Theorem resmgmhm2b 41800

Description: Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)

Hypothesis

Ref	Expression
resmgmhm2.u	⊢ 𝑈 = (𝑇 ↾s 𝑋)

Assertion

Ref	Expression
resmgmhm2b	⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))

Proof of Theorem resmgmhm2b

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

Step	Hyp	Ref	Expression
1		mgmhmrcl 41781	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2	1	simpld 475	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
3	2	adantl 482	. . . 4 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
4		resmgmhm2.u	. . . . . 6 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
5	4	submgmmgm 41795	. . . . 5 ⊢ (𝑋 ∈ (SubMgm‘𝑇) → 𝑈 ∈ Mgm)
6	5	ad2antrr 762	. . . 4 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑈 ∈ Mgm)
7	3, 6	jca 554	. . 3 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
8		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
10	8, 9	mgmhmf 41784	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
11	10	adantl 482	. . . . . . 7 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12		ffn 6045	. . . . . . 7 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
13	11, 12	syl 17	. . . . . 6 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
14		simplr 792	. . . . . 6 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ran 𝐹 ⊆ 𝑋)
15		df-f 5892	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹 ⊆ 𝑋))
16	13, 14, 15	sylanbrc 698	. . . . 5 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
17	4	submgmbas 41796	. . . . . . 7 ⊢ (𝑋 ∈ (SubMgm‘𝑇) → 𝑋 = (Base‘𝑈))
18	17	ad2antrr 762	. . . . . 6 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑋 = (Base‘𝑈))
19	18	feq3d 6032	. . . . 5 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋 ↔ 𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
20	16, 19	mpbid 222	. . . 4 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
21		eqid 2622	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
22		eqid 2622	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
23	8, 21, 22	mgmhmlin 41786	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
24	23	3expb 1266	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
25	24	adantll 750	. . . . . 6 ⊢ ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
26	4, 22	ressplusg 15993	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMgm‘𝑇) → (+g‘𝑇) = (+g‘𝑈))
27	26	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g‘𝑇) = (+g‘𝑈))
28	27	oveqd 6667	. . . . . 6 ⊢ ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
29	25, 28	eqtrd 2656	. . . . 5 ⊢ ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
30	29	ralrimivva 2971	. . . 4 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))
31	20, 30	jca 554	. . 3 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦))))
32		eqid 2622	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
33		eqid 2622	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
34	8, 32, 21, 33	ismgmhm 41783	. . 3 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑈) ↔ ((𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑈)(𝐹‘𝑦)))))
35	7, 31, 34	sylanbrc 698	. 2 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑈))
36	4	resmgmhm2 41799	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
37	36	ancoms 469	. . 3 ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
38	37	adantlr 751	. 2 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
39	35, 38	impbida 877	1 ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857   ↾_s cress 15858  +_gcplusg 15941  Mgmcmgm 17240   MgmHom cmgmhm 41777  SubMgmcsubmgm 41778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mgm 17242  df-mgmhm 41779  df-submgm 41780

This theorem is referenced by: (None)