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Mirrors > Home > MPE Home > Th. List > reslmhm2 | Structured version Visualization version GIF version |
Description: Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
Ref | Expression |
---|---|
reslmhm2.u | ⊢ 𝑈 = (𝑇 ↾s 𝑋) |
reslmhm2.l | ⊢ 𝐿 = (LSubSp‘𝑇) |
Ref | Expression |
---|---|
reslmhm2 | ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . 2 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
2 | eqid 2818 | . 2 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆) | |
3 | eqid 2818 | . 2 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇) | |
4 | eqid 2818 | . 2 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
5 | eqid 2818 | . 2 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
6 | eqid 2818 | . 2 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆)) | |
7 | lmhmlmod1 19734 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑈) → 𝑆 ∈ LMod) | |
8 | 7 | 3ad2ant1 1125 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑆 ∈ LMod) |
9 | simp2 1129 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑇 ∈ LMod) | |
10 | reslmhm2.u | . . . . 5 ⊢ 𝑈 = (𝑇 ↾s 𝑋) | |
11 | 10, 5 | resssca 16638 | . . . 4 ⊢ (𝑋 ∈ 𝐿 → (Scalar‘𝑇) = (Scalar‘𝑈)) |
12 | 11 | 3ad2ant3 1127 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → (Scalar‘𝑇) = (Scalar‘𝑈)) |
13 | eqid 2818 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
14 | 4, 13 | lmhmsca 19731 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑈) → (Scalar‘𝑈) = (Scalar‘𝑆)) |
15 | 14 | 3ad2ant1 1125 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → (Scalar‘𝑈) = (Scalar‘𝑆)) |
16 | 12, 15 | eqtrd 2853 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → (Scalar‘𝑇) = (Scalar‘𝑆)) |
17 | lmghm 19732 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑈) → 𝐹 ∈ (𝑆 GrpHom 𝑈)) | |
18 | 17 | 3ad2ant1 1125 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 GrpHom 𝑈)) |
19 | reslmhm2.l | . . . . 5 ⊢ 𝐿 = (LSubSp‘𝑇) | |
20 | 19 | lsssubg 19658 | . . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑋 ∈ (SubGrp‘𝑇)) |
21 | 20 | 3adant1 1122 | . . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝑋 ∈ (SubGrp‘𝑇)) |
22 | 10 | resghm2 18313 | . . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝑋 ∈ (SubGrp‘𝑇)) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
23 | 18, 21, 22 | syl2anc 584 | . 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
24 | eqid 2818 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈) | |
25 | 4, 6, 1, 2, 24 | lmhmlin 19736 | . . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦))) |
26 | 25 | 3expb 1112 | . . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦))) |
27 | 26 | 3ad2antl1 1177 | . . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦))) |
28 | simpl3 1185 | . . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → 𝑋 ∈ 𝐿) | |
29 | 10, 3 | ressvsca 16639 | . . . . 5 ⊢ (𝑋 ∈ 𝐿 → ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑈)) |
30 | 29 | oveqd 7162 | . . . 4 ⊢ (𝑋 ∈ 𝐿 → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦))) |
31 | 28, 30 | syl 17 | . . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦)) = (𝑥( ·𝑠 ‘𝑈)(𝐹‘𝑦))) |
32 | 27, 31 | eqtr4d 2856 | . 2 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥( ·𝑠 ‘𝑆)𝑦)) = (𝑥( ·𝑠 ‘𝑇)(𝐹‘𝑦))) |
33 | 1, 2, 3, 4, 5, 6, 8, 9, 16, 23, 32 | islmhmd 19740 | 1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 Scalarcsca 16556 ·𝑠 cvsca 16557 SubGrpcsubg 18211 GrpHom cghm 18293 LModclmod 19563 LSubSpclss 19632 LMHom clmhm 19720 |
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-1st 7678 df-2nd 7679 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-3 11689 df-4 11690 df-5 11691 df-6 11692 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-sca 16569 df-vsca 16570 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-sbg 18046 df-subg 18214 df-ghm 18294 df-mgp 19169 df-ur 19181 df-ring 19228 df-lmod 19565 df-lss 19633 df-lmhm 19723 |
This theorem is referenced by: reslmhm2b 19755 |
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