Theorem resmgmhm2b 18691

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 18672	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2	1	simpld 494	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝑆 ∈ Mgm)
3	2	adantl 481	. . . 4 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑆 ∈ Mgm)
4		resmgmhm2.u	. . . . . 6 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
5	4	submgmmgm 18686	. . . . 5 ⊢ (𝑋 ∈ (SubMgm‘𝑇) → 𝑈 ∈ Mgm)
6	5	ad2antrr 726	. . . 4 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑈 ∈ Mgm)
7	3, 6	jca 511	. . 3 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝑆 ∈ Mgm ∧ 𝑈 ∈ Mgm))
8		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2735	. . . . . . . . 9 ⊢ (Base‘𝑇) = (Base‘𝑇)
10	8, 9	mgmhmf 18675	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
11	10	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
12	11	ffnd 6707	. . . . . 6 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
13		simplr 768	. . . . . 6 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → ran 𝐹 ⊆ 𝑋)
14		df-f 6535	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹 ⊆ 𝑋))
15	12, 13, 14	sylanbrc 583	. . . . 5 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
16	4	submgmbas 18687	. . . . . . 7 ⊢ (𝑋 ∈ (SubMgm‘𝑇) → 𝑋 = (Base‘𝑈))
17	16	ad2antrr 726	. . . . . 6 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝑋 = (Base‘𝑈))
18	17	feq3d 6693	. . . . 5 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋 ↔ 𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
19	15, 18	mpbid 232	. . . 4 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
20		eqid 2735	. . . . . . . . 9 ⊢ (+g‘𝑆) = (+g‘𝑆)
21		eqid 2735	. . . . . . . . 9 ⊢ (+g‘𝑇) = (+g‘𝑇)
22	8, 20, 21	mgmhmlin 18677	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
23	22	3expb 1120	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
24	23	adantll 714	. . . . . 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)))
25	4, 21	ressplusg 17305	. . . . . . . 8 ⊢ (𝑋 ∈ (SubMgm‘𝑇) → (+g‘𝑇) = (+g‘𝑈))
26	25	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g‘𝑇) = (+g‘𝑈))
27	26	oveqd 7422	. . . . . 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)))
28	24, 27	eqtrd 2770	. . . . 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)))
29	28	ralrimivva 3187	. . . 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)))
30	19, 29	jca 511	. . 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))))
31		eqid 2735	. . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈)
32		eqid 2735	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
33	8, 31, 20, 32	ismgmhm 18674	. . 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)))))
34	7, 30, 33	sylanbrc 583	. 2 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑈))
35	4	resmgmhm2 18690	. . . 4 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
36	35	ancoms 458	. . 3 ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
37	36	adantlr 715	. 2 ⊢ (((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) ∧ 𝐹 ∈ (𝑆 MgmHom 𝑈)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))
38	34, 37	impbida 800	1 ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ⊆ wss 3926  ran crn 5655   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Basecbs 17228   ↾_s cress 17251  +_gcplusg 17271  Mgmcmgm 18616   MgmHom cmgmhm 18668  SubMgmcsubmgm 18669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mgm 18618  df-mgmhm 18670  df-submgm 18671

This theorem is referenced by: (None)