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Theorem resmhm 18833
Description: Restriction of a monoid homomorphism to a submonoid is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
Hypothesis
Ref Expression
resmhm.u 𝑈 = (𝑆s 𝑋)
Assertion
Ref Expression
resmhm ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MndHom 𝑇))

Proof of Theorem resmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl2 18801 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
2 resmhm.u . . . 4 𝑈 = (𝑆s 𝑋)
32submmnd 18826 . . 3 (𝑋 ∈ (SubMnd‘𝑆) → 𝑈 ∈ Mnd)
41, 3anim12ci 614 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑈 ∈ Mnd ∧ 𝑇 ∈ Mnd))
5 eqid 2737 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2737 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
75, 6mhmf 18802 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
85submss 18822 . . . . 5 (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
9 fssres 6774 . . . . 5 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
107, 8, 9syl2an 596 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
118adantl 481 . . . . . 6 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
122, 5ressbas2 17283 . . . . . 6 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1311, 12syl 17 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 = (Base‘𝑈))
1413feq2d 6722 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1510, 14mpbid 232 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
16 simpll 767 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
178ad2antlr 727 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑆))
18 simprl 771 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
1917, 18sseldd 3984 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 ∈ (Base‘𝑆))
20 simprr 773 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
2117, 20sseldd 3984 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦 ∈ (Base‘𝑆))
22 eqid 2737 . . . . . . . 8 (+g𝑆) = (+g𝑆)
23 eqid 2737 . . . . . . . 8 (+g𝑇) = (+g𝑇)
245, 22, 23mhmlin 18806 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2516, 19, 21, 24syl3anc 1373 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2622submcl 18825 . . . . . . . . 9 ((𝑋 ∈ (SubMnd‘𝑆) ∧ 𝑥𝑋𝑦𝑋) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
27263expb 1121 . . . . . . . 8 ((𝑋 ∈ (SubMnd‘𝑆) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
2827adantll 714 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
2928fvresd 6926 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
30 fvres 6925 . . . . . . . 8 (𝑥𝑋 → ((𝐹𝑋)‘𝑥) = (𝐹𝑥))
31 fvres 6925 . . . . . . . 8 (𝑦𝑋 → ((𝐹𝑋)‘𝑦) = (𝐹𝑦))
3230, 31oveqan12d 7450 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3332adantl 481 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3425, 29, 333eqtr4d 2787 . . . . 5 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
3534ralrimivva 3202 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
362, 22ressplusg 17334 . . . . . . . . 9 (𝑋 ∈ (SubMnd‘𝑆) → (+g𝑆) = (+g𝑈))
3736adantl 481 . . . . . . . 8 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (+g𝑆) = (+g𝑈))
3837oveqd 7448 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑥(+g𝑆)𝑦) = (𝑥(+g𝑈)𝑦))
3938fveqeq2d 6914 . . . . . 6 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4013, 39raleqbidv 3346 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (∀𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4113, 40raleqbidv 3346 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (∀𝑥𝑋𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4235, 41mpbid 232 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
43 eqid 2737 . . . . . . 7 (0g𝑆) = (0g𝑆)
4443subm0cl 18824 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑆) → (0g𝑆) ∈ 𝑋)
4544adantl 481 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g𝑆) ∈ 𝑋)
4645fvresd 6926 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑆)) = (𝐹‘(0g𝑆)))
472, 43subm0 18828 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑆) → (0g𝑆) = (0g𝑈))
4847adantl 481 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g𝑆) = (0g𝑈))
4948fveq2d 6910 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑆)) = ((𝐹𝑋)‘(0g𝑈)))
50 eqid 2737 . . . . . 6 (0g𝑇) = (0g𝑇)
5143, 50mhm0 18807 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
5251adantr 480 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹‘(0g𝑆)) = (0g𝑇))
5346, 49, 523eqtr3d 2785 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑈)) = (0g𝑇))
5415, 42, 533jca 1129 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ∧ ((𝐹𝑋)‘(0g𝑈)) = (0g𝑇)))
55 eqid 2737 . . 3 (Base‘𝑈) = (Base‘𝑈)
56 eqid 2737 . . 3 (+g𝑈) = (+g𝑈)
57 eqid 2737 . . 3 (0g𝑈) = (0g𝑈)
5855, 6, 56, 23, 57, 50ismhm 18798 . 2 ((𝐹𝑋) ∈ (𝑈 MndHom 𝑇) ↔ ((𝑈 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ ((𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ∧ ((𝐹𝑋)‘(0g𝑈)) = (0g𝑇))))
594, 54, 58sylanbrc 583 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  +gcplusg 17297  0gc0g 17484  Mndcmnd 18747   MndHom cmhm 18794  SubMndcsubmnd 18795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797
This theorem is referenced by:  resrhm  20601  dchrghm  27300
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