Theorem resmgmhm 18648

Description: Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)

Hypothesis

Ref	Expression
resmgmhm.u	⊢ 𝑈 = (𝑆 ↾s 𝑋)

Assertion

Ref	Expression
resmgmhm	⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇))

Proof of Theorem resmgmhm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgmhmrcl 18631	. . . 4 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2	1	simprd 495	. . 3 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑇 ∈ Mgm)
3		resmgmhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
4	3	submgmmgm 18645	. . 3 ⊢ (𝑋 ∈ (SubMgm‘𝑆) → 𝑈 ∈ Mgm)
5	2, 4	anim12ci 615	. 2 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝑈 ∈ Mgm ∧ 𝑇 ∈ Mgm))
6		eqid 2737	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
7		eqid 2737	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
8	6, 7	mgmhmf 18634	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
9	6	submgmss 18642	. . . . 5 ⊢ (𝑋 ∈ (SubMgm‘𝑆) → 𝑋 ⊆ (Base‘𝑆))
10		fssres 6708	. . . . 5 ⊢ ((𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ 𝑋 ⊆ (Base‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
11	8, 9, 10	syl2an 597	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
12	9	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
13	3, 6	ressbas2 17177	. . . . . 6 ⊢ (𝑋 ⊆ (Base‘𝑆) → 𝑋 = (Base‘𝑈))
14	12, 13	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → 𝑋 = (Base‘𝑈))
15	14	feq2d 6654	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ((𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇) ↔ (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇)))
16	11, 15	mpbid 232	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇))
17		simpll 767	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
18	9	ad2antlr 728	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑆))
19		simprl 771	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
20	18, 19	sseldd 3936	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑆))
21		simprr 773	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
22	18, 21	sseldd 3936	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑆))
23		eqid 2737	. . . . . . . 8 ⊢ (+g‘𝑆) = (+g‘𝑆)
24		eqid 2737	. . . . . . . 8 ⊢ (+g‘𝑇) = (+g‘𝑇)
25	6, 23, 24	mgmhmlin 18636	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
26	17, 20, 22, 25	syl3anc 1374	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
27	23	submgmcl 18644	. . . . . . . . 9 ⊢ ((𝑋 ∈ (SubMgm‘𝑆) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
28	27	3expb 1121	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubMgm‘𝑆) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
29	28	adantll 715	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
30		fvres 6861	. . . . . . 7 ⊢ ((𝑥(+g‘𝑆)𝑦) ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
31	29, 30	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
32		fvres 6861	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑥) = (𝐹‘𝑥))
33		fvres 6861	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑋 → ((𝐹 ↾ 𝑋)‘𝑦) = (𝐹‘𝑦))
34	32, 33	oveqan12d 7387	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
35	34	adantl 481	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
36	26, 31, 35	3eqtr4d 2782	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)))
37	36	ralrimivva 3181	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)))
38	3, 23	ressplusg 17223	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMgm‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
39	38	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (+g‘𝑆) = (+g‘𝑈))
40	39	oveqd 7385	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝑥(+g‘𝑆)𝑦) = (𝑥(+g‘𝑈)𝑦))
41	40	fveqeq2d 6850	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) ↔ ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦))))
42	14, 41	raleqbidv 3318	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (∀𝑦 ∈ 𝑋 ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦))))
43	14, 42	raleqbidv 3318	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑆)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦))))
44	37, 43	mpbid 232	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)))
45	16, 44	jca 511	. 2 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → ((𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦))))
46		eqid 2737	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
47		eqid 2737	. . 3 ⊢ (+g‘𝑈) = (+g‘𝑈)
48	46, 7, 47, 24	ismgmhm 18633	. 2 ⊢ ((𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇) ↔ ((𝑈 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ ((𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)((𝐹 ↾ 𝑋)‘(𝑥(+g‘𝑈)𝑦)) = (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)))))
49	5, 45, 48	sylanbrc 584	1 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903   ↾ cres 5634  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148   ↾_s cress 17169  +_gcplusg 17189  Mgmcmgm 18575   MgmHom cmgmhm 18627  SubMgmcsubmgm 18628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mgm 18577  df-mgmhm 18629  df-submgm 18630

This theorem is referenced by: (None)