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Theorem resmhm 17567
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 17547 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
2 resmhm.u . . . 4 𝑈 = (𝑆s 𝑋)
32submmnd 17562 . . 3 (𝑋 ∈ (SubMnd‘𝑆) → 𝑈 ∈ Mnd)
41, 3anim12ci 601 . 2 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑈 ∈ Mnd ∧ 𝑇 ∈ Mnd))
5 eqid 2771 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2771 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
75, 6mhmf 17548 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
85submss 17558 . . . . 5 (𝑋 ∈ (SubMnd‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
9 fssres 6211 . . . . 5 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
107, 8, 9syl2an 583 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
118adantl 467 . . . . . 6 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
122, 5ressbas2 16138 . . . . . 6 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1311, 12syl 17 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → 𝑋 = (Base‘𝑈))
1413feq2d 6170 . . . 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 750 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
178ad2antlr 706 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑆))
18 simprl 754 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
1917, 18sseldd 3753 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 ∈ (Base‘𝑆))
20 simprr 756 . . . . . . . 8 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
2117, 20sseldd 3753 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦 ∈ (Base‘𝑆))
22 eqid 2771 . . . . . . . 8 (+g𝑆) = (+g𝑆)
23 eqid 2771 . . . . . . . 8 (+g𝑇) = (+g𝑇)
245, 22, 23mhmlin 17550 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2516, 19, 21, 24syl3anc 1476 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2622submcl 17561 . . . . . . . . 9 ((𝑋 ∈ (SubMnd‘𝑆) ∧ 𝑥𝑋𝑦𝑋) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
27263expb 1113 . . . . . . . 8 ((𝑋 ∈ (SubMnd‘𝑆) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
2827adantll 693 . . . . . . 7 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
29 fvres 6350 . . . . . . 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 6350 . . . . . . . 8 (𝑥𝑋 → ((𝐹𝑋)‘𝑥) = (𝐹𝑥))
32 fvres 6350 . . . . . . . 8 (𝑦𝑋 → ((𝐹𝑋)‘𝑦) = (𝐹𝑦))
3331, 32oveqan12d 6815 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3433adantl 467 . . . . . 6 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3525, 30, 343eqtr4d 2815 . . . . 5 (((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
3635ralrimivva 3120 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
372, 22ressplusg 16201 . . . . . . . . . 10 (𝑋 ∈ (SubMnd‘𝑆) → (+g𝑆) = (+g𝑈))
3837adantl 467 . . . . . . . . 9 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (+g𝑆) = (+g𝑈))
3938oveqd 6813 . . . . . . . 8 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝑥(+g𝑆)𝑦) = (𝑥(+g𝑈)𝑦))
4039fveq2d 6337 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)))
4140eqeq1d 2773 . . . . . 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 3301 . . . . 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 3301 . . . 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 2771 . . . . . . 7 (0g𝑆) = (0g𝑆)
4645subm0cl 17560 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑆) → (0g𝑆) ∈ 𝑋)
4746adantl 467 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g𝑆) ∈ 𝑋)
48 fvres 6350 . . . . 5 ((0g𝑆) ∈ 𝑋 → ((𝐹𝑋)‘(0g𝑆)) = (𝐹‘(0g𝑆)))
4947, 48syl 17 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑆)) = (𝐹‘(0g𝑆)))
502, 45subm0 17564 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑆) → (0g𝑆) = (0g𝑈))
5150adantl 467 . . . . 5 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (0g𝑆) = (0g𝑈))
5251fveq2d 6337 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑆)) = ((𝐹𝑋)‘(0g𝑈)))
53 eqid 2771 . . . . . 6 (0g𝑇) = (0g𝑇)
5445, 53mhm0 17551 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
5554adantr 466 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹‘(0g𝑆)) = (0g𝑇))
5649, 52, 553eqtr3d 2813 . . 3 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → ((𝐹𝑋)‘(0g𝑈)) = (0g𝑇))
5715, 44, 563jca 1122 . 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 2771 . . 3 (Base‘𝑈) = (Base‘𝑈)
59 eqid 2771 . . 3 (+g𝑈) = (+g𝑈)
60 eqid 2771 . . 3 (0g𝑈) = (0g𝑈)
6158, 6, 59, 23, 60, 53ismhm 17545 . 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 572 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ (SubMnd‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wss 3723  cres 5252  wf 6026  cfv 6030  (class class class)co 6796  Basecbs 16064  s cress 16065  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502   MndHom cmhm 17541  SubMndcsubmnd 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544
This theorem is referenced by:  resrhm  19019  dchrghm  25202
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