5889 . . . 4 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑋) → (𝐹‘(0g𝑀)) ∈ (𝐹𝑋))
1812, 14, 16, 17syl3anc 1273 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → (𝐹‘(0g𝑀)) ∈ (𝐹𝑋))
1911, 18eqeltrrd 2309 . 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 531 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧𝑋)
2321, 22sseldd 3228 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑧 ∈ (Base‘𝑀))
24 simprr 533 . . . . . . . . . 10 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥𝑋)
2521, 24sseldd 3228 . . . . . . . . 9 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → 𝑥 ∈ (Base‘𝑀))
26 eqid 2231 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
27 eqid 2231 . . . . . . . . . 10 (+g𝑁) = (+g𝑁)
282, 26, 27mhmlin 13552 . . . . . . . . 9 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑧 ∈ (Base‘𝑀) ∧ 𝑥 ∈ (Base‘𝑀)) → (𝐹‘(𝑧(+g𝑀)𝑥)) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
2920, 23, 25, 28syl3anc 1273 . . . . . . . 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 13564 . . . . . . . . . . 11 ((𝑋 ∈ (SubMnd‘𝑀) ∧ 𝑧𝑋𝑥𝑋) → (𝑧(+g𝑀)𝑥) ∈ 𝑋)
32313expb 1230 . . . . . . . . . 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 5889 . . . . . . . . 9 ((𝐹 Fn (Base‘𝑀) ∧ 𝑋 ⊆ (Base‘𝑀) ∧ (𝑧(+g𝑀)𝑥) ∈ 𝑋) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
3530, 21, 33, 34syl3anc 1273 . . . . . . . 8 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ (𝑧𝑋𝑥𝑋)) → (𝐹‘(𝑧(+g𝑀)𝑥)) ∈ (𝐹𝑋))
3629, 35eqeltrrd 2309 . . . . . . 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 2605 . . . . 5 (((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) ∧ 𝑧𝑋) → ∀𝑥𝑋 ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋))
39 oveq2 6026 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹𝑧)(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)(𝐹𝑥)))
4039eleq1d 2300 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)(𝐹𝑥)) ∈ (𝐹𝑋)))
4140ralima 5896 . . . . . . 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 2605 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → ∀𝑧𝑋𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋))
46 oveq1 6025 . . . . . . 7 (𝑥 = (𝐹𝑧) → (𝑥(+g𝑁)𝑦) = ((𝐹𝑧)(+g𝑁)𝑦))
4746eleq1d 2300 . . . . . 6 (𝑥 = (𝐹𝑧) → ((𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4847ralbidv 2532 . . . . 5 (𝑥 = (𝐹𝑧) → (∀𝑦 ∈ (𝐹𝑋)(𝑥(+g𝑁)𝑦) ∈ (𝐹𝑋) ↔ ∀𝑦 ∈ (𝐹𝑋)((𝐹𝑧)(+g𝑁)𝑦) ∈ (𝐹𝑋)))
4948ralima 5896 . . . 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 13549 . . . 4 (𝐹 ∈ (𝑀 MndHom 𝑁) → 𝑁 ∈ Mnd)
5352adantr 276 . . 3 ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubMnd‘𝑀)) → 𝑁 ∈ Mnd)
543, 9, 27issubm 13557 . . 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 1206 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 1004   = wceq 1397  wcel 2202  wral 2510  wss 3200  ran crn 4726  cima 4728   Fn wfn 5321  wf 5322  cfv 5326  (class class class)co 6018  Basecbs 13084  +gcplusg 13162  0gc0g 13341  Mndcmnd 13501   MndHom cmhm 13542  SubMndcsubmnd 13543
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-map 6819  df-inn 9144  df-ndx 13087  df-slot 13088  df-base 13090  df-mhm 13544  df-submnd 13545
This theorem is referenced by:  rhmima  14268
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