Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  resmgmhm Structured version   Visualization version   GIF version

Theorem resmgmhm 41583
Description: Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
Hypothesis
Ref Expression
resmgmhm.u 𝑈 = (𝑆s 𝑋)
Assertion
Ref Expression
resmgmhm ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MgmHom 𝑇))

Proof of Theorem resmgmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmrcl 41566 . . . 4 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
21simprd 477 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑇 ∈ Mgm)
3 resmgmhm.u . . . 4 𝑈 = (𝑆s 𝑋)
43submgmmgm 41580 . . 3 (𝑋 ∈ (SubMgm‘𝑆) → 𝑈 ∈ Mgm)
52, 4anim12ci 588 . 2 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝑈 ∈ Mgm ∧ 𝑇 ∈ Mgm))
6 eqid 2609 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
7 eqid 2609 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
86, 7mgmhmf 41569 . . . . 5 (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
96submgmss 41577 . . . . 5 (𝑋 ∈ (SubMgm‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
10 fssres 5968 . . . . 5 ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
118, 9, 10syl2an 492 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋):𝑋⟶(Base‘𝑇))
129adantl 480 . . . . . 6 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
133, 6ressbas2 15704 . . . . . 6 (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
1412, 13syl 17 . . . . 5 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → 𝑋 = (Base‘𝑈))
1514feq2d 5930 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ((𝐹𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
1611, 15mpbid 220 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇))
17 simpll 785 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
189ad2antlr 758 . . . . . . . 8 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑋 ⊆ (Base‘𝑆))
19 simprl 789 . . . . . . . 8 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥𝑋)
2018, 19sseldd 3568 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑥 ∈ (Base‘𝑆))
21 simprr 791 . . . . . . . 8 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦𝑋)
2218, 21sseldd 3568 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → 𝑦 ∈ (Base‘𝑆))
23 eqid 2609 . . . . . . . 8 (+g𝑆) = (+g𝑆)
24 eqid 2609 . . . . . . . 8 (+g𝑇) = (+g𝑇)
256, 23, 24mgmhmlin 41571 . . . . . . 7 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2617, 20, 22, 25syl3anc 1317 . . . . . 6 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2723submgmcl 41579 . . . . . . . . 9 ((𝑋 ∈ (SubMgm‘𝑆) ∧ 𝑥𝑋𝑦𝑋) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
28273expb 1257 . . . . . . . 8 ((𝑋 ∈ (SubMgm‘𝑆) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
2928adantll 745 . . . . . . 7 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥(+g𝑆)𝑦) ∈ 𝑋)
30 fvres 6102 . . . . . . 7 ((𝑥(+g𝑆)𝑦) ∈ 𝑋 → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
3129, 30syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (𝐹‘(𝑥(+g𝑆)𝑦)))
32 fvres 6102 . . . . . . . 8 (𝑥𝑋 → ((𝐹𝑋)‘𝑥) = (𝐹𝑥))
33 fvres 6102 . . . . . . . 8 (𝑦𝑋 → ((𝐹𝑋)‘𝑦) = (𝐹𝑦))
3432, 33oveqan12d 6546 . . . . . . 7 ((𝑥𝑋𝑦𝑋) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3534adantl 480 . . . . . 6 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
3626, 31, 353eqtr4d 2653 . . . . 5 (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
3736ralrimivva 2953 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
383, 23ressplusg 15764 . . . . . . . . . 10 (𝑋 ∈ (SubMgm‘𝑆) → (+g𝑆) = (+g𝑈))
3938adantl 480 . . . . . . . . 9 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (+g𝑆) = (+g𝑈))
4039oveqd 6544 . . . . . . . 8 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝑥(+g𝑆)𝑦) = (𝑥(+g𝑈)𝑦))
4140fveq2d 6092 . . . . . . 7 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)))
4241eqeq1d 2611 . . . . . 6 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4314, 42raleqbidv 3128 . . . . 5 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (∀𝑦𝑋 ((𝐹𝑋)‘(𝑥(+g𝑆)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
4414, 43raleqbidv 3128 . . . 4 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘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))))
4537, 44mpbid 220 . . 3 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))
4616, 45jca 552 . 2 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ((𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦))))
47 eqid 2609 . . 3 (Base‘𝑈) = (Base‘𝑈)
48 eqid 2609 . . 3 (+g𝑈) = (+g𝑈)
4947, 7, 48, 24ismgmhm 41568 . 2 ((𝐹𝑋) ∈ (𝑈 MgmHom 𝑇) ↔ ((𝑈 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ ((𝐹𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹𝑋)‘(𝑥(+g𝑈)𝑦)) = (((𝐹𝑋)‘𝑥)(+g𝑇)((𝐹𝑋)‘𝑦)))))
505, 46, 49sylanbrc 694 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MgmHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  wss 3539  cres 5030  wf 5786  cfv 5790  (class class class)co 6527  Basecbs 15641  s cress 15642  +gcplusg 15714  Mgmcmgm 17009   MgmHom cmgmhm 41562  SubMgmcsubmgm 41563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mgm 17011  df-mgmhm 41564  df-submgm 41565
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator