Theorem resmgmhm 46082

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 46065	. . . 4 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2	1	simprd 496	. . 3 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑇 ∈ Mgm)
3		resmgmhm.u	. . . 4 ⊢ 𝑈 = (𝑆 ↾s 𝑋)
4	3	submgmmgm 46079	. . 3 ⊢ (𝑋 ∈ (SubMgm‘𝑆) → 𝑈 ∈ Mgm)
5	2, 4	anim12ci 614	. 2 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝑈 ∈ Mgm ∧ 𝑇 ∈ Mgm))
6		eqid 2736	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
7		eqid 2736	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
8	6, 7	mgmhmf 46068	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
9	6	submgmss 46076	. . . . 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 596	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋):𝑋⟶(Base‘𝑇))
12	9	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → 𝑋 ⊆ (Base‘𝑆))
13	3, 6	ressbas2 17120	. . . . . 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 231	. . 3 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋):(Base‘𝑈)⟶(Base‘𝑇))
17		simpll 765	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
18	9	ad2antlr 725	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑋 ⊆ (Base‘𝑆))
19		simprl 769	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ 𝑋)
20	18, 19	sseldd 3945	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ (Base‘𝑆))
21		simprr 771	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
22	18, 21	sseldd 3945	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ (Base‘𝑆))
23		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝑆) = (+g‘𝑆)
24		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝑇) = (+g‘𝑇)
25	6, 23, 24	mgmhmlin 46070	. . . . . . 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 1371	. . . . . 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 46078	. . . . . . . . 9 ⊢ ((𝑋 ∈ (SubMgm‘𝑆) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
28	27	3expb 1120	. . . . . . . 8 ⊢ ((𝑋 ∈ (SubMgm‘𝑆) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥(+g‘𝑆)𝑦) ∈ 𝑋)
29	28	adantll 712	. . . . . . 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 7376	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (((𝐹 ↾ 𝑋)‘𝑥)(+g‘𝑇)((𝐹 ↾ 𝑋)‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
35	34	adantl 482	. . . . . 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 2786	. . . . 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 3197	. . . 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 17171	. . . . . . . . 9 ⊢ (𝑋 ∈ (SubMgm‘𝑆) → (+g‘𝑆) = (+g‘𝑈))
39	38	adantl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (+g‘𝑆) = (+g‘𝑈))
40	39	oveqd 7374	. . . . . . 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 3319	. . . . 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 3319	. . . 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 231	. . 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 512	. 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 2736	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
47		eqid 2736	. . 3 ⊢ (+g‘𝑈) = (+g‘𝑈)
48	46, 7, 47, 24	ismgmhm 46067	. 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 583	1 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3910   ↾ cres 5635  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Basecbs 17083   ↾_s cress 17112  +_gcplusg 17133  Mgmcmgm 18495   MgmHom cmgmhm 46061  SubMgmcsubmgm 46062

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mgm 18497  df-mgmhm 46063  df-submgm 46064

This theorem is referenced by: (None)