Theorem resghm2b 13335

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 13318	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2	1	a1i 9	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp))
3		ghmgrp1 13318	. . 3 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑈) → 𝑆 ∈ Grp)
4	3	a1i 9	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑈) → 𝑆 ∈ Grp))
5		subgsubm 13269	. . . . . 6 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑋 ∈ (SubMnd‘𝑇))
6		resghm2.u	. . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
7	6	resmhm2b 13064	. . . . . 6 ⊢ ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
8	5, 7	sylan 283	. . . . 5 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
9	8	adantl 277	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
10		subgrcl 13252	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑇 ∈ Grp)
11	10	adantr 276	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑇 ∈ Grp)
12		ghmmhmb 13327	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
13	11, 12	sylan2 286	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
14	13	eleq2d 2263	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑇)))
15	6	subggrp 13250	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑈 ∈ Grp)
16	15	adantr 276	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑈 ∈ Grp)
17		ghmmhmb 13327	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑈 ∈ Grp) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
18	16, 17	sylan2 286	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
19	18	eleq2d 2263	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑈) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
20	9, 14, 19	3bitr4d 220	. . 3 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
21	20	expcom 116	. 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝑆 ∈ Grp → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈))))
22	2, 4, 21	pm5.21ndd 706	1 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164   ⊆ wss 3154  ran crn 4661  ‘cfv 5255  (class class class)co 5919   ↾_s cress 12622   MndHom cmhm 13032  SubMndcsubmnd 13033  Grpcgrp 13075  SubGrpcsubg 13240   GrpHom cghm 13313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-submnd 13035  df-grp 13078  df-minusg 13079  df-subg 13243  df-ghm 13314

This theorem is referenced by:  ghmghmrn  13336  resrhm2b  13748