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Mirrors > Home > MPE Home > Th. List > resghm2b | Structured version Visualization version GIF version |
Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
resghm2.u | ⊢ 𝑈 = (𝑇 ↾s 𝑋) |
Ref | Expression |
---|---|
resghm2b | ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmgrp1 18298 | . . 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 18298 | . . 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 18239 | . . . . . 6 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑋 ∈ (SubMnd‘𝑇)) | |
6 | resghm2.u | . . . . . . 7 ⊢ 𝑈 = (𝑇 ↾s 𝑋) | |
7 | 6 | resmhm2b 17975 | . . . . . 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 482 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈))) |
10 | subgrcl 18222 | . . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑇 ∈ Grp) | |
11 | 10 | adantr 481 | . . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑇 ∈ Grp) |
12 | ghmmhmb 18307 | . . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇)) | |
13 | 11, 12 | sylan2 592 | . . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇)) |
14 | 13 | eleq2d 2895 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑇))) |
15 | 6 | subggrp 18220 | . . . . . . 7 ⊢ (𝑋 ∈ (SubGrp‘𝑇) → 𝑈 ∈ Grp) |
16 | 15 | adantr 481 | . . . . . 6 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → 𝑈 ∈ Grp) |
17 | ghmmhmb 18307 | . . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑈 ∈ Grp) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈)) | |
18 | 16, 17 | sylan2 592 | . . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈)) |
19 | 18 | eleq2d 2895 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑈) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈))) |
20 | 9, 14, 19 | 3bitr4d 312 | . . 3 ⊢ ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈))) |
21 | 20 | expcom 414 | . 2 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝑆 ∈ Grp → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))) |
22 | 2, 4, 21 | pm5.21ndd 381 | 1 ⊢ ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ↾s cress 16472 MndHom cmhm 17942 SubMndcsubmnd 17943 Grpcgrp 18041 SubGrpcsubg 18211 GrpHom cghm 18293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-subg 18214 df-ghm 18294 |
This theorem is referenced by: ghmghmrn 18315 cayley 18471 pj1ghm2 18759 dpjghm2 19115 reslmhm2b 19755 m2cpmghm 21280 |
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