Theorem resghm2b 19217

Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)

Hypothesis

Ref	Expression
resghm2.u	⊢ 𝑈 = (𝑇 ↾s 𝑋)

Assertion

Ref	Expression
resghm2b	⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))

Proof of Theorem resghm2b

Step	Hyp	Ref	Expression
1		ghmgrp1 19201	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2	1	a1i 11	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp))
3		ghmgrp1 19201	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑈) → 𝑆 ∈ Grp)
4	3	a1i 11	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑈) → 𝑆 ∈ Grp))
5		subgsubm 19131	. . . . . 6 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑋 ∈ (SubMnd‘𝑇))
6		resghm2.u	. . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
7	6	resmhm2b 18800	. . . . . 6 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
8	5, 7	sylan 580	. . . . 5 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
9	8	adantl 481	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
10		subgrcl 19114	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑇 ∈ Grp)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑇 ∈ Grp)
12		ghmmhmb 19210	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
13	11, 12	sylan2 593	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
14	13	eleq2d 2820	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑇)))
15	6	subggrp 19112	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑈 ∈ Grp)
16	15	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑈 ∈ Grp)
17		ghmmhmb 19210	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑈 ∈ Grp) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
18	16, 17	sylan2 593	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
19	18	eleq2d 2820	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑈) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
20	9, 14, 19	3bitr4d 311	. . 3 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
21	20	expcom 413	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝑆 ∈ Grp → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈))))
22	2, 4, 21	pm5.21ndd 379	1 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926  ran crn 5655  ‘cfv 6531  (class class class)co 7405   ↾_s cress 17251   MndHom cmhm 18759  SubMndcsubmnd 18760  Grpcgrp 18916  SubGrpcsubg 19103   GrpHom cghm 19195

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-rmo 3359  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-1st 7988  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-0g 17455  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-subg 19106  df-ghm 19196

This theorem is referenced by:  ghmghmrn  19218  cayley  19395  pj1ghm2  19685  dpjghm2  20047  resrhm2b  20562  reslmhm2b  21012  m2cpmghm  22682