Theorem resghm2b 19148

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 19132	. . 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 19132	. . 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 19062	. . . . . 6 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑋 ∈ (SubMnd‘𝑇))
6		resghm2.u	. . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋)
7	6	resmhm2b 18731	. . . . . 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 19045	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑇 ∈ Grp)
11	10	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑇 ∈ Grp)
12		ghmmhmb 19141	. . . . . 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 2814	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑇)))
15	6	subggrp 19043	. . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑈 ∈ Grp)
16	15	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑈 ∈ Grp)
17		ghmmhmb 19141	. . . . . 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 2814	. . . 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 2109   ⊆ wss 3911  ran crn 5632  ‘cfv 6499  (class class class)co 7369   ↾_s cress 17176   MndHom cmhm 18690  SubMndcsubmnd 18691  Grpcgrp 18847  SubGrpcsubg 19034   GrpHom cghm 19126

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-submnd 18693  df-grp 18850  df-minusg 18851  df-subg 19037  df-ghm 19127

This theorem is referenced by:  ghmghmrn  19149  cayley  19328  pj1ghm2  19618  dpjghm2  19980  resrhm2b  20522  reslmhm2b  20993  m2cpmghm  22664