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Theorem resmhm 17280
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 17260 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
2 resmhm.u . . . 4 𝑈 = (𝑆s 𝑋)
32submmnd 17275 . . 3 (𝑋 ∈ (SubMnd‘𝑆) → 𝑈 ∈ Mnd)
41, 3anim12ci 590 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑈 ∈ Mnd ∧ 𝑇 ∈ Mnd))
5 eqid 2621 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2621 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
75, 6mhmf 17261 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
85submss 17271 . . . . 5 (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
9 fssres 6027 . . . . 5 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
107, 8, 9syl2an 494 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
118adantl 482 . . . . . 6 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
122, 5ressbas2 15852 . . . . . 6 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1311, 12syl 17 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 = (Base‘𝑈))
1413feq2d 5988 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1510, 14mpbid 222 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
16 simpll 789 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
178ad2antlr 762 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑆))
18 simprl 793 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
1917, 18sseldd 3584 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 ∈ (Base‘𝑆))
20 simprr 795 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
2117, 20sseldd 3584 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦 ∈ (Base‘𝑆))
22 eqid 2621 . . . . . . . 8 (+g𝑆) = (+g𝑆)
23 eqid 2621 . . . . . . . 8 (+g𝑇) = (+g𝑇)
245, 22, 23mhmlin 17263 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2516, 19, 21, 24syl3anc 1323 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2622submcl 17274 . . . . . . . . 9 ((𝑋 ∈ (SubMnd‘𝑆) ∧ 𝑥𝑋𝑦𝑋) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
27263expb 1263 . . . . . . . 8 ((𝑋 ∈ (SubMnd‘𝑆) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
2827adantll 749 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
29 fvres 6164 . . . . . . 7 ((𝑥(+g𝑆)𝑦) ∈ 𝑋 → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
3028, 29syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
31 fvres 6164 . . . . . . . 8 (𝑥𝑋 → ((𝐹𝑋)‘𝑥) = (𝐹𝑥))
32 fvres 6164 . . . . . . . 8 (𝑦𝑋 → ((𝐹𝑋)‘𝑦) = (𝐹𝑦))
3331, 32oveqan12d 6623 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3433adantl 482 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3525, 30, 343eqtr4d 2665 . . . . 5 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
3635ralrimivva 2965 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
372, 22ressplusg 15914 . . . . . . . . . 10 (𝑋 ∈ (SubMnd‘𝑆) → (+g𝑆) = (+g𝑈))
3837adantl 482 . . . . . . . . 9 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (+g𝑆) = (+g𝑈))
3938oveqd 6621 . . . . . . . 8 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑥(+g𝑆)𝑦) = (𝑥(+g𝑈)𝑦))
4039fveq2d 6152 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)))
4140eqeq1d 2623 . . . . . 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))))
4213, 41raleqbidv 3141 . . . . 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))))
4313, 42raleqbidv 3141 . . . 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))))
4436, 43mpbid 222 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
45 eqid 2621 . . . . . . 7 (0g𝑆) = (0g𝑆)
4645subm0cl 17273 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑆) → (0g𝑆) ∈ 𝑋)
4746adantl 482 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g𝑆) ∈ 𝑋)
48 fvres 6164 . . . . 5 ((0g𝑆) ∈ 𝑋 → ((𝐹𝑋)‘(0g𝑆)) = (𝐹‘(0g𝑆)))
4947, 48syl 17 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑆)) = (𝐹‘(0g𝑆)))
502, 45subm0 17277 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑆) → (0g𝑆) = (0g𝑈))
5150adantl 482 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g𝑆) = (0g𝑈))
5251fveq2d 6152 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑆)) = ((𝐹𝑋)‘(0g𝑈)))
53 eqid 2621 . . . . . 6 (0g𝑇) = (0g𝑇)
5445, 53mhm0 17264 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
5554adantr 481 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹‘(0g𝑆)) = (0g𝑇))
5649, 52, 553eqtr3d 2663 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑈)) = (0g𝑇))
5715, 44, 563jca 1240 . 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)))
58 eqid 2621 . . 3 (Base‘𝑈) = (Base‘𝑈)
59 eqid 2621 . . 3 (+g𝑈) = (+g𝑈)
60 eqid 2621 . . 3 (0g𝑈) = (0g𝑈)
6158, 6, 59, 23, 60, 53ismhm 17258 . 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))))
624, 57, 61sylanbrc 697 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wss 3555  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  s cress 15782  +gcplusg 15862  0gc0g 16021  Mndcmnd 17215   MndHom cmhm 17254  SubMndcsubmnd 17255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257
This theorem is referenced by:  resrhm  18730  dchrghm  24881
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